Biostatistics and data handling

Class 12 Biology, unit 11, book pages 210–236

Practise 52 MCQs Take the unit quiz Review 55 cards

Biostatistics applies statistics to data from living organisms so that biological, medical and agricultural problems can be studied and solved. This unit teaches the averages (mean, median, mode), the measures of spread (standard deviation, range, percentile and quartiles), how to present data as bar charts, pie charts and error bars, and how to design a fair experiment with experimental and control groups and independent, dependent and controlled variables.

Learning outcomes (8)

Define biostatistics and its use.
Define mean, median, mode, standard deviation, range, percentile.
Calculate mean, median, mode, standard deviation, range, percentile from a given set of data.
Sketch a bar chart for a given set of data.
Sketch error bars based off of range or standard deviation for a given set of data on a bar chart.
Evaluate the appropriate type of figure or chart for a given set of data and/or experiment (bar chart, pie chart, xy axis data figure etc).
Make the appropriate chart with proper title, labeled axes, legend, axes units.
Design an appropriate experiment with control group and dependent, independent and control variables.

11.1.1Biostatistics, data and data set

Biostatistics applies statistical principles and techniques to analyse, interpret and present the findings of research in the fields of biology, medicine and health sciences. It deals mainly with data collected from living organisms and then presents the findings in an appropriate way, so that plans or solutions can be made from them.

Biostatistics: The application of statistical principles and techniques to analyse, interpret and present research findings in the biological, medical and health sciences.
Data: In biostatistics, the facts, measurements or observations related to living organisms that are collected to analyse and draw conclusions about biological phenomena. Examples are heights, weights, enzyme activity levels, population counts or the rate of any disease.
Data set: A collection of related data values, usually organised systematically (in a table) for analysis. Each row typically represents a subject or experimental unit, each column represents a variable, and each entry is an individual sample measurement of one characteristic.

A data set arranged as a table: each row is one subject or experimental unit (here a plant) and each column is one variable. The highlighted row is one subject.

Example from the book: a table recording the heights and weights of 100 plants grown under different availability of light or moisture conditions is a data set.

Where biostatistics is applied

Medicine, public health and clinical research.
Agriculture: calculating and assessing crop yields and the growth of livestock (poultry and dairy farming).
Epidemiology: understanding disease outbreaks, health trends, possible preventive measures and the effectiveness of specific disease treatments.

11.1.1Components of biostatistics studies

For the effective application of biostatistics, the following components (steps) are followed.

The five components (steps) followed for the effective application of biostatistics, in order.

Components of a biostatistics study.
Component	What happens
Identification of problem	It is the very first requirement to solve any problem affecting living organisms, especially human beings.
Collecting and analysing data	Data are collected before the experiments, and then the outcome of the experiments is analysed thoroughly.
Designing experiments	For the solution of any problem associated with living organisms, biologists use biostatistics to plan biological experiments specifically for the requirements and durations.
Interpreting results	After the experiments, the results are used to interpret the reasons, and the possible future effects of the biological problem are estimated.
Developing new tools	Interpretations of experimental results lead to new predictive tools and plans for dealing effectively with the problem, for example health related issues.

11.1.2Uses of biostatistics

The book lists seven uses of biostatistics. Each is shown below with what biostatistics does in that area.

Seven uses of biostatistics.
No.	Use	What biostatistics does
1	Need assessments of agriculture and cattle farming for a growing population	Assess the demand for agriculture and dairy farming according to the rate of population growth; helps the government to assess the need to import or export food items.
2	Evaluation of efficacy of medical instruments and pharmacological drugs	Design clinical trials and controlled studies to assess efficacy and safety of newly designed drugs, treatment plans or medical instruments; findings show improvements or side effects; decide optimal doses.
3	Monitoring of epidemiological studies and policy development	Monitor and analyse data about the spread of an epidemic disease; identify risk factors, pattern and rate of disease; help control and prevent outbreaks (e.g. Hepatitis, Polio, COVID-19); guide governments on population management, healthcare plans and funding.
4	Management of public health and population growth	Estimate health trends from birth rates, death rates and prevalence of diseases; guide health initiatives and allocation of resources; monitor and optimise hospital performance (patient numbers, availability of doctors and medicine, effectiveness of treatment).
5	Genetic diseases	Analyse inheritance patterns of genetic diseases in a population (e.g. numbers of thalassaemia and muscular dystrophy patients); helps in guessing risk factors and behaviour of genetic disorders.
6	Pollution indicators and environmental protection	Analyse pollution level, its causes and impact on population health; design policies to reduce health risks from environmental hazards (e.g. causes of smog, identification of affected areas and solutions such as plantation in Punjab, especially Lahore).
7	Survival analysis	Predict and note the survival rate of patients after a particular treatment; estimate life expectancy and chances of success of medical treatments (e.g. five year survival after cancer treatment is considered a successful treatment); survival data indicate the effectiveness of health plans.

11.2Average and its typesOften examined

In daily life the following statements are common.

The heartbeat of a human is 72 beats per minute.
The death rate due to cancer in Pakistan is 25 per thousand patients.
The production of wheat in Punjab is 2000 kg per acre.
The price of bananas in the market is Rs. 120 per dozen.
The food consumption of a human is 1 kg per day.
The rainfall in Islamabad is 1500 mm per year.

None of these statements is found exactly correct. The heart does not beat 72 times in every minute: it may be 90 per minute while running and 60 per minute during sleep. In one hospital only 5 cancer patients may die out of 1000, while in another 60 die out of 1000, so 25 per 1000 is only about right. Wheat production may be 1500 kg per acre in one district and 2400 kg in another. These statements are approximate statements in specific situations. In statistics they are called average statements.

Average (measure of central tendency): A single value calculated to represent the whole data. It expresses the central idea of the observations and may be calculated for a data sample (such as patients) or a population (such as migrating birds). The proper type of average is chosen by the expert for the data in hand.

Average is very important in agriculture, health and poultry, and experts at national and international level discuss the findings of studies in averages.

Types of averages

The five commonly used averages.
Type	Studied in this chapter?
Arithmetic mean	Yes (11.2.1)
Median	Yes (11.2.2)
Mode	Yes (11.2.3)
Geometric mean	No, only named
Harmonic mean	No, only named

Ungrouped data and grouped data

Ungrouped and grouped data.
	Ungrouped data	Grouped data
Meaning	Raw data as individual values, not organised into groups or categories.	Data organised into classes or intervals with their frequencies.
Suitable for	Small data sets.	Large data sets: easier to summarise and analyse.
Book example	Blood pressure readings of 10 patients: 120, 125, 130, 122, 118, 135, 128, 124, 129, 121.	Blood pressure ranges and number of patients: 110–119 mm Hg: 2 patients; 120–129 mm Hg: 5 patients; 130–139 mm Hg: 3 patients.

Frequency distribution: A way of organising data in the form of a table or graph to display the number of occurrences (frequencies) of different outcomes or data values in a data set. It shows how often values occur, summarises large data sets, helps to identify patterns and is used to construct histograms and other statistical charts.

Two types of frequency distribution.
Type of frequency distribution	What it does
Ungrouped frequency distribution	Lists each individual value and its frequency.
Grouped frequency distribution	Groups data into intervals (classes) and shows the frequency for each class.

11.2.1Arithmetic meanOften examined

Mean (arithmetic mean): The sum of all the values of a data set divided by the total number of values in the data set. It is the single value calculated to represent the whole set of values. The symbol $\overset{x}{ˉ}$ (read "x bar") represents the sample mean.

\overset{x}{ˉ} = \frac{Sum of values}{Total number of values} = \frac{Σ x}{n}

Mean

Formulae for the mean.
	Direct method for ungrouped data	Direct method for grouped data
Formula	$\overset{x}{ˉ} = \frac{Σ x}{n}$	$\overset{x}{ˉ} = \frac{Σ f x}{Σ f}$
Symbols	$x$ = value of the data set; $n$ = number of values	$f$ = frequency; $x$ = class mark = $\frac{lower limit + upper limit}{2}$

Calculating the mean

Write down the values, or for grouped data the classes with their frequencies $f$ .
For grouped data, find the class mark $x$ of every class: (lower limit + upper limit) divided by 2.
For grouped data, multiply each frequency by its class mark to get $f x$ .
Add the values ( $Σ x$ ) or add the $f x$ column ( $Σ f x$ ) and the frequencies ( $Σ f$ ).
Divide: by $n$ for ungrouped data, or $Σ f x$ by $Σ f$ for grouped data.

Example 1 (ungrouped data): polio patients per month

A team of the World Health Organization assessed the prevalence of polio in Pakistan. The number of confirmed polio patients detected each month of 2022 and 2023 is given. Calculate the mean number of polio patients per month in each year and compare the prevalence in the two years.

Polio patients detected each month.
Sr. no.	Month of sampling	Polio patients in 2022	Polio patients in 2023
1	January	9	7
2	February	13	9
3	March	15	13
4	April	19	17
5	May	22	18
6	June	25	17
7	July	20	15
8	August	22	14
9	September	18	11
10	October	13	9
11	November	10	8
12	December	6	6
	Total	192	144

Mean in 2022 = \frac{192}{12} = 16 Mean in 2023 = \frac{144}{12} = 12

Difference in polio patients per month between 2022 and 2023 = 16 − 12 = 4. The prevalence of polio decreased in 2023 compared with 2022 by 4 patients per month.

Example 2 (grouped data): body mass of 100 students

The United Nations World Food Programme (UN-WFP) assessed the impact of food quality on the body mass of 100 students of different age groups. Find the arithmetic mean of the frequency distribution of body mass before the start of the experiment.

Calculation of the mean for grouped data.
Mass (lbs)	Frequency ( $f$ )	$x = \frac{lower + upper}{2}$	$f x$
35–39	13	37	13 × 37 = 481
40–44	15	42	15 × 42 = 630
45–49	28	47	28 × 47 = 1316
50–54	17	52	17 × 52 = 884
55–59	12	57	12 × 57 = 684
60–64	10	62	10 × 62 = 620
65–69	5	67	5 × 67 = 335
Total	100		4950

\overset{x}{ˉ} = \frac{Σ f x}{Σ f} = \frac{4950}{100} = 49.5 lbs

Frequency distribution of the body mass of 100 students (book Example 2). The dashed accent line marks the mean, 49.5 lbs, which falls within the tallest classes.

11.2.2MedianOften examined

Median: The middle value of a data set arranged in ascending or descending order of magnitude. It divides the data set into two equal parts: one part has values less than the middle value and the other part has values greater than it. It is denoted by $\tilde{x}$ (read "x tilde" or "x snake").

It is necessary to arrange the values in ascending or descending order first. The method of finding the median is simple, and the median is not affected by changes in the extreme values of the data set.

Median for ungrouped data

Median of ungrouped data.
Number of values	Median
Odd	The middle value: $\tilde{x} =$ value of the $(\frac{n + 1}{2})^{th}$ item ( $n$ = number of items)
Even	The average of the two middle values: $\tilde{x} = \frac{( \frac{n}{2} ) ^{th} value + ( \frac{n}{2} + 1 ) ^{th} value}{2}$

Example 1: odd number of values

In an experiment to check the effect of a specific fertiliser on plant growth, the heights (in cm) of 13 plants in the experimental data set were recorded: 58, 67, 65, 70, 68, 62, 63, 46, 64, 49, 55, 66, 72. Find the median height.

