MathsICSE Class 8 · Data Handling

Grouping Data, Class Intervals and Bar Graphs (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

To group data, choose equal class intervals (such as $0$-$10$, $10$-$20$) in which the upper limit belongs to the next class, count observations with tally marks, and form a frequency table. A bar graph shows frequencies by bars of equal width on a uniform scale. The class mark is the midpoint of an interval.

Tally marks become bar heights; the tallest bar is the modal class, 20 goes into 20-30, and the class mark 15 is the midpoint of 10-20.

Illustration drawn for QuestionBuddy.

Understand Grouping Data and Bar Graphs

The CISCE curriculum asks you to arrange ungrouped data into groups, represent grouped data by bar graphs, and construct and interpret bar graphs. It also says mean, median and mode are applied to data with many observations that may need grouping.

Class intervals. Choose a class size (width) such as $10$ so that the range is covered by 4 to 8 classes. Use a consistent rule: the upper limit is not included in the class, so $20$ is counted in the class $20$ - $30$ .

Tally and frequency. Go through the data once, put a tally mark in the right class for each value, and count tallies in groups of five. The frequencies must add up to the total number of observations, which is a quick check.

Class mark. Class mark $= \frac{lower limit + upper limit}{2}$ . For $10$ - $20$ it is $15$ .

Mean of grouped data (extension). Assume each observation in a class equals its class mark: $\overset{x}{ˉ} = \frac{\sum f x}{\sum f}$ . It is an estimate because the exact values are lost. The modal class is the class with the highest frequency.

Bar graphs. Draw equal-width bars with equal gaps along a baseline, choose a scale that fits the largest value (for example $1$ cm for $5$ students), label both axes and give the graph a title. The height of each bar is $\frac{value}{scale}$ .

Reading a bar graph. Compare heights, find differences, totals and percentages ( $\frac{part}{total} \times 100$ ).

Misleading graphs. If the vertical scale does not start at $0$ , small differences look huge. Always read the scale before comparing bars.

Grouping Data and Bar Graphs: formulas to remember

Class mark: $\frac{lower limit + upper limit}{2}$; Midpoint of a class.
Grouped mean: $\overset{x}{ˉ} = \frac{\sum f x}{\sum f}$; x is the class mark.
Bar height: $height = \frac{value}{scale}$; e.g. 1 cm for 5 units.

How to solve Grouping Data and Bar Graphs questions

Fix the class size, tally every value once, check the total, then draw or read the graph.

1Find the smallest and largest values and choose a class size.
2Write the class intervals with the upper limit excluded.
3Tally the data and write frequencies; check $\sum f = n$ .
4Choose a uniform scale and draw equal-width bars.
5Label axes and title; read values using the scale.

Grouping Data and Bar Graphs in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Exam scores in groups

Suppose 20 students scored marks grouped as 0-10 (4 students), 10-20 (6), 20-30 (7) and 30-40 (3).

Estimate the mean mark using class marks.

Try it first, then show the solutionSolution

Class marks: $5, 15, 25, 35$ .
$\sum f x = 20 + 90 + 175 + 105 = 390$ ; $\sum f = 20$ ; mean $= 19.5$ .

Answer: About 19.5 marks.

Memory hook: Replace each class by its midpoint, then take the weighted average.

Tricks to remember Grouping Data and Bar Graphs

Short memory aids. Say each one aloud once, then use it on a question the same day.

Upper limit goes up
A value equal to the upper limit belongs to the next class.
Use it when: Tally accuracy.
Frequencies add to n
If your frequencies do not total the number of values, you missed or double-counted something.
Use it when: Checking a table.

Grouping Data and Bar Graphs: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Class size · 1 mark · MCQBoard level

In the class interval 10-20, the class size is:
(a) 5 (b) 10 (c) 15 (d) 20

Try it first, then show the solutionSolution

Class size $= 20 - 10 = 10$ .

Answer: (b)

Example 2: Making a frequency table · 3 marks · Short answerBoard level

The marks of 20 students are: 7, 12, 25, 18, 33, 9, 14, 27, 21, 38, 15, 5, 29, 11, 36, 24, 17, 31, 8, 22. Form a frequency table with classes 0-10, 10-20, 20-30, 30-40.

Try it first, then show the solutionSolution

0-10: 7, 9, 5, 8, so 4.
10-20: 12, 18, 14, 15, 11, 17, so 6.
20-30: 25, 27, 21, 29, 24, 22, so 6. 30-40: 33, 38, 36, 31, so 4.
Total $4 + 6 + 6 + 4 = 20$ .

Answer: Frequencies 4, 6, 6, 4

Example 3: Mean from a grouped table · 3 marks · Short answerHarder board level

The classes 0-10, 10-20, 20-30, 30-40 have frequencies 4, 6, 7, 3. Find the modal class and the estimated mean.

