JEE Mains Maths · Statistics

Frequency Distributions

Mean, variance and median of data given as a frequency table, with single values or classes, including an unknown frequency and the mean deviation of a table.

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Why this matters

Seventeen PYQs, eleven of them multiple choice, and four from 2026. Twelve find the mean or variance of a table, or an unknown frequency in it — two of these are probability distributions read the same way; five find the median of a table or its mean deviation. Two ideas cover the page.

Concept 1 of 2: Mean and variance of a frequency table

A frequency table is a list with repeats. The total

N=\sum f_i

, and the sums

\sum f_ix_i

and

\sum f_ix_i^2

take the place of

\sum x

and

\sum x^2

. For classes, each class is read as its midpoint. An unknown frequency sits in

N

as well as in the sums, so the equation for it is often a quadratic. A probability distribution is the same table with

N=1

Definition

$N=\sum f_i$ , $\bar x=\frac{1}{N}\sum f_ix_i$ .
$\sigma^2=\frac{1}{N}\sum f_ix_i^2-\bar x^2$ .
Classes: $x_i$ is the class mark (midpoint).
Probabilities: $\mu=\sum p_ix_i$ , $\sigma^2=\sum p_ix_i^2-\mu^2$ .
Shifting to $d_i=x_i-a$ keeps the numbers small and the variance the same.

Frequency table

\bar x=\frac{\sum f_ix_i}{\sum f_i},\qquad \sigma^2=\frac{\sum f_ix_i^2}{\sum f_i}-\bar x^2

Worked example

Find the variance of the values

1,2,3,4

with frequencies

2,3,4,1

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 2 Apr 2026 Shift 1 · Q58Moderate

Example 1 · Statistics · Frequency Distributions

If the mean of the data

Class	Frequency
5-10	2
10-15	$k$
15-20	28
20-25	54
25-30	$k + 1$
30-35	5

is 21, then

k

is one of the roots of the equation :

The unknown frequency is in N too

When a frequency is

k

, the total

N=\sum f_i

also contains

k

. Clear the fraction before solving; if the equation is a quadratic, keep only the root that is a whole, non-negative frequency.

Drill 11 more on mean and variance of a frequency table

Concept 2 of 2: Median and mean deviation of a table

For the median, run the cumulative frequencies until they reach

\frac N2

. With single values, that value is the median. With classes, the median lies inside that class, and you place it by the fraction of the class still needed. For the mean deviation, weight each distance

|x_i-A|

by its frequency.

Definition

Single values: the median is where the cumulative frequency first reaches $\frac N2$ (average the two middle values when $N$ is even and they differ).
Classes: $l$ is the lower limit of the median class, $F$ the cumulative frequency before it, $f$ its frequency, $h$ its width.
Mean deviation about $A$ : $\frac{1}{N}\sum f_i|x_i-A|$ .

Grouped median

M=l+\frac{\frac{N}{2}-F}{f}\times h

Worked example

The classes

0-10

10-20

20-30

30-40

have frequencies

5,8,4,3

. Find the median.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 5 Apr 2026 Shift 1 · Q58Moderate

Example 2 · Statistics · Frequency Distributions

The mean deviation about the mean for the data

$x_{i}$	5	7	9	10	12	15
$f_{i}$	8	6	2	2	2	6

is equal to :

F stops before the median class

In the grouped median,

F

is the cumulative frequency of the classes before the median class, not up to and including it. Using the larger figure puts the median too low.

Drill 4 more on median and mean deviation of a table

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Mean and variance of a frequency table
Frequency table
$\bar x=\frac{\sum f_ix_i}{\sum f_i},\qquad \sigma^2=\frac{\sum f_ix_i^2}{\sum f_i}-\bar x^2$
Median and mean deviation of a table
Grouped median
$M=l+\frac{\frac{N}{2}-F}{f}\times h$

Watch out for (2)

The unknown frequency is in N too
→ Mean and variance of a frequency table
F stops before the median class
→ Median and mean deviation of a table

Test yourself on Statistics

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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