Arrange in ascending order: 46, 49, 55, 58, 62, 63, 64, 65, 66, 67, 68, 70, 72.
Count the values: $n = 13$ (odd).
Position of the median: $\frac{n + 1}{2} = \frac{13 + 1}{2} = 7$ , the 7th value.
The 7th value is 64, so the median height is 64 cm.

Example 2: even number of values

The numbers of fruits produced on each of 16 plants of an experimental group were recorded: 3, 5, 18, 21, 15, 10, 8, 12, 13, 7, 11, 14, 9, 16, 20, 24. Find the median number of fruits per plant.

Arrange in ascending order: 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 24.
Count the values: $n = 16$ (even).
The middle values are the $\frac{n}{2} = 8$ th and the $\frac{n}{2} + 1 = 9$ th: 12 and 13.
Median $= \frac{12 + 13}{2} = \frac{25}{2} = 12.5$ fruits per plant.

Finding the median of ordered data. (a) Odd number of values: the middle value (7th of 13). (b) Even number of values: the average of the two middle values (8th and 9th of 16). The small grey numbers give the position.

Activity (to be solved by the students)

Calculate the median of the following heights of 22 plants in an experimental group used to assess the effect of less water availability: 8, 9, 10, 18, 10, 8, 9, 7, 8, 11, 23, 19, 17, 15, 7, 12, 14, 12, 11, 14, 15, 13.

Arrange in ascending order: 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 17, 18, 19, 23.
$n = 22$ (even), so the middle values are the 11th and 12th.
The 11th value is 11 and the 12th value is 12.
Median $= \frac{11 + 12}{2} = 11.5$ .

Median for grouped data (discontinuous or unconnected values)

First obtain the cumulative frequency (cf) of the whole data set, which is the running total of the frequencies. The class value against the $\frac{n + 1}{2}$ th cumulative frequency is the median for an odd number of data; for an even number, take the average of the class values against the $\frac{n}{2}$ th and $(\frac{n}{2} + 1)$ th cumulative frequencies.

Example for an odd number of data: flowers on 21 rose plants

Number of flowers on 21 rose plants.
Class (no. of flowers per plant)	Frequency (no. of plants = $n$ )	Cumulative frequency
1	3	3
2	5	8
3	6	14
4	4	18
5	3	21

Number of plants $n = 21$ . Median position $= \frac{n + 1}{2} = \frac{21 + 1}{2} = 11$ . The cumulative frequency 8 ends at class 2 and the next cumulative frequency is 14, so the 11th value lies in the class with cumulative frequency 14, which is class 3. The median is 3 flowers per plant.

Example for an even number of data: heights of 80 plants

Height of 80 plants.
Class (plant height in cm)	Frequency (no. of plants = $n$ )	Cumulative frequency
119	4	4
120	9	13
121	14	27
122	18	45
123	15	60
124	13	73
125	7	80

Number of plants $n = 80$ . The middle values are the $\frac{80}{2} = 40$ th and $\frac{80}{2} + 1 = 41$ st. Cumulative frequency 27 ends at 121 and the next is 45, so both the 40th and the 41st values lie in the class with cumulative frequency 45, which is 122. Median $= \frac{122 + 122}{2} = 122$ cm.

Worked example: why the median resists extreme values

Six pots gave 12, 9, 14, 10, 15 and 12 germinated seeds. One value, 15, is wrongly entered as 60 (an extreme value). Compare how the mean and the median change.

Data	Mean	Median
9, 10, 12, 12, 14, 15	$\frac{72}{6} = 12$	$\frac{12 + 12}{2} = 12$
9, 10, 12, 12, 14, 60	$\frac{117}{6} = 19.5$	$\frac{12 + 12}{2} = 12$

The mean rises from 12 to 19.5 because every value, including the extreme one, enters the sum. The median is still 12 because it depends only on the middle positions, not on the size of the end values.

Conclusion: this is why the book says the median is not affected by abnormally large or small values, while the mean can be pulled towards them.

mean changes (12 to 19.5); median stays 12

Advantages of median

It is easy to calculate and is located exactly.
It is not affected by abnormally large and small values in the data set.
It can be used in quantitative measurements.

Disadvantages of median

The median of two or more data sets cannot be calculated by using or adding the medians of the individual data sets.
The median value may not necessarily be represented in the central data.
It cannot be used where specific values, e.g. weight and age, are used.

11.2.3ModeOften examined

Mode: The value in a data set that appears the maximum number of times; the most frequent or repeated value. It is denoted by $\overset{x}{^}$ (read "x hat" or "x cap").

Number of modes.
A data set may have	Called
One mode	Unimodal
More than one mode	Bimodal (two) or multimodal (many)
No mode, all values occur the same number of times	Amodal

Formula for the mode of grouped data

\overset{x}{^} = l + [\frac{( f _{m} - f _{1} ) \times h}{( f _{m} - f _{1} ) + ( f _{m} - f _{2} )}]

Mode of grouped data

Symbols in the formula.
Symbol	Meaning
$l$	Lower class boundary of the modal group
$f_{m}$	Maximum frequency of a group (frequency of the modal group)
$f_{1}$	Frequency before $f_{m}$ (previous group)
$f_{2}$	Frequency after $f_{m}$ (next group)
$h$	Class interval (width) of the modal group

Class interval and modal group

Class interval: A range of values into which data are grouped in a frequency distribution table. Each interval has a lower and an upper boundary (e.g. 120–129). All intervals are typically of equal width.
Modal group (modal class): The class interval with the highest frequency in a grouped frequency distribution. It shows the most common range of values in the data set.

The class intervals are 110–119, 120–129 and 130–139. The highest frequency is 5, so 120–129 is the modal group.
Blood pressure (mm Hg)	Frequency ( $f$ )
110–119	2
120–129	5
130–139	3

Examples for calculating mode

Mode of ungrouped data.
Example	Data	Mode
#01	The letters of the word BIOLOGY	O (appears twice)
#02	3, 5, 6, 6, 9, 10, 9, 6, 6	6 (appears four times)
#03	3, 5, 5, 6, 9, 9, 10, 12	5 and 9 (bimodal)

Example #04 (frequency distribution): find the mode of the following data.

Data for Example #04.
Groups	Frequency ( $f$ )	Class boundary
10–19	5	9.5–19.5
20–29	25 ( $f_{1}$ )	19.5–29.5
30–39	40 ( $f_{m}$ ), modal group	29.5–39.5
40–49	20 ( $f_{2}$ )	39.5–49.5
50–59	10	49.5–59.5

Frequency distribution of book Example #04. The tallest bar (30–39, frequency 40) is the modal group; its neighbours give

f_{1} = 25

(before) and

f_{2} = 20

(after).

l = 29.5, f_{m} = 40, f_{1} = 25, f_{2} = 20, h = 10

\overset{x}{^} = 29.5 + \frac{( 40 - 25 ) \times 10}{( 40 - 25 ) + ( 40 - 20 )} = 29.5 + \frac{150}{35} = 33.786

Worked example: mode of a grouped frequency distribution

The lengths of 50 leaves (mm) are grouped as: 0–9 (4 leaves), 10–19 (12), 20–29 (20), 30–39 (8), 40–49 (6). Find the mode.

Group	Frequency	Class boundary
0–9	4	−0.5–9.5
10–19	12 ( $f_{1}$ )	9.5–19.5
20–29	20 ( $f_{m}$ )	19.5–29.5
30–39	8 ( $f_{2}$ )	29.5–39.5
40–49	6	39.5–49.5

Step 1: the highest frequency is 20, so the modal group is 20–29. Then $f_{m} = 20$ , $f_{1} = 12$ , $f_{2} = 8$ .

Step 2: the lower class boundary of the modal group is $l = 19.5$ and the class width is $h = 10$ .

Step 3: $\overset{x}{^} = 19.5 + \frac{( 20 - 12 ) \times 10}{( 20 - 12 ) + ( 20 - 8 )} = 19.5 + \frac{80}{20} = 19.5 + 4 = 23.5$ .

The mode lies inside the modal group (19.5 to 29.5) and is pulled towards the side of the larger neighbouring frequency (12 is greater than 8), as expected.

\overset{x}{^} = 23.5 mm

11.2.4Standard deviationOften examined

Standard deviation (SD): A measure of the dispersion (spread) of data values around the mean. It states the extent to which data values in a data set deviate from the mean, and shows how much a set of data varies from its mean value. It is expressed in the same units as the data.

Figure 11.1: Normal curve showing the mean and standard deviation (SD). About 34.1% of the data lie within one SD on each side of the mean, 13.6% between one and two SD, 2.3% between two and three SD and 0.1% beyond three SD.

Standard deviation tells how close the data values are to the mean.

Meaning of low and high standard deviation.
Standard deviation	What it means
Low (small)	Most data values are clustered tightly around the mean.
High (large)	Most data values are widely spread and not grouped around the mean; the greater the difference of the values from the mean, the greater the SD.
Zero	All values are equal to the mean, so there is no difference between the scores and the mean.

Two data sets with the same mean. The tall narrow curve (accent) has a small standard deviation; the low wide curve (dashed) has a large one.

Examples of data for which a biologist may calculate SD are the height of each individual in a sample population, the number of hepatitis infected patients in each city of Pakistan, or the number of fruits on each plant in a garden under investigation.

The standard deviation is represented by the Greek letter sigma ( $σ$ ) for a population and by the Latin letter $s$ for a sample. A higher sigma value means the data are further from the mean. SD and variance are related: SD is the square root of the variance, and the variance is sigma squared ( $σ^{2}$ ). SD is easier to interpret because it is expressed in the same units as the data.

Book illustration: each of 40 students scored 77 marks in a biology quiz, so the mean score is also 77. All scores equal the mean, so the SD is zero. If a few students scored 75, the SD is no longer zero but is very small, much less than one sigma.

Calculation of standard deviation

Steps (the book lists them as i to vi)

Determine the mean (arithmetic average).
Subtract the mean from each score.
Square the result for each score.
Add the squared results together.
Divide this sum by the number of scores (biased) or by the number of scores minus 1 (unbiased).
Take the square root of this number: this is the standard deviation.

Book example: COVID-19 cases in four cities of Pakistan in 2021. The four cities are treated as a sample from the whole of Pakistan, so the sum of squares is divided by $n - 1$ (unbiased).

Calculation of SD for COVID-19 cases.
City	Confirmed COVID cases ( $X$ )	$X - \overset{x}{ˉ}$ (mean = 60)	$(X - \overset{x}{ˉ})^{2}$
Karachi	90	90 − 60 = 30	900
Lahore	80	80 − 60 = 20	400
Peshawar	40	40 − 60 = −20	400
Quetta	30	30 − 60 = −30	900
Total	240		2600

Deviation of each city from the mean in the book's COVID-19 example. Each arrow runs from the mean (dashed, accent) to the value and is labelled with

X - \overset{x}{ˉ}

and its square; the four squares add up to 2600.

s = \frac{Σ ( X - x ˉ ) ^{2}}{n - 1} = \frac{2600}{3} = 866.7 \approx 29.4

Standard deviation of a population and of a sample.
	Population	Sample
Formula	$σ = \frac{Σ ( X - μ ) ^{2}}{N}$	$s = \frac{Σ ( X - x ˉ ) ^{2}}{n - 1}$
Symbols	$X$ = value in the data distribution; $μ$ = population mean; $N$ = total number of observations	$X$ = value in the data distribution; $\overset{x}{ˉ}$ = sample mean; $n$ = total number of observations

Calculating SD by hand is complex and the risk of error is high, so statisticians usually use a calculator or computer software.

