MathsICSE Class 10 · Measures of Central Tendency

Mode and Modal Class: Raw, Arrayed and Grouped Data (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

The mode is the value that occurs most often. In raw data, count each value; in a frequency table, it is the $x$ with the highest frequency. For grouped data, the modal class is the class with the highest frequency. Data can have one mode, more than one, or none.

The modal class is the class with the highest frequency: here 21–30 with 12. Name the class, not the frequency.

Illustration drawn for QuestionBuddy.

Understand Mode and Modal Class

The CISCE 2027 syllabus asks for the mode of raw and arrayed data, and the modal class of grouped data (continuous and discontinuous). Reading the mode itself from a histogram is in the next chapter.

Raw data. Count how many times each value occurs (a tally helps). The value with the greatest count is the mode.

Unimodal: one value occurs most often.
Bimodal (or multimodal): two or more values share the greatest count; give all of them.
No mode: every value occurs the same number of times (for example, each once).

Arrayed data. In a frequency table of single values, the mode is the $x$ with the highest frequency $f$ . The answer is the $x$ , never the frequency itself.

Modal class. For grouped data, the modal class is the class with the highest frequency. The exact mode of grouped data needs a histogram (next chapter); here you only name the class.

Discontinuous classes. For classes like 21–30, 31–40, …, you can name the modal class directly as 31–40. If a question asks for true class limits (boundaries), they are 30.5–40.5: subtract 0.5 from each lower limit and add 0.5 to each upper limit.

Unknowns. If the mode is given, ask what value of the unknown makes that value the most frequent, and check that no other value ties with it. If a frequency contains an unknown, write the inequality 'this frequency is greater than every other'.

Mode, median and mean together. The modal class and the median class need not be the same class. Find the modal class from the frequencies, and the median class from the cumulative frequencies ( $\frac{N}{2}$ th observation).

When the mode is the right average. The mode is the only average that works for categories (sizes, colours, brands) and is the best choice for 'most popular' questions.

Mode and Modal Class: formulas to remember

Mode of raw data: $mode = value with the greatest count$; May be more than one value, or none.
Mode of arrayed data: $mode = x with the highest f$; Answer is the $x$ value.
Modal class: $class with the highest frequency$; Grouped data, continuous or discontinuous.
Class boundaries (discontinuous): $21 - 30 \to 20.5 - 30.5$; Half the gap between classes on each side.

How to solve Mode and Modal Class questions

Count, compare, then answer with the value (or class), not the count.

1Raw data: tally each value or arrange the data in order so equal values sit together.
2Find the greatest count and list every value with that count.
3Arrayed data: scan the $f$ column for the largest frequency and write the matching $x$ .
4Grouped data: the class with the largest frequency is the modal class; write it with its limits.
5With an unknown, test which value makes the given mode unique, and check for ties.
6If the median class is also asked, build the c.f. column separately.

Mode and Modal Class 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.

Planning stock with the mode

Suppose a café owner notes how many cups of tea each of 10 customers orders: 1, 2, 1, 1, 3, 2, 1, 4, 2, 1.

What is the most common order, and why is this average useful to the owner?

Try it first, then show the solutionSolution

Counts: 1 cup (5 times), 2 cups (3), 3 cups (1), 4 cups (1).
Mode $= 1$ cup.

Answer: 1 cup; it tells the owner the order size to prepare for most often

Memory hook: Most popular = mode.

Modal class of waiting times

Imagine a school canteen records the waiting time of 40 students in the lunch queue: 0–4 minutes (9 students), 4–8 (16), 8–12 (11), 12–16 (4).

Which waiting-time class is most common?

Try it first, then show the solutionSolution

The highest frequency is 16.
Modal class: 4–8 minutes.

Answer: 4–8 minutes

Memory hook: Tallest frequency, modal class.

Tricks to remember Mode and Modal Class

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

MOde = MOst
Both start with 'mo': the mode is the value that occurs most.
Use it when: Keeping mode apart from median and mean.
Answer with x, not f
The frequency tells you which one; the answer is the value beside it.
Use it when: Arrayed data and modal class.
Check for a tie
Before writing one mode, look again for an equal count elsewhere.
Use it when: Bimodal data and unknown-value questions.

Mode and Modal Class: solved examples

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

Example 1: Mode of raw data · 1 mark · MCQBoard level

The mode of 6, 9, 4, 9, 6, 9, 2, 6, 9 is:
(a) 4
(b) 6
(c) 9
(d) 6 and 9

Try it first, then show the solutionSolution

9 occurs 4 times, 6 occurs 3 times, the rest once.
Mode $= 9$ .

