CDS Mathematics · Statistics

Mean, Median and Mode of Grouped Data

For class-interval data the mean uses class mid-points, the median interpolates inside the class holding the N/2th value, and the mode interpolates inside the class with the largest frequency.

All Statistics notes CDS Mathematics strategy

Why this matters

Nineteen PYQs, the most of any page, usually two or three items on one table. Three formulas and one habit do it: find any missing frequency first, convert inclusive classes to boundaries, then apply the formula for the measure asked.

Concept 1 of 3: Mean of grouped data and missing frequencies

Inside a class we do not know the individual values, so each one is represented by the class mid-point. The mean is then a frequency-weighted mean of the mid-points.

Definition

Mid-point of a class = $\dfrac{\text{lower} + \text{upper}}{2}$ .
$\bar x = \dfrac{\sum f m}{\sum f}$ .
A missing frequency: set the formula equal to the stated mean and solve (with the stated total if there are two unknowns).
'Less than' or 'more than' tables: first turn them into class frequencies.

Grouped mean

\bar x = \dfrac{\sum f m}{\sum f}, \quad m = \text{class mid-point}

Worked example

Classes

0

–

10

10

–

20

20

–

30

30

–

40

have frequencies

3, 5, f, 2

, and the mean is

19

. Find

f

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q53Moderate

Example 1 · Statistics · Mean, Median and Mode of Grouped Data

Consider the following distribution :

Class	Frequency
0 - 20	17
20 - 40	28
40 - 60	32
60 - 80	f
80 - 100	19

If the mean of the above distribution is 50, what is the value of f ?

An open last class

When the last class has no upper limit, the mean is only possible by assuming a width. Take the width of the other classes; that is the reading the options are built on.

Drill 7 more on mean of grouped data and missing frequencies

Concept 2 of 3: Median of grouped data

Find the class that holds the

\dfrac N2

th value, then assume its values are spread evenly across the class. The median is the lower boundary plus the fraction of the class you need to walk through.

Definition

Median $= l + \dfrac{\frac N2 - cf}{f}\times h$ , where $l$ is the lower boundary of the median class, $cf$ the cumulative frequency BEFORE it, $f$ its frequency and $h$ its width.
Inclusive classes ( $18$ – $26$ , $27$ – $35$ ) must first become boundaries ( $17.5$ – $26.5$ , $26.5$ – $35.5$ ).
A given median gives one equation for a missing frequency.

Grouped median

\text{Median} = l + \dfrac{\tfrac N2 - cf}{f}\,h

Worked example

Classes

0

–

10

10

–

20

20

–

30

30

–

40

with frequencies

5, 8, 12, 5

. Find the median.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (II) 2024 — Elementary Mathematics · Q54Moderate

Example 2 · Statistics · Mean, Median and Mode of Grouped Data

A frequency distribution is as follows :

Marks	18-26	27-35	36-44	45-53	54-62	63-71	72-80
Number of students	5	7	10	15	8	3	2

What is the median of the distribution ?

Use boundaries, not printed limits

With inclusive classes the median class

45

–

53

starts at

44.5

. Using

45

shifts the answer by

0.5

, and that shifted value is usually among the options.

Drill 6 more on median of grouped data

Concept 3 of 3: Mode of grouped data

The mode lies in the class with the largest frequency, pulled toward whichever neighbour is larger. The formula measures that pull.

Definition

Mode $= l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\times h$ , where $f_1$ is the modal class frequency, $f_0$ the one before, $f_2$ the one after.
If the neighbours are equal, the mode is the mid-point of the modal class.
Solve for any missing frequency first; it may change which class is modal.

Grouped mode

\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h

Diagram · mode = the tallest bar

The mode is the value with the highest frequency — category D here. Data can have two modes (bimodal) or none (all equal); the mode is the only average that also works for non-numeric categories.

Worked example

Classes

0

–

10

10

–

20

20

–

30

30

–

40

with frequencies

3, 9, 15, 6

. Find the mode.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q96Moderate

Example 3 · Statistics · Mean, Median and Mode of Grouped Data

Consider the following grouped data :

Class	Frequency
0 - 10	4
10 - 20	8
20 - 30	15
30 - 40	10
40 - 50	3

What is the mode of the above distribution ?

Two times f₁ in the denominator

The denominator is

(f_1 - f_0) + (f_1 - f_2) = 2f_1 - f_0 - f_2

. Writing

f_1 - f_0 - f_2

gives a negative or tiny denominator and a mode outside the class.

Drill 3 more on mode of grouped data

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 (3)

Mean of grouped data and missing frequencies
Grouped mean
$\bar x = \dfrac{\sum f m}{\sum f}, \quad m = \text{class mid-point}$
Median of grouped data
Grouped median
$\text{Median} = l + \dfrac{\tfrac N2 - cf}{f}\,h$
Mode of grouped data
Grouped mode
$\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h$

Watch out for (3)

An open last class
→ Mean of grouped data and missing frequencies
Use boundaries, not printed limits
→ Median of grouped data
Two times f₁ in the denominator
→ Mode of grouped data

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