Worked example: standard deviation of two samples

Two plots each have five plants. The fruits per plant are plot A: 6, 8, 10, 12, 14 and plot B: 9, 10, 10, 10, 11. Both are samples. Find the mean and the SD of each, and say which plot is more uniform.

Plot A $X$	$X - \overset{x}{ˉ}$	$(X - \overset{x}{ˉ})^{2}$	Plot B $X$	$X - \overset{x}{ˉ}$	$(X - \overset{x}{ˉ})^{2}$
6	−4	16	9	−1	1
8	−2	4	10	0	0
10	0	0	10	0	0
12	2	4	10	0	0
14	4	16	11	1	1
Sum 50		40	Sum 50		2

Means: plot A $\overset{x}{ˉ} = 50/5 = 10$ ; plot B $\overset{x}{ˉ} = 50/5 = 10$ . Both means are equal.

Plot A: $s = 40/ (5 - 1) = 10 \approx 3.16$ fruits.

Plot B: $s = 2/ (5 - 1) = 0.5 \approx 0.71$ fruits.

Conclusion: the means are the same but the SD of B is much smaller, so the values of plot B are clustered tightly around the mean and plot B is more uniform. If the five plants were the whole population, divide by $N = 5$ instead: $σ_{A} = 8 \approx 2.83$ and $σ_{B} = 0.4 \approx 0.63$ .

s_{A} \approx 3.16, s_{B} \approx 0.71

11.2.5Range

Range: The difference between the highest and the lowest values in a set of data (the difference between the highest observation and the lowest observation). It is also called the range of data values or range of observation, and it is a measure of the spread or variability of a data set obtained from an experiment.

Calculation of range

Arrange the values or observations in ascending order, from lowest to highest.
Identify the highest (maximum) value.
Identify the lowest (minimum) value.
Subtract: range = highest value − lowest value.

\text{Range}=\text{Highest (maximum) value}-\text{Lowest (minimum) value}

Range

Book example: heights of plants after fertiliser treatment

A student collected the heights (cm) of a sample of plants in a garden after treatment with fertiliser.

Calculation of range.
Step	Result
1. Raw data	10, 15, 18, 21, 25, 32, 41, 28, 54, 35, 26, 23, 33, 38, 40, 19, 33, 43, 8, 13
2. Arranged in ascending order	8, 10, 13, 15, 18, 19, 21, 23, 25, 26, 28, 32, 33, 33, 35, 38, 40, 41, 43, 54
3. Highest value	54 cm
4. Lowest value	8 cm
5. Range	54 − 8 = 46 cm

The range is 46 cm, meaning there is a 46 cm difference between the smallest and the tallest plants in the sample.

Heights of the 20 sampled plants plotted on a number line. The range is the distance from the lowest value (8 cm) to the highest value (54 cm).

Importance of range in biology

The range helps scientists understand the variability within a population. This is useful in studies of genetics, ecology and physiology, where variability may indicate environmental influences or adaptations of organisms in a given population.

Interpreting the range.
Range	What it indicates
Small	The organisms responded to the treatment at about the same level; individuals are more similar. This is good news for the researcher because the treatment is almost equally effective for all organisms, and best when the low range is also at a high effectivity level.
Middle (intermediate)	The most difficult for the researcher, because no clear-cut information can be derived from the data. The researcher must re-plan to proceed with the experiment.
Large	A high level of variation among individuals.

Limitations of range

The range statistic may be deceiving because it accounts only for the highest and lowest values. For example, if almost all plants responded well to a fertiliser treatment with greater productivity and only one or two plants showed the lowest productivity, looking at the range alone may be misleading.

11.2.6Percentile and quartilesOften examined

Percentile: A value that indicates the percentage of data in a set that is below a certain value. For example, the 25th percentile is the value below which 25% of the data points lie. A percentile is the value or score on the ordered array of data that indicates the percentage of the scores that fall at or below that value.

Percentiles divide the ordered data into 100 separate units, so a data point can be at any number between 0 and 100, for example the 99th, 44th or 9th percentile. Percentiles are used to interpret and understand data and are often used to report test scores and biometric measurements.

Book example: the 25th percentile of the number of fruits on the plants in a garden is 39 fruits per plant. This means that 25% of the plants of that garden produced fewer than 39 fruits.

The 25th percentile is also known as the first quartile; the 50th percentile is also the median.
The 99th percentile is the location point above which almost no scores lie: a plant at the 99th percentile produced very well compared with the rest of the plants in the pool.
In research and biometry, percentile ranks are used to interpret an individual raw value by comparing it with the values of other individuals in the study group. If a student height of 77 cm translates to a percentile rank of 89, then 89% of the students in that population have a height of 77 cm or less.

Figure 11.2: Percentile calculation in a normal bell shaped curve of data. Each vertical line is one standard deviation; the cumulative percentage gives the percentile rank at that point (for example, 84.1% of values lie below +1σ). Selected percentiles are shown.

Quartiles (further helping material)

Quartile: A special type of percentile. A quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents one quarter of the population. Each quartile defines a specific section of the ordered data distribution.

The three quartiles.
Quartile	Also called	Meaning
First quartile (Q_1)	Lower quartile, 25th percentile	The point on the ordered array where 25% of the values are below it and 75% are above it. The range of the first quartile accounts for the bottom one quarter (lowest 25%) of whatever is measured.
Second quartile (Q_2)	Median, 50th percentile	The midpoint of the data: 50% of the values are above and 50% below.
Third quartile (Q_3)	Upper quartile, 75th percentile	The point where 25% of the values are above it and 75% are below it.

Interquartile range (IQR): The difference between the upper and lower quartiles (Q_3-Q_1). It is often used to characterise the bulk of the population.

Book example: quartiles of the number of fruits on 16 orange plants in a small garden. In ascending order the values are 11, 13, 14, 15, 18, 18, 19, 19, 23, 23, 23, 24, 24, 25, 25, 26. The book marks the first quartile at 15, the second quartile as 21 (between 19 and 23) and the third quartile at 24. Quartiles divide the data so that the researcher can analyse the results by ordered groups rather than by the entire data pool.

The 16 ordered fruit counts of the book quartile table, with the three quartiles marked. Q1, Q2 and Q3 divide the data into four equal parts, each holding one quarter of the plants.

Steps to calculate a percentile

Steps

Arrange data: sort the data values in ascending order.
Calculate position: use the formula for the position of the Pth percentile (below), where P is the desired percentile (for example 25 for the 25th percentile) and N is the number of observations.

\text{Position}=\dfrac{P}{100}\times(N+1)

Position of the Pth percentile

The percentile (percentile rank) of a data point can also be calculated from the data:

P=\dfrac{n}{N}\times100\%

P is the percentile, n is the number of data points below the data point of interest and N is the total number of data points in the data set.

Worked example: quartiles, IQR and percentile rank

The heights (cm) of 15 plants in ascending order are 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 30, 33. Find Q_1, Q_2, Q_3 and the IQR, and the percentile rank of a plant 21 cm tall.

Step 1: the data are already in ascending order and N=15, so N+1=16.

Quartile	Percentile P	Position \frac{P}{100}\times16	Value
Q_1	25	4th	17
Q_2 (median)	50	8th	22
Q_3	75	12th	27

Step 2: \text{IQR}=Q_3-Q_1=27-17=10 cm.

Step 3: the value 21 is the 7th value, so n=6 values are below it. P=\dfrac{n}{N}\times100\%=\dfrac{6}{15}\times100\%=40\%.

Meaning: about 40% of the plants are shorter than 21 cm. The middle 50% of the plants lie between 17 cm and 27 cm.

The 15 ordered plant heights of the worked example. The 4th, 8th and 12th values are the quartiles (positions 25%, 50% and 75% of N+1=16). The dashed box is the plant of height 21 cm.

Q_1=17,\ Q_2=22,\ Q_3=27,\ \text{IQR}=10;\ P=40\%

11.3Graphic presentation of data and the simple bar chart

Graphic presentation of scientific data: need and importance

Biostatistical values (data) are documented in the form of figures and are usually recorded in tables. Numerical or arithmetic figures are not attractive, and a large number of figures in a table is not easily understood, so it is difficult to get a clear picture from numerical data. Diagrams and graphs are a more attractive way of presenting the clear and complete picture of data. Data presented as diagrams, pictures, charts and graphs also leave a long lasting impression on the mind of the observer.

Several types of graphs, pictures and diagrams are used to represent data, including the bar chart, rectangles and the pie chart.

Simple bar diagram or simple bar chart

Simple bar chart: A chart whose bars represent a single data series without further classification of the characteristic. One bar represents only one value or figure, and there may be many bars for many values. Bars of equal width are drawn, but their lengths vary: the height (length) of a bar represents the magnitude of a quantity or the degree of the value in the data.

The parts of a labelled bar chart: title, axis labels with units, a scale, bars of equal width with equal gaps, and names of the categories.

11.4Constructing bar charts and pie chartsOften examined

Sketching or constructing bar charts and pie charts needs a proper title, labelled axes, a legend and axes units.

Steps to sketch a bar chart

Take graph paper with suitable size and scale for the requirement and type of data.
Draw two lines on the graph, perpendicular to each other and intersecting at zero.
Label the axes: the horizontal line is the X-axis and the vertical line is the Y-axis.
Choose a uniform width of bars and a uniform gap between the bars along the X-axis.
Label the X-axis with the names of the data items whose values are to be presented.
Choose a suitable scale to determine and specify the heights of the bars for the given values along the Y-axis.
Label the Y-axis scale with the specific unit of the data value to grade the length of the bars.
Calculate the heights of the bars according to the scale chosen and mark them as dots.
Draw the bars carefully and recheck all the entries.

Bars may be drawn vertically or horizontally, but normally vertical bars are used.

Steps to make a pie chart manually

Gather your data: a list of categories and their values (for example category A: 20, B: 30, C: 50).
Calculate total and percentages: add the values to get the total, then divide each value by the total and multiply by 100 (total = 100, so A = 20%, B = 30%, C = 50%).
Convert percentages to degrees: a circle is 360°, so multiply each percentage by 3.6 (A: 20\times3.6=72^\circ, B: 108^\circ, C: 180^\circ).
Draw a circle with a compass and mark the centre.
Measure and draw each segment: start at the top of the circle (12 o'clock position), use a protractor to measure each angle from the centre and draw lines from the centre to the edge to divide the circle into sections.
Label each segment with the category name and percentage, inside or near each section.
Add colour (optional) to each segment for clarity.

Bar charts and pie charts can also be constructed by using computer software.