Try it first, then show the solutionSolution

Highest frequency $7$ , so modal class is 20-30.
Class marks $5, 15, 25, 35$ : $\sum f x = 20 + 90 + 175 + 105 = 390$ .
$\sum f = 20$ , mean $= \frac{390}{20} = 19.5$ .

Answer: Modal class 20-30; mean 19.5

Example 4: Reading a bar graph · 4 marks · Short answerHarder board level

A survey of 120 students gives favourite sports: Cricket 45, Football 30, Hockey 15, Tennis 10, Badminton 20. A bar graph uses a scale of 1 cm for 5 students. Find the height of each bar and the percentage who chose cricket.

Try it first, then show the solutionSolution

Heights: $9, 6, 3, 2, 4$ cm.
Check total: $45 + 30 + 15 + 10 + 20 = 120$ .
Cricket $= \frac{45}{120} \times 100 = 37.5%$ .

Answer: Heights 9, 6, 3, 2, 4 cm; cricket 37.5%

Example 5: Double bar graph · 4 marks · Case-basedHOTS / competency

Marks of Section A: Maths 28, Science 24, English 20, Hindi 16. Section B: Maths 21, Science 30, English 18, Hindi 20 (a double bar graph). Find (i) the total for each section, (ii) the subject with the largest difference between sections, (iii) the best subject overall.

Try it first, then show the solutionSolution

(i) A: $28 + 24 + 20 + 16 = 88$ ; B: $21 + 30 + 18 + 20 = 89$ .
(ii) Differences: Maths $7$ , Science $6$ , English $2$ , Hindi $4$ ; largest is Maths.
(iii) Combined: Maths $49$ , Science $54$ , English $38$ , Hindi $36$ ; best is Science. Total $177 = 88 + 89$ .

Answer: 88 and 89; Maths (7); Science (54)

Where students lose marks in Grouping Data and Bar Graphs

The slips examiners see most often, and what to write instead.

Mistake: Putting the upper limit value in the same class
The upper limit belongs to the next class.
Wrong working
20 is counted in 10-20.
Right working
20 is counted in 20-30.
Do this: Write the rule on top of the table.
Usually costs: Frequencies and mean.
Mistake: Drawing bars of unequal width or an unlabelled scale
Bar graphs need uniform bars and a stated scale.
Wrong working
Bar for cricket drawn wider and no scale written.
Right working
Equal widths and "Scale: 1 cm = 5 students".
Do this: Write the scale on the graph.
Usually costs: Graph marks.

Exam tips

Check that frequencies add up to the number of observations.
Write the scale on the axis.
Use class marks for the mean of grouped data.

Practice questions on Grouping Data and Bar Graphs

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The number of observations in a class is called its:
(a) frequency (b) class mark (c) range (d) class size
Hint
What does the tally count?
Answer
(a)
Q2 · 2 marks · Short answerBoard level
Find the class mark and the class size of the interval 40-50.
Hint
Midpoint and width.
Answer
Class mark $= \frac{40 + 50}{2} = 45$ ; class size $= 10$ .
Q3 · 3 marks · Short answerHarder board level
Form a frequency table for 3, 8, 12, 5, 17, 9, 14, 20, 11, 6, 15, 2, 19, 10, 13 using classes 0-5, 5-10, 10-15, 15-20, 20-25.
Hint
Remember the upper limit rule.
Answer
0-5: 3, 2 gives 2. 5-10: 8, 5, 9, 6 gives 4. 10-15: 12, 14, 11, 10, 13 gives 5. 15-20: 17, 15, 19 gives 3. 20-25: 20 gives 1. Total $15$ .
Q4 · 3 marks · Short answerHarder board level
The classes 0-20, 20-40, 40-60, 60-80 have frequencies 5, 10, 10, 5. Estimate the mean.
Hint
Class marks 10, 30, 50, 70.
Answer
$\sum f x = 50 + 300 + 500 + 350 = 1200$ ; $\sum f = 30$ ; mean $= 40$ .
Q5 · 4 marks · Long answerHOTS / competency
A bar graph of a company's sales for three years shows bars of height 2 cm, 5 cm and 8 cm, but the vertical axis starts at 50 (sales 52, 55 and 58 units). Explain why the graph is misleading and find the true percentage rise from the first to the third year.
Hint
Compare the bar ratio with the true ratio.
Answer
Bars suggest the third year is $\frac{8}{2} = 4$ times the first. True sales: $58 \div 52 \approx 1.12$ . True rise $= \frac{58 - 52}{52} \times 100 \approx 11.5%$ . The truncated axis exaggerates the difference.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Class mark of 20-30?Show answer
25.
2.Which class does the value 20 belong to in 10-20, 20-30?Show answer
20-30.
3.Formula for grouped mean?Show answer
Sum of f x divided by sum of f.
4.Why can a bar graph mislead?Show answer
If the axis does not start at 0.

Grouping Data and Bar Graphs: frequently asked questions

Why is the grouped mean only an estimate?

Because each class is replaced by its midpoint.

How many classes should I use?

Usually 4 to 8, with equal width.