Answer: (c)

Example 2: Bimodal data · 1 mark · Short answerBoard level

Find the mode of 12, 15, 12, 18, 15, 20, 11.

Try it first, then show the solutionSolution

12 occurs twice and 15 occurs twice; every other value once.
Two values share the greatest count, so the data are bimodal.

Answer: 12 and 15

Example 3: Mode, median and mean of arrayed data · 3 marks · Short answerHarder board level

The number of children in 50 families is shown. Find the mode, the median and the mean.

Children	0	1	2	3	4	5
Families	5	12	17	9	4	3

Try it first, then show the solutionSolution

Mode: the highest frequency is 17, for 2 children. Mode $= 2$ .
Median: $N = 50$ ; c.f. 5, 17, 34, …; the 25th and 26th values are both 2. Median $= 2$ .
Mean: $Σ f x = 0 + 12 + 34 + 27 + 16 + 15 = 104$ ; mean $= \frac{104}{50} = 2.08$ .

Answer: Mode 2, median 2, mean 2.08

Example 4: Unknown value from the mode · 3 marks · Short answerHarder board level

The data 4, 6, 8, $x$ , 6, 8, 10, $x$ , 12 have a single mode, 8. Find $x$ and the median.

Try it first, then show the solutionSolution

Without the $x$ 's, 6 and 8 each occur twice.
If $x = 8$ , 8 occurs four times: a single mode 8 ✓. If $x = 6$ , the mode is 6. If $x$ is 4, 10 or 12, that value occurs three times and becomes the only mode. Any other new $x$ occurs twice and ties with 6 and 8, so there is no single mode.
So $x = 8$ . In order: 4, 6, 6, 8, 8, 8, 8, 10, 12; median $=$ 5th value $= 8$ .

Answer: $x = 8$ ; median 8

Example 5: Modal class and median class differ · 3 marks · Short answerHarder board level

For the distribution below, find the modal class and the median class.

Marks	21–30	31–40	41–50	51–60	61–70
Students	8	15	11	16	5

Try it first, then show the solutionSolution

Highest frequency 16: modal class 51–60 (boundaries 50.5–60.5).
$N = 55$ , $\frac{N}{2} = 27.5$ . c.f.: 8, 23, 34, 50, 55.
The first c.f. $\geq 27.5$ is 34: median class 41–50.

Answer: Modal class 51–60; median class 41–50

Example 6: Case study: a missing frequency · 5 marks · Case-basedHOTS / competency

A survey records the daily pocket money (₹) of some students.

₹	0–10	10–20	20–30	30–40	40–50
Students	3	8	$p$	10	5

(a) For which values of $p$ is 20–30 the only modal class?
(b) The mean is ₹26. Find $p$ .
(c) For this $p$ , find the median class.

Try it first, then show the solutionSolution

(a) $p$ must exceed every other frequency, the largest of which is 10: $p > 10$ , i.e. $p \geq 11$ .
(b) Class marks 5, 15, 25, 35, 45: $Σ f x = 15 + 120 + 25 p + 350 + 225 = 710 + 25 p$ , $Σ f = 26 + p$ . $710 + 25 p = 26 (26 + p) = 676 + 26 p$ , so $p = 34$ (and $34 > 10$ , consistent with (a)).
(c) $N = 60$ , $\frac{N}{2} = 30$ ; c.f. 3, 11, 45, …; median class 20–30.

Answer: (a) $p \geq 11$ (b) $p = 34$ (c) 20–30

Where students lose marks in Mode and Modal Class

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

Mistake: Giving the frequency as the mode
The mode is a data value; the frequency only tells you which one.
Wrong working
Mode $= 17$ (the highest frequency)
Right working
Mode $= 2$ (the $x$ with frequency 17)
Do this: Circle the highest $f$ , then write the $x$ above it.
Usually costs: The answer mark.
Mistake: Missing a second mode
When two values tie for the greatest count, both are modes.
Wrong working
Mode of 12, 15, 12, 18, 15, 20, 11 is 12
Right working
Modes 12 and 15
Do this: After finding the greatest count, scan the whole list again.
Usually costs: 1 mark in a short answer.
Mistake: Mixing up the modal class and the median class
One comes from the highest frequency, the other from the c.f. at $\frac{N}{2}$ .
Wrong working
Median class = class with the highest frequency
Right working
Median class = first class with c.f. $\geq \frac{N}{2}$
Do this: Do them in separate lines with their own reasons.
Usually costs: The median-class mark.
Mistake: Grouping discrete data before finding the mode
Grouping a discrete table into classes loses the exact values.
Wrong working
Turning $x$ = 10, 11, …, 16 into classes 10–12, 12–14, …
Right working
Read the mode straight from the $x$ and $f$ columns
Do this: Use classes only when the question gives classes.
Usually costs: The CISCE examiners' comments on 2025 Q8(ii) report this error for the mean of a discrete table; it costs marks here in the same way.