Example 1: cotton production of Bt-cotton varieties

Draw a simple bar chart to represent the cotton (weight in kg) produced per acre of land from six Bt-cotton varieties in 2023.

Cotton production of six Bt-cotton varieties per acre in 2023.
Cotton variety	IR-NIBGE-370	RH-647	MNH-886	FH-142	IUB-2013	FH-Lalazar
Production per acre (kg)	800	750	700	650	600	500
Production per acre (%)	20%	18.75%	17.5%	16.25%	15%	12.5%

Simple bar chart of the cotton production per acre of six Bt-cotton varieties in 2023 (book Example 1). Each bar is labelled with its value and percentage.

Simple pie chart of the same cotton data. Each slice is labelled with the variety, its production and its percentage; the slices are measured clockwise from the top.

Trend line and its purpose

A trend line shows the overall direction or pattern in the data over time or across categories. In bar graphs, trend lines are usually drawn as a dotted line across the tops of the bars to show an increasing, decreasing or constant trend. A trend line leads to statements such as: "There is an increasing trend", "The trend line shows a gradual decline", "The trend remains fairly constant", or "There is a sharp rise or drop between...".

Example 2: prevalence of polio in Pakistan, 2019 to 2023

Draw a simple bar chart to represent the prevalence of polio, showing the number of confirmed polio cases in Pakistan in the last five years.

Polio cases in Pakistan (total 2650).
Year	2019	2020	2021	2022	2023
No. of polio cases	550	350	450	700	600
Percentage of polio cases	20.76%	13.21%	16.98%	26.42%	22.64%

Polio cases in Pakistan (total 2650).
Year	No. of polio cases	Percentage of polio cases
2019	550	20.76%
2020	350	13.21%
2021	450	16.98%
2022	700	26.42%
2023	600	22.64%

Simple bar chart (book Example 2) of confirmed polio cases in Pakistan, 2019 to 2023, with a dashed trend line across the tops of the bars. The line shows an overall increasing trend.

Pie chart of polio cases in Pakistan in the last five years: each slice shows the year, its percentage and its number of cases.

Worked example: drawing a pie chart

In a class of 40 students the blood groups are A: 10, B: 12, AB: 4 and O: 14. Construct a pie chart with the percentage and angle of each slice.

Blood group	Students	Percentage (value ÷ 40 × 100)	Angle (% × 3.6°)
A	10	25%	90°
B	12	30%	108°
AB	4	10%	36°
O	14	35%	126°
Total	40	100%	360°

Step 1: total = 10 + 12 + 4 + 14 = 40. Step 2: percentage of each group as in the table. Step 3: angle = percentage × 3.6°. Check: the angles must add up to 360° (90 + 108 + 36 + 126 = 360).

Step 4: draw a circle, start at the 12 o'clock position and use a protractor to mark 90°, 108°, 36° and 126° in turn. Step 5: label each slice with its name, percentage and angle, and give the chart a title such as "Blood groups of 40 students".

Pie chart of the worked example: each slice angle is 3.6^\circ for every 1% of the total, starting at the 12 o'clock position and moving clockwise.

90^\circ,\ 108^\circ,\ 36^\circ,\ 126^\circ

11.5Error bars on a bar chartOften examined

Error bars show the variability of data, such as the range or the standard deviation, on a bar chart. To sketch them, follow three steps.

Steps

Calculate the error values. Range-based error bars: find the minimum and maximum values of each data point or category; the error bar length is half the range, extending above and below the mean or median value. Standard deviation (SD) error bars: calculate the SD of each category; the error bars extend above and below the mean by one standard deviation.
Draw the bar chart with the values representing the mean (or other central tendency) of each category.
Add error bars. Vertical error bars are lines centred on each bar, extending upward and downward by the calculated range or SD. Use caps on the ends of each line for a clean representation.

Length of the error bar.
Type of error bar	Error bar length
Based on range	\dfrac{\text{Max value}-\text{Min value}}{2} (above and below the mean)
Based on standard deviation	Equal to the standard deviation (\pm SD)

Calculation for standard deviation: book example

Three categories of data points in a biological experiment: calculate the mean and the SD of each category, then add error bars equal to the SD on the bar of each mean.

Mean, range, SD and error bar lengths.
Category	Data points	Mean	Range: (max − min) / 2	SD	Length of error bars (\pm SD)
A	10, 12, 15, 11, 13	12.2	2.5	1.92	±1.92
B	20, 21, 22, 19, 18	20.0	2	1.58	±1.58
C	30, 33, 32, 31, 29	31.0	2	1.58	±1.58

Bar chart of the mean of each category with error bars of one standard deviation above and below the top of the bar (caps on both ends of each line).

11.6Experimental group and control groupOften examined

While designing experiments, two main types of groups are used to get a better and clearer outcome. Control and experimental groups are used to compare the results and evaluate the acceptance of the findings of an experiment.

Experimental group (treatment group): The group that tests and experiences a change in a specific variable that is being tested. It is also known as the treatment group because the scientist applies a specific change or treatment to it to observe the effects of that treatment.
Control group: The group kept under unchanged normal conditions, without any specific treatment as applied in the experimental group. It serves as a standard or reference to compare the effects of the treatment in the experimental group.

Book example: if students of class 12 test the effect of a fertiliser on plant growth, plants in the experimental group receive the fertiliser, while the control group consists of plants grown without any added fertiliser.

The experimental group (left) receives the treatment; the control group (right) is kept under normal conditions without it. Arrows show the added fertiliser.

Experimental group and control group.
Point	Experimental group	Control group
Treatment	Receives the specific change or treatment being tested	Receives no treatment
Conditions	A specific variable is changed	Kept under unchanged, normal conditions
Purpose	To observe the effects of the treatment	Serves as the standard or reference for comparison
Fertiliser experiment	Plants receive the fertiliser	Plants grown without added fertiliser

11.6.3Independent, dependent and controlled variablesOften examined

There are three main types of variables in scientific investigations: independent, dependent and controlled variables.

The three types of variables. The independent variable is changed, the dependent variable is measured, and the controlled variables are held constant so that any change in the dependent variable is due to the independent variable only.

Independent variable: The variable that the experimenter (researcher) changes or manipulates in the experiment. It is the variable that the scientist thinks will cause an effect during the experiment, so it is also known as the "input variable" or "cause variable". In the fertiliser experiment it is the presence or absence of fertiliser.
Dependent variable: A variable that depends on, or is affected by, the independent variable. It is what the researcher measures or observes to determine the effect of the independent variable, so it is also known as the outcome variable or effect variable. In the fertiliser experiment it might be plant height, number of leaves or overall growth rate.
Controlled variable: A factor that must be kept constant during an experiment across both the experimental and control groups so that it does not influence the results. Also known as constant variables or controlled factors. Their purpose is to ensure that any change observed in the dependent variable is due to the independent variable and not to other factors. In the fertiliser experiment they might be the amount of sunlight, temperature, water, type of plant and soil quality.

Comparison of the three variables.
	Independent variable	Dependent variable	Controlled variable
Other names	Input variable, cause variable	Outcome variable, effect variable	Constant variable, controlled factor
Role	Changed by the researcher	Measured or observed by the researcher	Kept constant in all groups
Fertiliser example	Presence or absence of fertiliser	Plant height, number of leaves, growth rate	Sunlight, temperature, water, type of plant, soil quality

Further examples (book box): temperature and photosynthesis

Further examples of variables.
Variable	Temperature and the rate of photosynthesis in plants
Independent	Temperature: the researcher manipulates it to see how it affects the rate of photosynthesis. An experiment should have only one independent variable at a time; if more than one is changed, the researcher cannot know which caused the change in the dependent variable.
Dependent	The rate of photosynthesis, which is affected by changes in temperature. It is crucial to keep a continuous record of dependent variables, and they should be measurable quantitatively so they can be expressed and compared in numerical values.
Controlled	Type of plant, duration and intensity of light and the amount of \ce{CO2}. Keeping these constant ensures that changes in the rate of photosynthesis are due to temperature alone.

Worked example: identify the variables

A student studies whether the concentration of salt in the water given to bean seeds changes the number of seeds that germinate in one week. She uses four dishes with 0%, 1%, 2% and 3% salt solution, 20 seeds in each dish, the same amount of water, the same temperature and the same light. Name the independent, dependent and controlled variables and the control group.

Item	Answer	Reason
Independent variable	Salt concentration (0%, 1%, 2%, 3%)	It is the factor the researcher changes (input or cause).
Dependent variable	Number of seeds germinated in one week	It is measured to see the effect of salt (outcome or effect).
Controlled variables	Type and number of seeds, amount of water, temperature, light, size of dish	They are kept constant in every dish so that they cannot affect germination.
Control group	The dish with 0% salt	It receives no treatment and is the reference for comparison.
Experimental groups	The dishes with 1%, 2% and 3% salt	They receive the treatment being tested.

Check: only one factor (salt) differs between dishes, so any difference in germination can be linked to salt.

11.6.6Experimental designOften examined

To investigate the effect of light intensity on plant growth and the effect of different amounts of water on plant growth, two separate experiments are designed. The book adds a third experiment on pH and enzyme activity. Each states an objective, a hypothesis, the materials, a procedure and the variables.

Experiment 1: effect of light intensity on plant growth

Experiment 1.
Part	Details
Objective	To investigate how different light intensities affect the growth of a specific plant (e.g. bean plants).
Hypothesis	Plants exposed to higher light intensities will grow more compared with plants under lower light intensities.
Materials required	20 potted bean plants (similar age and size), a light source with adjustable intensity, ruler, water and watering can, soil of the same composition, thermometers.

Procedure

Divide the 20 potted plants into five groups of four plants each.
Place each group under a different light intensity (100%, 75%, 50%, 25% and 10%).
Make sure all groups receive the same amount of water and are kept at a constant temperature.
Measure the height of the plants every three days continuously for four weeks.
Record and analyse the growth data.

Figure 11.3: Experimental design to check the effect of light intensity on plant growth. Five groups of bean plants are placed under light of 100%, 75%, 50%, 25% and 10% intensity (shown above each group).

Variables of Experiment 1.
Variable or group	Experiment 1
Independent variable	Light intensity (100%, 75%, 50%, 25% and 10%)
Dependent variable	Plant growth, measured by the height of the plants (cm) and the number of leaves
Control variables	Type of plant, type of soil, amount of water, temperature, duration of light, amount of \ce{CO2}, pot size and initial plant size
Control group	Plants grown under the normal light intensity (e.g. 100%)

Results (Figure 11.4 of the book): photographs of the plants of each group at the end of the experiment show that the growth of the plants changes with the light intensity, with the best growth under the highest intensity and weaker, thinner plants under the lowest.

Experiment 2: effect of different amounts of water on plant growth

Experiment 2.
Part	Details
Objective	To investigate how different amounts of watering affect the growth of a specific plant (e.g. potato plants).
Hypothesis	Plants given more water (1000 ml per week) will grow more compared with plants given lower amounts of water.
Materials required	20 potato plants (similar age and size) in larger pots, water and a graduated watering can, a light source of the same intensity for all plants, ruler, soil of the same composition, thermometer.

