Class 10 Maths Chapter 13: Statistics

Statistics is Chapter 13 of NCERT Class 10 Maths. It teaches you how to find the three measures of central tendency, mean, mode and median, for grouped data arranged in class intervals. Watch the five video lessons below, then use these notes, formulas and fully solved examples to practise the steps that carry marks in the board exam.

What you will learn

Finding the class mark of each class interval
Three ways to find the mean: direct method, assumed mean method and step deviation method
Finding the mode of grouped data using the modal class
Finding the median of grouped data using cumulative frequency
The empirical relationship between mean, median and mode

Video Lessons

Statistics - Mean of Grouped Data

Statistics - Mode of Grouped Data

Statistics - Median of Grouped Data

Statistics - Revision Part 1

Statistics - Revision Part 2

Key Concepts and Formulas

Class Mark and Grouped Data

In grouped data, individual values are not known, so each class is represented by its middle value. We assume all observations in a class are centred at this point.

Class mark xᵢ = (lower limit + upper limit) ÷ 2
Class size h = upper limit − lower limit
n = Σfᵢ

If classes are not continuous (like 10-19, 20-29), make them continuous by adjusting the limits by half the gap before finding median or mode.

Mean of Grouped Data

All three methods give the same answer. Choose the one that makes the arithmetic easiest.

Direct:        x̄ = Σfᵢxᵢ ÷ Σfᵢ
Assumed mean:  dᵢ = xᵢ − a
               x̄ = a + Σfᵢdᵢ ÷ Σfᵢ
Step deviation: uᵢ = (xᵢ − a) ÷ h
               x̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ)
where a = assumed mean (a central class mark)

Step deviation is best when all class sizes are equal and the numbers are large.

Mode of Grouped Data

The mode is the value around which observations cluster the most. In grouped data, the class with the highest frequency is called the modal class.

Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h
where l = lower limit of modal class
f₁ = frequency of modal class
f₀ = frequency of class before it
f₂ = frequency of class after it
h = class size

Frequency table highlighting the modal class and f₀, f₁, f₂ for finding the mode of grouped data, Class 10 Maths Statistics — Modal class and its neighbours give the mode

Median of Grouped Data

The median is the middle value of the data. To locate it you need the cumulative frequency (cf), which is a running total of frequencies. Find n/2 and pick the class whose cumulative frequency is just greater than n/2; this is the median class.

Median = l + [(n/2 − cf) ÷ f] × h
where l = lower limit of median class
cf = cumulative frequency of class before it
f = frequency of median class
h = class size

Cumulative frequency table locating the median class for finding the median of grouped data, Class 10 Maths Chapter 13 — Locate the median class using n/2 and cf

Empirical Relationship

For a moderately asymmetrical distribution, the three measures are approximately linked. Use it to find one measure when the other two are given.

3 Median = Mode + 2 Mean

Solved Examples

Example 1: Classes 0-10, 10-20, 20-30, 30-40, 40-50 have frequencies 5, 8, 15, 12, 10. Find the mean by the direct method.

n = 5 + 8 + 15 + 12 + 10 = 50
Class marks x = 5, 15, 25, 35, 45
fx = 25, 120, 375, 420, 450
Σfx = 1390
Mean = 1390 ÷ 50
∴ Mean = 27.8

Example 2: Find the mean of the same data by the step deviation method.

a = 25, h = 10
u = −2, −1, 0, 1, 2
fu = −10, −8, 0, 12, 20
Σfu = 14
Mean = 25 + 10 × (14 ÷ 50)
= 25 + 2.8
∴ Mean = 27.8 (same as direct method)

Example 3: Find the mode of the same data.

Modal class = 20-30 (highest f = 15)
l = 20, f₁ = 15, f₀ = 8, f₂ = 12, h = 10
Mode = 20 + [(15 − 8) ÷ (30 − 8 − 12)] × 10
= 20 + (7 ÷ 10) × 10
∴ Mode = 27

Example 4: Find the median of the same data.

cf = 5, 13, 28, 40, 50
n/2 = 25, so median class = 20-30
l = 20, cf = 13, f = 15, h = 10
Median = 20 + [(25 − 13) ÷ 15] × 10
= 20 + 8
∴ Median = 28

Example 5: For a distribution, the mean is 24 and the median is 26. Find the mode.

3 Median = Mode + 2 Mean
Mode = 3 × 26 − 2 × 24
= 78 − 48
∴ Mode = 30

Example 6: Classes 0-20, 20-40, 40-60, 60-80 have frequencies 4, p, 10, 6 and the mean is 45. Find p.

Class marks = 10, 30, 50, 70
Σfx = 40 + 30p + 500 + 420 = 960 + 30p
n = 20 + p
(960 + 30p) ÷ (20 + p) = 45
960 + 30p = 900 + 45p
15p = 60
∴ p = 4

Common Mistakes to Avoid

Taking the frequency of the class before or after the modal class from the wrong side. f₀ is always the class just before, f₂ the class just after.
In the median formula, using the cumulative frequency of the median class itself instead of the class before it.
Choosing the modal class by looking at class marks or cumulative frequencies. The modal class is the one with the highest ordinary frequency.
Skipping the table. Always draw a neat table with columns for xᵢ, fᵢ, fᵢxᵢ (or uᵢ, fᵢuᵢ) and cf, since CBSE marking gives credit for each step.

Quick Revision

Class mark = (lower + upper) ÷ 2
x̄ = a + h × (Σfu ÷ Σf)
Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h
Median = l + [(n/2 − cf) ÷ f] × h
3 Median = Mode + 2 Mean