Exam tips

Section A 1-mark items may ask for the mode of a short list or the modal class of a table; count carefully and check for ties.
When both the modal class and the median class are asked, give a reason for each: 'highest frequency 16' and 'c.f. first reaches 27.5'.
If the question asks for the mode of grouped data as a number, you need a histogram (next chapter); otherwise give the modal class.
For discontinuous classes, name the class as printed unless boundaries are asked for.

Practice questions on Mode and Modal Class

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 modal class of the following distribution is:
Class 0–5 5–10 10–15 15–20
Frequency 4 11 7 3
(a) 0–5
(b) 5–10
(c) 10–15
(d) 15–20
Hint
Highest frequency.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
Find the mode and the median.
$x$ 10 11 12 13 14
$f$ 3 7 12 8 5
Hint
$N = 35$ : the median is the 18th value.
Answer
Mode 12 (frequency 12). c.f. 3, 10, 22, …: the 18th value is 12, so median $= 12$ .
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): A set of data can have more than one mode.
Reason (R): Two or more values may share the highest frequency.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Think of bimodal data.
Answer
(c)
Q4 · 3 marks · Short answerHOTS / competency
The mean of 3, 5, $x$ , 7, 5, 9, $x$ , 11 is 6.5. Find $x$ and the mode of the data.
Hint
Sum $= 40 + 2 x$ .
Answer
$40 + 2 x = 52$ , so $x = 6$ . Data: 3, 5, 6, 7, 5, 9, 6, 11; 5 and 6 each occur twice, so the modes are 5 and 6.
Q5 · 2 marks · Short answerHarder board level
Find the modal class and the median class.
Class 1–10 11–20 21–30 31–40 41–50
Frequency 9 14 12 18 7
Hint
$\frac{N}{2} = 30$ .
Answer
Modal class 31–40. c.f. 9, 23, 35, …: median class 21–30.
Q6 · 3 marks · Short answerHarder board level
For the distribution below, find the mode, the median and the mean (to 2 decimal places).
$x$ 1 2 3 4 5 6
$f$ 2 4 5 9 6 4
Hint
$N = 30$ .
Answer
Mode 4. c.f. 2, 6, 11, 20, …: 15th and 16th values are 4, median 4. $Σ f x = 115$ , mean $= \frac{115}{30} \approx 3.83$ .
Q7 · 3 marks · Short answerHOTS / competency
In the table below, the mode is 3 and it is the only mode. Find the least whole-number value of $k$ .
$x$ 1 2 3 4 5
$f$ 4 7 $2 k$ 6 $k + 3$
Hint
$2 k$ must beat 7, 6 and $k + 3$ .
Answer
$2 k > 7$ gives $k > 3.5$ ; $2 k > k + 3$ gives $k > 3$ . So $k \geq 4$ . Check $k = 4$ : frequencies 4, 7, 8, 6, 7, and 8 is the single largest ✓. Least $k = 4$ .

Class	0–5	5–10	10–15	15–20
Frequency	4	11	7	3

$x$	10	11	12	13	14
$f$	3	7	12	8	5

Class	1–10	11–20	21–30	31–40	41–50
Frequency	9	14	12	18	7

$x$	1	2	3	4	5	6
$f$	2	4	5	9	6	4

$x$	1	2	3	4	5
$f$	4	7	$2 k$	6	$k + 3$

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.Mode of 2, 3, 3, 5, 5, 7?Show answer
3 and 5 (bimodal)
2.Mode of a frequency table?Show answer
The $x$ with the highest frequency.
3.Modal class?Show answer
The class with the highest frequency.
4.Boundaries of the class 31–40?Show answer
30.5–40.5
5.Can data have no mode?Show answer
Yes, if every value occurs equally often.

Mode and Modal Class: frequently asked questions

What is the mode in statistics?

The mode is the value that occurs most often in a set of data. A data set may have one mode, more than one mode, or no mode.

What is the modal class?

In grouped data, the modal class is the class with the highest frequency. The ICSE Class 10 syllabus asks you to identify it from the table.

Is the modal class always the same as the median class?

No. The modal class has the highest frequency; the median class contains the (N/2)th observation. They can be different classes.