Figure 11.5: Experimental design to check the effect of the amount of water on plant growth. Five groups of potato plants receive 1000, 800, 600, 400 and 200 ml of water per week.

Procedure

Divide the 20 potato plants, potted in larger pots of the same size, into five groups of four plants each.
Place all five groups under light of the same, higher intensity (100%) and at a constant temperature (30 °C).
Give a different amount of water per week to each group of plants (1000 ml, 800 ml, 600 ml, 400 ml and 200 ml).
Measure the height (cm) of the plants and the number of leaves on each group after every three days continuously for six weeks.
Record and analyse the data of plant growth.

Variables of Experiment 2.
Variable or group	Experiment 2
Independent variable	Amount of water given per week to each group of plants (1000, 800, 600, 400 and 200 ml)
Dependent variable	Plant growth, measured by the height of the plants (cm) and the number of leaves
Control variables	Type and species of plant, type of soil, duration and intensity of light, temperature, amount of \ce{CO2}, pot size and initial plant size
Control group	Plants watered with 1000 ml per week

Results (Figure 11.6 of the book): the photograph shows plants of the five groups, labelled by the amount of water per week, with the tallest growth at 1000 ml and the poorest at 200 ml.

Experiment 3: effect of different pH levels on enzyme activity

Experiment 3.
Part	Details
Objective	To examine how pH levels affect the activity of the enzyme catalase in breaking down hydrogen peroxide.
Hypothesis	Catalase will have the highest activity at a neutral pH and will be less active in acidic or alkaline conditions.
Materials required	Hydrogen peroxide solution, catalase (potato or liver tissue can be used as a source), test tubes, pH buffer solutions (pH 4, pH 7, pH 10), stopwatch, measuring cylinder, thermometer.

Procedure

Place an equal amount of catalase in each test tube.
Add 10 mL of hydrogen peroxide to each test tube.
Add different pH buffers to the tubes to achieve the desired specific pH level in each test tube.
Start the stopwatch as soon as hydrogen peroxide is added, and observe the time taken for bubbles to form or the amount of gas produced after 1 minute.
Repeat three times for each pH and calculate the average.

Variables of Experiment 3.
Variable or group	Experiment 3
Independent variable	pH level (pH 4, pH 7, pH 10)
Dependent variable	Enzyme activity (rate of reaction measured by gas produced, or reaction time)
Control variables	Amount of hydrogen peroxide, amount of catalase, temperature and test tube size
Control group	Reaction at neutral pH (pH 7)

These experiments give a hands-on and easily observable understanding of the scientific method and of the effects of variables on biological processes.

Formula sheet

Mean of ungrouped data

\bar{x}=\dfrac{\Sigma x}{n}

x = value of the data set; n = number of values

Unit: same as the data

Mean of grouped data

\bar{x}=\dfrac{\Sigma fx}{\Sigma f}

f = frequency; x = class mark =\dfrac{\text{lower limit}+\text{upper limit}}{2}

Unit: same as the data

Median position (odd number of values)

\tilde{x}=\text{value of the }\left(\dfrac{n+1}{2}\right)^{\text{th}}\text{ item}

n = number of values (odd), data in ascending or descending order

Unit: same as the data

Median (even number of values)

\tilde{x}=\dfrac{\left(\frac{n}{2}\right)^{\text{th}}\text{ value}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ value}}{2}

n = number of values (even)

Unit: same as the data

Mode of grouped data

\hat{x}=l+\left[\dfrac{(f_m-f_1)\times h}{(f_m-f_1)+(f_m-f_2)}\right]

l = lower class boundary of the modal group; f_m = maximum frequency; f_1 = frequency before f_m; f_2 = frequency after f_m; h = class interval

Unit: same as the data

Standard deviation of a population

\sigma=\sqrt{\dfrac{\Sigma(X-\mu)^2}{N}}

X = value; \mu = population mean; N = total number of observations

Unit: same as the data

Standard deviation of a sample

s=\sqrt{\dfrac{\Sigma(X-\bar{x})^2}{n-1}}

X = value; \bar{x} = sample mean; n = total number of observations

Unit: same as the data

Range

\text{Range}=\text{highest value}-\text{lowest value}

maximum and minimum values of the data

Unit: same as the data

Position of the Pth percentile

\text{Position}=\dfrac{P}{100}\times(N+1)

P = desired percentile; N = number of observations

Unit: position in the ordered data

Percentile rank

P=\dfrac{n}{N}\times100\%

n = number of data points below the data point of interest; N = total number of data points

Unit: per cent

Interquartile range

\text{IQR}=Q_3-Q_1

Q_3 = upper quartile; Q_1 = lower quartile

Unit: same as the data

Range-based error bar length

\dfrac{\text{Max value}-\text{Min value}}{2}

maximum and minimum values of a category

Unit: same as the data

Pie chart angle

\text{angle}=\text{percentage}\times3.6^\circ

percentage = value \div total \times100; 100% = 360^\circ

Unit: degrees

Key terms

Biostatistics: Application of statistical principles and techniques to analyse, interpret and present research findings in biology, medicine and health sciences.
Data: Facts, measurements or observations related to living organisms, collected to analyse and draw conclusions about biological phenomena.
Data set: A collection of related data values organised systematically, usually in a table: rows are subjects and columns are variables.
Average: A single value calculated to represent the whole data; also called a measure of central tendency.
Average statement: An approximate statement in a specific situation, such as a heartbeat of 72 beats per minute.
Arithmetic mean: The sum of all values of a data set divided by the number of values; symbol \bar{x}.
Median: The middle value of data arranged in ascending or descending order; symbol \tilde{x}.
Mode: The value that appears the maximum number of times in a data set; symbol \hat{x}.
Unimodal: A data set with one mode.
Bimodal: A data set with two modes.
Multimodal: A data set with more than two modes.
Amodal: A data set with no mode, because all values occur the same number of times.
Ungrouped data: Raw data in the form of individual values that have not been organised into groups; suitable for small data sets.
Grouped data: Data organised into classes or intervals with their frequencies; makes large data sets easier to summarise.
Frequency distribution: A table or graph that displays the number of occurrences (frequencies) of different outcomes or values in a data set.
Class interval: A range of values into which data are grouped in a frequency table, with a lower and an upper boundary.
Modal group (modal class): The class interval with the highest frequency.
Cumulative frequency: The running total of the frequencies, used to find the median of grouped data.
Standard deviation: A measure of the spread of data values around the mean; the square root of the variance.
Variance: The square of the standard deviation, \sigma^2.
Population standard deviation (\sigma): Standard deviation using all values of the population, dividing the sum of squares by N.
Sample standard deviation (s): Standard deviation of a sample, dividing the sum of squares by n-1.
Normal curve: Bell shaped curve of data with the mean at the centre, used to show standard deviations and percentiles.
Range: The difference between the highest and lowest values in a set of data.
Percentile: A value that indicates the percentage of data in a set that is below a certain value.
Quartile: One of the three values that divide the sorted data into four equal parts.
First quartile (Q_1): The 25th percentile: 25% of the values are below it and 75% above it.
Second quartile (Q_2): The 50th percentile, also called the median.
Third quartile (Q_3): The 75th percentile: 25% of the values are above it and 75% below it.
Interquartile range (IQR): The difference between the upper and lower quartiles.
Bar chart: A chart of bars of equal width whose heights show the magnitude of each value; a simple bar chart shows one characteristic.
Pie chart: A circle divided into slices whose angles are proportional to the percentage of each category.
Trend line: A line, often dotted across the tops of the bars, that shows the overall direction or pattern in the data.
Error bars: Lines with caps on a bar chart that show variability such as the standard deviation or half the range.
Experimental group: The group that receives the specific change or treatment being tested; also called the treatment group.
Control group: The group kept under normal conditions without the treatment; the standard for comparison.
Independent variable: The variable the researcher changes in the experiment; also called the input variable or cause variable.
Dependent variable: The variable that is measured to see the effect of the independent variable; also called the outcome variable or effect variable.
Controlled variable: A factor kept constant in the experimental and control groups; also called a constant variable or controlled factor.
Hypothesis: A statement of the expected outcome of an experiment, for example that plants under higher light will grow more.
Epidemiology: The study of disease outbreaks, risk factors, patterns and rates of disease in a population, in which biostatistics is used.
Survival analysis: Use of biostatistics to predict the survival rate of patients after a particular treatment and to estimate life expectancy.

Multiple choice questionsp. 234–235

In an experiment examining the effects of pH on the activity of the enzyme, the independent variable will be

ApHAnswer
BTemperature
CSubstrate
DReaction time

Why

The experimenter deliberately sets the pH (pH 4, pH 7 and pH 10 in the book's Experiment 3), so pH is the independent variable. The enzyme activity (gas produced or reaction time) is what is measured, so it is the dependent variable (option D). Temperature and the amounts of catalase and hydrogen peroxide are kept constant, so they are controlled variables (options B and C).

The variable that the experimenter changes in the experiment to check its effect during the experiment is called

ADependant variable
BIndependent variableAnswer
CControl variable
DExperimental variable

Why

The independent variable (also called the input or cause variable) is the one the researcher changes or manipulates to see its effect. The dependent variable (A) is only measured, and a control variable (C) is kept constant. "Experimental variable" (D) is not one of the three variables named in the book.

Variables that the researcher measures to determine the effect of the independent variable are also known as

AOutcome variablesAnswer
BInput variables
CCause variables
DControl variable

Why

The variable that is measured depends on the independent variable, so it is the dependent variable, which the book also calls the outcome variable or effect variable. Input variables and cause variables (B and C) are other names of the independent variable, the one that is changed.

To which of the following fields of life sciences is biostatistics not applicable?

AHealth
BChemical engineeringAnswer
CAgriculture
DMedicine

Why

The book applies biostatistics to living organisms in medicine, public health and agriculture (crop yield, dairy and poultry farming). Chemical engineering is not a field of life sciences, so biostatistics is not applied there.

The 25th percentile is the value ........... of the data points lie.

AAbove which 25%
BBelow which 25%Answer
C3/4 th
D25

Why

A percentile gives the percentage of data that is below a value. The 25th percentile is the value below which 25\% of the data points lie (it is also the first quartile). The value above which 25\% lie is the 75th percentile, so option A is wrong.

Biostatistics is not involved in

AEpidemiological studies
BAgriculture
CPetroleum industryAnswer
DPublic health

Why

Epidemiological studies, agriculture and public health are all uses of biostatistics listed in the book. The petroleum industry does not deal with data from living organisms, so biostatistics is not involved in it.

The factors that must be kept constant during an experiment across both the experimental and control groups are known as

AControl variablesAnswer
BOutcome variables
CInput variables
DCause variables

Why

Controlled (control or constant) variables are the factors kept the same in the experimental and control groups, so that any change in the dependent variable is due only to the independent variable. Outcome variables are measured, and input and cause variables are changed.

All of the following are types of averages except

AArithmetic mean
BMedian
CMode
DProduct of data valuesAnswer

Why

The book names the types of average as arithmetic mean, median, mode, geometric mean and harmonic mean. The product of data values is not an average.

The symbol for mean is

A\bar{x} (read as "x bar")Answer
B\tilde{x} (read as "x tilde")
C\hat{x} (read as x-hat)
D"h" (read as "highly stable")

Why

The mean is written \bar{x} ("x bar"). The median is \tilde{x} ("x tilde") and the mode is \hat{x} ("x hat").

The formula of "mean" for un-grouped data is

A\dfrac{\sum x}{n}Answer
B\dfrac{\sum fx}{\sum f}
C\dfrac{\sum f_1}{\sum f_2}
D\dfrac{n+1}{2}

Why

For ungrouped data the mean is the sum of the values divided by the number of values: \bar{x}=\dfrac{\sum x}{n}. Option B is the formula for grouped data, and option D is the position of the median value, not a mean.

Find the mode from this dataset: 46, 49, 55, 58, 62, 66, 64, 65, 66, 67, 68, 70, 72

A46
B62
C66Answer
D72

Why

The mode is the value that occurs most often. In this dataset 66 appears twice and every other value appears once, so the mode is 66.

The first step in genetic engineering is

AIsolation of the gene of interestAnswer
BInsertion of the gene into a vector
CTransfer of recombinant DNA into the host organism
DExpression of the gene

Why

The gene that is wanted must first be found and isolated. Only then can it be inserted into a vector, the recombinant DNA transferred into the host organism, and the gene expressed. The later options cannot happen before the gene is isolated.

Note. This question belongs to genetic engineering and is not taught in the biostatistics unit.

Biostatistics is similar to maths because of

ACalculationsAnswer
BCounting birds
CUsed of symbol pie
DIt is also useful in engineering studies

Why

Biostatistics uses mathematical calculations (means, medians, standard deviation, percentages) on biological data, which is what it shares with maths. Counting birds and the pie symbol are not reasons for the similarity.

If the number of diseased persons is greater in the 2nd year of study than in the 1st year, then

AHeight of the bar will be greater in 1st year
BWidth of the bar will be greater in 1st year
CHeight of the bar will be smaller in 1st yearAnswer
DWidth of the bar will be greater in 2nd year

Why

In a bar chart the height of a bar shows the magnitude of the value, while all bars have equal width. More diseased persons in year 2 means a taller bar in year 2, so the bar of year 1 is smaller in height. Width does not change with the value, so B and D are wrong.

In a bar chart of the number of deaths due to hepatitis in Pakistan in the last 10 years, what will we write on the x-axis of the graph?

AYearsAnswer
BNumber of deaths
CAny of the value
DCauses of hepatitis

Why

The categories whose values are compared go on the horizontal (x) axis, so the 10 years are written there. The measured quantity, the number of deaths, is on the vertical (y) axis with its scale.

Short answer questionsp. 236

13 marks

How is standard deviation different from range?

Solution

Both measure the spread of data, but they are calculated differently and tell different things.

Point	Standard deviation (SD)	Range
Definition	A measure of how far the data values spread around the mean	The difference between the highest and the lowest value in the data
Formula	s=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n-1}} for a sample	\text{Range}=\text{highest value}-\text{lowest value}
Values used	Every value in the dataset (through its distance from the mean)	Only the two extreme values
Calculation	Long, usually done with a calculator or software	Very simple, one subtraction

Because the range ignores all the values between the extremes, it may be misleading, whereas the standard deviation uses every value.

23 marks

What are the demerits of relying on range to plan future scientific experiments?

Solution

The range is only the difference between the highest and the lowest value, so it has these limitations.

It accounts only for the highest and lowest values and gives no information about the values between them or how they are distributed.
It may be deceiving: if almost all plants respond well to a treatment and only one or two plants show the lowest productivity, the range is large although the treatment worked for most plants.
A middle-sized range gives no clear-cut information, so the researcher must re-plan the experiment. Looking at the range alone may therefore lead to a wrong plan.

So another measure, such as the standard deviation, should be considered before planning future studies.

33 marks

Why is a control group important for scientific experiments?

Solution

The control group is kept under normal, unchanged conditions without the treatment given to the experimental group. Its importance is:

It is the standard or reference with which the results of the experimental group are compared.
The difference between the two groups shows the effect of the treatment alone, so the findings of the experiment can be judged and accepted.
Example: in a fertiliser experiment the control plants grow without fertiliser. If the treated plants grow taller, the extra growth is due to the fertiliser.

43 marks

Define mean, median and mode.

Solution

Mean: The sum of all the values of a data set divided by the total number of values. It is one value that represents the whole data set and is written \bar{x} ("x bar").
Median: The middle value of a data set arranged in ascending or descending order. It divides the data into two equal parts and is written \tilde{x} ("x tilde").
Mode: The value that occurs the greatest number of times in a data set. A set may have one mode, more than one mode or none. It is written \hat{x} ("x hat").

Example for the data 2,3,3,5,7: \bar{x}=\dfrac{2+3+3+5+7}{5}=\dfrac{20}{5}=4, the median is the 3rd value, 3, and the mode is 3.

53 marks

Write down the formulae of median and mode.

Solution

Median

Ungrouped data, odd n: \tilde{x}=\left(\dfrac{n+1}{2}\right)^{\text{th}} value.
Ungrouped data, even n: \tilde{x}=\dfrac{\left(\frac{n}{2}\right)^{\text{th}}\text{ value}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ value}}{2}.
Grouped data (using cumulative frequency): for odd n the class value against the \left(\dfrac{n+1}{2}\right)^{\text{th}} cumulative frequency; for even n the average of the class values against the \dfrac{n}{2} and \dfrac{n}{2}+1 cumulative frequencies.

Mode

For ungrouped data the mode is the most repeated value. For grouped data:

\hat{x}=l+\dfrac{(f_m-f_1)\times h}{(f_m-f_1)+(f_m-f_2)}

where l is the lower class boundary of the modal group, f_m the maximum (modal) frequency, f_1 the frequency before it, f_2 the frequency after it and h the class interval of the modal group.

63 marks

What is the importance of mean calculation?

Solution

The mean is the most common average, and calculating it is important because:

It gives a single value that represents the whole data set, so a large amount of data can be described in one number.
It expresses the central idea of the observations, so two groups can be compared easily (for example, the mean number of polio patients per month was 16 in 2022 and 12 in 2023, a fall of 4 patients per month).
It is widely used in agriculture, health and poultry studies, and experts at national and international level discuss the findings of their studies in terms of averages.

73 marks

Why is the value of the mean not completely reliable to get the correct picture?

Solution

The mean is only an approximate statement, because it is calculated from all the values and so it hides the differences between them.

No individual value is exactly the mean. The heartbeat is said to be 72 beats per minute, but it may be 60 in sleep and 90 in running; wheat yield may be 1500\,\mathrm{kg} per acre in one district and 2400\,\mathrm{kg} in another.
Extreme values change the mean (the median is not affected by them). For the plant heights 10,11,12,13\,\mathrm{cm} the mean is 11.5\,\mathrm{cm}, but if one very tall plant of 50\,\mathrm{cm} is added the mean becomes \dfrac{96}{5}=19.2\,\mathrm{cm}, which is larger than four of the five plants.
The mean says nothing about spread. Two groups can have the same mean and very different variation, so the mean must be read with the range or the standard deviation.

83 marks

Write the formula for calculation of mode.

Solution

For grouped data the mode is calculated from the modal group (the class with the highest frequency):

\hat{x}=l+\dfrac{(f_m-f_1)\times h}{(f_m-f_1)+(f_m-f_2)}

l: lower class boundary of the modal group
f_m: maximum frequency (frequency of the modal group)
f_1: frequency of the group before the modal group
f_2: frequency of the group after the modal group
h: class interval (width) of the modal group

For ungrouped data no formula is needed: the mode is the value that is repeated most often (for 3,5,6,6,9,10,9,6,6 the mode is 6).

Note. The book prints "lowercase boundary of model group" for l; it should read "lower class boundary of the modal group".

93 marks

How is a bar chart different from a pie chart?

Solution

Point	Bar chart	Pie chart
Shape and drawing	Separate rectangles of equal width on x and y axes, with equal gaps	A circle divided into sectors from the centre, with no axes
What shows the value	The height (length) of each bar	The angle (and so the area) of each sector, \text{angle}=\text{percentage}\times3.6^\circ
Best use	Comparing the actual values of categories or years, with the trend seen across bars	Showing each category as a part (percentage) of the whole of 100\%

Example: for the cotton data a bar chart reads 800\,\mathrm{kg} from the bar height, while the pie chart shows the same value as a 20\% sector of 72^\circ.

103 marks

Write one similarity and one difference between a rectangle graph and a pie graph.

Solution

The rectangle graph is the simple bar chart, whose bars are rectangles.

Similarity: both present the same set of data in a picture, instead of a table, so the data are easier to see and understand.
Difference: a bar chart shows the value by the height of a bar against a scale on the y-axis, whereas a pie chart shows each value as a sector of a circle, with the angle proportional to its percentage of the total.

Example: the cotton production data can be drawn as six bars of different heights or as six sectors of one circle.

113 marks

How is graphical representation of data more important than tabular representation?

Solution

Biostatistical data are recorded in tables, but graphs and diagrams are better because:

A table with a large number of figures is not easily understandable, and it is difficult to get a clear picture from numbers, whereas a graph gives a clear and complete picture of the data.
Diagrams and graphs are more attractive than figures, so they interest the reader more.
Information presented as charts and graphs leaves a long-lasting impression on the mind of the observer, and different types (bar chart, pie chart) can be chosen to suit the data.

123 marks

Find out the standard deviation of the values in the following dataset: 33, 55, 66, 84, 91, 72, 68, 87, 78, 66, 82, 46, 71, 31, 38

Solution

The data are a sample, so the sample formula of the book is used:

s=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n-1}}

Count the values and find the mean.
n=15,\qquad \sum x=968,\qquad \bar{x}=\dfrac{\sum x}{n}=\dfrac{968}{15}=64.53
Subtract the mean from each value and square the result (table below).
Add the squares.
\sum(x-\bar{x})^2=5441.73
Divide by n-1 (unbiased, because the data are treated as a sample).
s^2=\dfrac{5441.73}{15-1}=\dfrac{5441.73}{14}=388.70
Take the square root.
s=\sqrt{388.70}=19.72

x	x-\bar{x}	(x-\bar{x})^2
33	-31.53	994.35
55	-9.53	90.88
66	1.47	2.15
84	19.47	378.95
91	26.47	700.48
72	7.47	55.75
68	3.47	12.02
87	22.47	504.75
78	13.47	181.35
66	1.47	2.15
82	17.47	305.08
46	-18.53	343.48
71	6.47	41.82
31	-33.53	1124.48
38	-26.53	704.02
\sum x=968	\sum=0	\sum=5441.73

If the 15 values were the whole population, \sigma=\sqrt{5441.73/15}=19.05.

s\approx19.72

Note. The data have no unit in the question; if they are marks or counts, the standard deviation has the same unit as the data.

133 marks

Calculate the median of the values given in the dataset of the above question.

Solution

Step 1: arrange the values in ascending order.

31,\ 33,\ 38,\ 46,\ 55,\ 66,\ 66,\ \mathbf{68},\ 71,\ 72,\ 78,\ 82,\ 84,\ 87,\ 91

Step 2: n=15 is odd, so the median is the value at position \dfrac{n+1}{2}.

\text{position}=\dfrac{15+1}{2}=\dfrac{16}{2}=8

Step 3: the 8th value in the ordered list is 68.

\tilde{x}=68

Note. The book says "of above question 13" but means question 12, the dataset of 15 values.

Extensive answer questionsp. 236

16 marks

What are the advantages and disadvantages of median?

Solution

Meaning of median

The median is the middle value of a data set arranged in ascending or descending order. It divides the data into two equal parts: half of the values are smaller than it and half are larger. For n values the median is the \left(\dfrac{n+1}{2}\right)^{\text{th}} value if n is odd, and the average of the \dfrac{n}{2} and \left(\dfrac{n}{2}+1\right) values if n is even.

Finding the median: the middle value for an odd number of values (a), and the average of the two middle values for an even number (b).

Advantages

Easy to calculate and located exactly. Arrange the values in order and pick the middle one; no long calculation is needed.
Not affected by abnormally large or small values. Only the position matters, so one extreme value does not change it. For 12,13,14,15,16 the median is 14. If 16 is replaced by 90 the median is still 14, but the mean rises from 14 to \dfrac{12+13+14+15+90}{5}=28.8.
Can be used for quantitative measurements, such as heights of plants or number of fruits per plant.

Disadvantages

The median of two or more datasets cannot be found by using or adding the medians of the individual datasets. Set A: 1,2,3 has median 2. Set B: 10,11,12,13,14 has median 12. Combined, the eight values 1,2,3,10,11,12,13,14 have median \dfrac{10+11}{2}=10.5, not \dfrac{2+12}{2}=7.
The median may not necessarily represent the central data. For the 16 plants of the book the median is 12.5 fruits, a value that no plant actually produced.
It cannot be used where specific values, such as weight and age, are used (the book's statement), because the exact values are ignored and only their order is used.

Advantages	Disadvantages
Easy to calculate, located exactly	Medians of datasets cannot be combined
Not affected by extreme values	May not represent the central data
Usable for quantitative measurements	Not suitable where specific values (weight, age) are used

26 marks

Describe the detailed procedure to add error bars in the graph.

Solution

What an error bar is

An error bar is a line drawn above and below the top of a bar to show the variability of the data. Its length is based either on the range or on the standard deviation (SD) of each category.

Procedure

Calculate the error value for each category. For range-based error bars find the minimum and maximum values; the length of the error bar is half of the range. For SD error bars calculate the standard deviation; the error bar extends one SD above and below the mean.
\begin{aligned}\text{range-based error}&=\dfrac{\text{max value}-\text{min value}}{2}\\ \text{SD error}&=s=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n-1}}\end{aligned}
Draw the bar chart. The height of each bar is the mean (or other central tendency) of that category, with labelled axes and a scale starting at zero.
Add the error bars. Draw a vertical line centred on the top of each bar, extending upward and downward by the calculated error value, and finish each end with a small cap.
Label the chart, and say in the title or legend whether the bars show the range or the SD.

Worked example (the book's data, error bars from SD)

Three categories A, B and C of a biological experiment have five values each. For category A the working is shown in full.

x (A)	x-\bar{x}	(x-\bar{x})^2
10	-2.2	4.84
12	-0.2	0.04
15	2.8	7.84
11	-1.2	1.44
13	0.8	0.64
\sum x=61		\sum=14.80

Mean of A.
\bar{x}=\dfrac{10+12+15+11+13}{5}=\dfrac{61}{5}=12.2
Standard deviation of A (sample formula, n-1=4).
s=\sqrt{\dfrac{14.80}{4}}=\sqrt{3.70}=1.92
Range-based error of A.
\dfrac{15-10}{2}=2.5
The same working for B (mean 20.0, \sum(x-\bar{x})^2=10, s=\sqrt{10/4}=1.58) and C (mean 31.0, \sum(x-\bar{x})^2=10, s=1.58).

Category	Mean	Range-based error	SD	Error bar (± SD)
A	12.2	2.5	1.92	12.2 ± 1.92 (10.28 to 14.12)
B	20.0	2	1.58	20.0 ± 1.58 (18.42 to 21.58)
C	31.0	2	1.58	31.0 ± 1.58 (29.42 to 32.58)

Bar chart of the three means with error bars of one standard deviation above and below each mean.

34 marks

Complete the following table by filling in the values of cumulative frequency.

Solution

The cumulative frequency (cf) of a class is the frequency of that class added to the frequencies of all the classes before it. Each cumulative frequency is the previous cumulative frequency plus the frequency of the class.

First class: the cumulative frequency equals its frequency.
\text{cf}_1=30
Second class: add its frequency to the previous cumulative frequency.
\text{cf}_2=30+50=80
Third class.
\text{cf}_3=80+60=140
Fourth class.
\text{cf}_4=140+40=180
Fifth class.
\text{cf}_5=180+30=210

Class (no. of flowers/plant)	Frequency (no. of plants = n)	Cumulative frequency
1	30	30
2	50	80
3	60	140
4	40	180
5	30	210

Check: the last cumulative frequency must equal the total frequency, 30+50+60+40+30=210 plants.

Use of the table: the median

The table gives the median of the data by the book's method. The number of plants is n=210 (even).

\text{position}=\dfrac{n}{2}\ \text{and}\ \dfrac{n}{2}+1=105\ \text{and}\ 106

The 105th and the 106th values both lie in the class with cumulative frequency 140 (this class starts after 80), which is the class 3 flowers per plant. The median is \dfrac{3+3}{2}=3 flowers per plant.

\text{cf}=30,\ 80,\ 140,\ 180,\ 210

46 marks

Give detailed steps to construct the bar chart.

Solution

Simple bar chart

A simple bar chart represents one characteristic of the data. Each bar shows one value, all bars have the same width, and the height (length) of the bar shows the magnitude of the quantity. Bars are normally drawn vertically.

Steps

Take graph paper with a suitable size and scale for the type of data.
Draw two lines perpendicular to each other, meeting at zero: the horizontal line is the X-axis and the vertical line is the Y-axis.
Label the axes. Write the names of the data items (categories) along the X-axis, and the name of the measured quantity with its unit along the Y-axis.
Choose a uniform width for the bars and a uniform gap between them along the X-axis.
Choose a suitable scale on the Y-axis, so that the largest value fits on the paper, and mark equal steps from zero.
Calculate the height of each bar from the scale and mark it with a dot.
\text{height of bar}=\dfrac{\text{value}}{\text{value per cm of the scale}}
Draw the bars carefully up to the dots, then recheck all the entries and give the chart a title.

Example (the polio data of the book)

Year	2019	2020	2021	2022	2023
No. of polio cases	550	350	450	700	600

Scale: 1\,\mathrm{cm} on the Y-axis represents 100 cases, with the axis running from 0 to 800. The bar heights are:

2019:\dfrac{550}{100}=5.5\,\mathrm{cm},\quad 2020:3.5\,\mathrm{cm},\quad 2021:4.5\,\mathrm{cm},\quad 2022:7.0\,\mathrm{cm},\quad 2023:6.0\,\mathrm{cm}

A simple bar chart with its parts labelled: title, axis titles, scale from zero, equal bars and gaps, and values on the bars.

54 marks

What do the letters l, fm, f1, f2 and h stand for in the given formula? For which calculation is this formula used? l+\left[\dfrac{(f_m-f_1)\times h}{(f_m-f_1)+(f_m-f_2)}\right]

Solution

The formula and its use

The formula is used to calculate the mode of grouped data, that is, data arranged in classes with frequencies. The result is written \hat{x}.

\hat{x}=l+\left[\dfrac{(f_m-f_1)\times h}{(f_m-f_1)+(f_m-f_2)}\right]

Meaning of the letters

Letter	Meaning
l	lower class boundary of the modal group (the class with the highest frequency)
f_m	maximum frequency, the frequency of the modal group
f_1	frequency of the group just before the modal group (previous frequency)
f_2	frequency of the group just after the modal group (next frequency)
h	class interval (width) of the modal group

Worked example (the book's frequency distribution)

Group	Frequency	Class boundary
10–19	5	9.5–19.5
20–29	25 (f_1)	19.5–29.5
30–39	40 (f_m)	29.5–39.5 (modal group)
40–49	20 (f_2)	39.5–49.5
50–59	10	49.5–59.5

Find the modal group: the class with the highest frequency, 30–39 (frequency 40).
Read the letters from the table.
l=29.5,\quad f_m=40,\quad f_1=25,\quad f_2=20,\quad h=39.5-29.5=10
Substitute in the formula.
\hat{x}=29.5+\dfrac{(40-25)\times10}{(40-25)+(40-20)}=29.5+\dfrac{150}{35}
Calculate.
\hat{x}=29.5+4.29=33.79\ (\text{the book gives }33.786)

The modal group (accent) with the quantities of the formula: l is the lower boundary, h the width, fm the highest frequency, f1 and f2 the frequencies on either side.

\hat{x}=33.79

Note. The book prints "lowercase boundary of model group" for l; it should read "lower class boundary of the modal group".

66 marks

Sketch and label the pie chart by using the data given in the table below about the deaths of vulture birds due to pollution in the last five years. Also label the data values in the pie chart. Years 2019, 2020, 2021, 2022, 2023; No. of deaths 250, 400, 275, 350, 175

Solution

Steps

Total the values.
250+400+275+350+175=1450\ \text{deaths}
Convert each value to a percentage of the total.
\text{percentage}=\dfrac{\text{value}}{\text{total}}\times100
Convert each percentage to an angle, because a circle is 360^\circ and 100\%\times3.6=360^\circ.
\text{angle}=\text{percentage}\times3.6^\circ
Draw a circle with a compass and mark its centre. Starting at the 12 o'clock position, measure each angle with a protractor and draw the radii.
Label each sector with its year, its value and its percentage, add a title, and check that the angles add up to 360^\circ.

Year	Deaths	Percentage of total	Angle
2019	250	\dfrac{250}{1450}\times100=17.2\%	17.2\times3.6=62.1^\circ
2020	400	\dfrac{400}{1450}\times100=27.6\%	27.6\times3.6=99.3^\circ
2021	275	\dfrac{275}{1450}\times100=19.0\%	19.0\times3.6=68.3^\circ
2022	350	\dfrac{350}{1450}\times100=24.1\%	24.1\times3.6=86.9^\circ
2023	175	\dfrac{175}{1450}\times100=12.1\%	12.1\times3.6=43.4^\circ
Total	1450	100\%	360^\circ

Pie chart of the deaths of vultures due to pollution, 2019 to 2023. The largest sector (2020) is in accent; each sector shows its percentage, and the label gives the year, the number of deaths and the angle.

The 2020 sector is the largest (27.6\%), so the largest number of vulture deaths was in 2020, and the 2023 sector is the smallest (12.1\%).

Note. Angles are rounded to one decimal place, so the percentages add to 100.0\% and the angles to 360.0^\circ.

76 marks

Draw a table and then sketch and label the bar chart by using the data of height (in cm) collected from 200 students of 12th class of your college.

Solution

The real data are collected by measuring the students. Here the following results are used as an illustration; the method is the same for any measurements.

Step 1: Record the data in a table

The 200 heights are grouped into classes of equal width (5\,\mathrm{cm}) and the number of students (frequency) in each class is counted. Each percentage is \dfrac{f}{200}\times100.

Height class (cm)	Number of students	Percentage
145–149	8	4%
150–154	24	12%
155–159	46	23%
160–164	56	28%
165–169	38	19%
170–174	20	10%
175–179	8	4%
Total	200	100%

Check: 8+24+46+56+38+20+8=200.

Step 2: Choose the scale

The tallest bar is 56 students, so the Y-axis runs from 0 to 60 with 1\,\mathrm{cm} representing 10 students. The classes are written along the X-axis in equal slots.

Step 3: Draw the chart

Draw perpendicular axes meeting at zero; label the X-axis "Height of students (cm)" and the Y-axis "Number of students".
Draw seven bars of the same width with equal gaps, each as high as its frequency on the scale (for example \dfrac{56}{10}=5.6\,\mathrm{cm} for 160–164\,\mathrm{cm}).
Write the value on each bar and give a title.

Bar chart of the heights of 200 students (illustrative data). The tallest bar is the class 160–164 cm.

86 marks

Compare and contrast control variables, dependent variables and independent variables.

Solution

Variables in an experiment

There are three main types of variables in a scientific investigation. In an experiment there should be only one independent variable at a time, otherwise the researcher cannot tell which variable caused the change in the dependent variable.

The three kinds of variable: the independent variable is changed and affects the dependent variable, while the controlled variables are held constant.

Point	Independent variable	Dependent variable	Control (controlled) variable
Meaning	The variable that the experimenter changes or manipulates	The variable that depends on, or is affected by, the independent variable	A factor that must be kept constant in the experimental and control groups
Other names	Input variable, cause variable	Outcome variable, effect variable	Constant variable, controlled factor
Role	Believed to cause the effect	Measured or observed to find the effect; it must be measurable so that it can be compared as numbers	Makes the experiment fair, so that a change is due to the independent variable only
Light experiment	Light intensity (100%, 75%, 50%, 25%, 10%)	Plant growth (height in cm, number of leaves)	Type of plant, soil, amount of water, temperature, duration of light, pot size
pH experiment	pH level (pH 4, pH 7, pH 10)	Enzyme activity (gas produced or reaction time)	Amount of hydrogen peroxide, amount of catalase, temperature, test tube size

Similarity

All three are factors of the same experiment, and all three must be identified when an experiment is designed with a control group.

96 marks

Describe the usage of percentile along with its demerits.

Solution

Percentile

A percentile is a value that indicates the percentage of the data in a set that is below that value. It divides the ordered data into 100 units, so a percentile can be any number from 0 to 100. The 25th percentile is the value below which 25\% of the data points lie.

Usage of percentile

To report and interpret test scores and biometric measurements (heights, number of fruits per plant).
To compare an individual with the study group. A percentile rank interprets an individual's raw value by comparing it with the values of the other individuals. If a height of 77\,\mathrm{cm} has a percentile rank of 89, then 89\% of the students are 77\,\mathrm{cm} or shorter.
To find quartiles, which divide the ordered data into four equal parts: the 25th percentile is the first quartile Q_1, the 50th percentile is the second quartile (the median) Q_2, and the 75th percentile is the third quartile Q_3. The interquartile range is Q_3-Q_1.
To analyse data in groups of ordered parts rather than by focusing on the whole data pool.

Percentiles on a normal bell-shaped curve: the cumulative percentage under each standard deviation is the percentile at that point, and the mean is the 50th percentile.

Calculation

\text{Position}=\dfrac{P}{100}\times(N+1),\qquad P=\dfrac{n}{N}\times100\%

The first formula gives the position of the Pth percentile in the ordered data (N is the number of observations). The second gives the percentile P of a value, where n is the number of data points below it and N is the total number of data points.

Example (the book's orange tree data, N=16 fruit counts in order: 11,13,14,15,18,18,19,19,23,23,23,24,24,25,25,26).

Position of the 25th percentile.
\dfrac{25}{100}\times(16+1)=4.25
The 4th value is 15, so the first quartile is marked at 15 (the book marks this value).
Median (50th percentile): position \dfrac{50}{100}\times17=8.5, between the 8th and 9th values 19 and 23.
\dfrac{19+23}{2}=21
The 75th percentile is at position \dfrac{75}{100}\times17=12.75, close to the 12th and 13th values, both 24.
Percentile of the value 19 (the 7th and 8th values): n=6 values lie below the first 19.
P=\dfrac{6}{16}\times100\%=37.5\%

The three quartiles marked on the ordered data of the book: Q1 = 15, Q2 = 21 and Q3 = 24.

Demerits of percentile

A percentile gives only the position (rank) of a value in the data, not its size or how far it is from the other values.
The data must first be arranged in order, and the position has to be worked out for each percentile, so it is long to do by hand for a large data set.
A percentile depends on the study group with which the value is compared; the same height of 77\,\mathrm{cm} has a different percentile rank in a different group of students.

Note. In the 16-value table the book marks the first quartile at 15 (the 4th value) and the third quartile at 24.

106 marks

Describe stepwise details to make the appropriate chart with proper title, labelled axes, legend and axes units from a fictitious data set.

Solution

Fictitious data set

A student grows bean plants under three light intensities and measures the mean height of the plants after 2 weeks and after 4 weeks.

Light intensity (% of full light)	Mean height after 2 weeks (cm)	Mean height after 4 weeks (cm)
100	6	14
50	5	10
10	3	5

Steps

Identify the variables. Light intensity is the independent variable (categories on the X-axis) and plant height is the dependent variable (measured on the Y-axis). There are two sets of values (2 weeks and 4 weeks), so a legend will be needed.
Choose the appropriate chart. Categories whose values are compared are shown in a bar chart; parts of a whole are shown in a pie chart. Here the heights of two groups are compared for each light intensity, so a bar chart with two bars per category is appropriate.
Write a proper title that says what is shown, for example "Effect of light intensity on bean plant height".
Draw the axes at right angles, meeting at zero. Write the categories on the X-axis and mark the scale on the Y-axis.
Label the axes with names and units: X-axis "Light intensity (% of full light)" and Y-axis "Mean plant height (cm)".
Choose the Y-axis scale in equal steps from zero to just above the largest value (14\,\mathrm{cm}): 0 to 16\,\mathrm{cm} in steps of 4\,\mathrm{cm}.
Draw bars of equal width with equal gaps between the groups, one bar for each value, shading the two series differently.
Add a legend that explains the shading ("After 2 weeks", "After 4 weeks"), write the values on the bars and recheck all entries.

A complete chart: title, labelled axes with units, equal scale from zero, two shaded series and a legend.

Checklist of the parts

Part	What it must show
Title	What the chart is about
Axes	Perpendicular, meeting at zero
Axis labels and units	Variable name and unit, such as (cm) or (%)
Scale	Equal steps, starting at zero
Legend	Meaning of each shading or colour when there is more than one series

FBISE 2026 questions

Questions from the FBISE HSSC-II 2026 (1st Annual) papers that belong to this unit, answered the way an examiner expects.

MCQ 162026 Annual I, Set B, MCQ 161 mark

Range of the data 10, 12, 15, 11, 13, is:

A15
B2.5
C5Answer
D10

Why

Range is the difference between the highest and the lowest value: Range = highest - lowest =15-10=5. The data in order are 10, 11, 12, 13, 15.

The tempting options are the end values themselves: 15 is only the highest value and 10 is only the lowest. The range is the gap between them, not either of them. 2.5 is half of the range and has no meaning here.

Q.2 (iv)2026 Annual I, Set B, Q.2 (iv)1+2 = 3 marks

Calculate Median and Mean of given data: 20, 14, 6, 8, 9, 12, 16, 4.

Solution

Median (1 mark)

Arrange the data in ascending order: 4, 6, 8, 9, 12, 14, 16, 20.
Count the values: n=8, which is even, so the median is the average of the \frac{n}{2}=4th and the \frac{n}{2}+1=5th values.
The 4th value is 9 and the 5th value is 12.
Median.
\tilde{x}=\dfrac{9+12}{2}=\dfrac{21}{2}=10.5

Mean (2 marks)

Add all the values.
Divide the sum by the number of values.
\Sigma x=4+6+8+9+12+14+16+20=89
Mean.
\bar{x}=\dfrac{\Sigma x}{n}=\dfrac{89}{8}=11.125\approx 11.13

\text{Median}=10.5,\quad \bar{x}=11.125\approx 11.13

MCQ 172026 Annual I, Set D, MCQ 171 mark

For grouped data, Mean=

A\frac{\sum x}{f}
B\frac{\sum fx}{\sum f}Answer
C\frac{\sum f}{f}
D\frac{\bar{X}}{f}

Why

For grouped data the mean is \bar{x}=\dfrac{\Sigma fx}{\Sigma f}, where f is the frequency and x is the class mark. Each class mark is multiplied by its frequency, the products are added, and the sum is divided by the total frequency.

Option A divides by a single f instead of the total \Sigma f and uses \Sigma x without the frequencies, which is the formula for ungrouped data (\Sigma x/n) written wrongly. Options C and D are not formulas for the mean.

Q.2 (iv)2026 Annual I, Set D, Q.2 (iv)1+2 = 3 marks

Calculate Median and Mean of given data: 3, 21, 6, 18, 9, 15, 12.

Solution

Median (1 mark)

Arrange the data in ascending order: 3, 6, 9, 12, 15, 18, 21.
Count the values: n=7, which is odd, so the median is the value at position \frac{n+1}{2}.
\dfrac{n+1}{2}=\dfrac{7+1}{2}=4
The 4th value is 12, so the median is 12.

Mean (2 marks)

Add all the values.
\Sigma x=3+6+9+12+15+18+21=84
Divide by the number of values.
\bar{x}=\dfrac{\Sigma x}{n}=\dfrac{84}{7}=12

\text{Median}=12,\quad \bar{x}=12

Exam focus

Calculate the median and mean of a small data set (sort first; odd and even n). (short)
Asked in both 2026 sets as a 3-mark numerical (Set B Q.2 (iv), Set D Q.2 (iv)); the book works median and mean examples and short question 13.
Go to answer
Write the formula of the mean for ungrouped and grouped data and state what each symbol means. (mcq)
Set D MCQ 17 asked the grouped mean; book exercise MCQ 9 and 10 and short question 5 ask the symbols and formulae.
Go to answer
Find the range of a data set and explain what a small, middle or large range means. (mcq)
Set B MCQ 16 asked the range of five values; book short questions 1 and 2 ask the demerits of the range.
Go to answer
Advantages and disadvantages of the median (three points each). (long)
Not asked in 2026 but book long question 1 and the median is calculated in both 2026 sets.
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Standard deviation and how it differs from range. (short)
No paper question in 2026, but short questions 1 and 12 of the exercise ask it.
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Steps to construct a bar chart and a pie chart, and to add error bars. (long)
No paper question in 2026; book long questions 2, 4, 6, 7 and 10 are all about charts.
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Compare and contrast independent, dependent and controlled variables; why a control group matters. (long)
No paper question in 2026; book long question 8 and short question 3.
Go to answer