Class 11 Maths NCERT Solutions Chapter 13 Statistics | Boundless Maths
Chapter 13 Class 11 Maths NCERT Solutions · Unit IV · Statistics & Probability

Class 11 Maths NCERT Solutions Chapter 13: Statistics

Free, step-by-step NCERT Solutions for both exercises of this chapter — mean deviation about the mean and the median for ungrouped, discrete, and continuous data, and variance and standard deviation with both the direct and shortcut step-deviation methods — plus the Miscellaneous Exercise on recovering missing observations and correcting wrongly recorded data. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

Mean, median and mode describe where a data set is centred, but two very different-looking data sets can share the exact same mean. This chapter introduces measures of dispersion — numbers that describe how spread out or bunched-together the data actually is around that central value. Starting from the simple range, the chapter builds up to mean deviation and, finally, to variance and standard deviation — the most widely used measure of spread in statistics, and the one every later application (including probability distributions) is built on.

3Exercises (incl. Misc.)
28Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 13 — Overview

This Class 11 Maths NCERT Solutions Chapter 13 hub covers Statistics, the chapter that moves beyond measures of central tendency (mean, median, mode) into measures of dispersion — numbers that describe how scattered a data set is. The simplest of these is the range, the gap between the maximum and minimum values, but it ignores everything in between. Mean deviation improves on this by averaging the absolute distance of every single observation from a chosen central value, usually the mean or the median, calculated separately for ungrouped data, discrete frequency distributions, and continuous (grouped) frequency distributions.

Exercise 13.2 then introduces variance and standard deviation, obtained by squaring each deviation from the mean instead of taking its absolute value — a change that makes the resulting measure far more useful for further algebraic and statistical work. Both a direct calculation method and a faster shortcut (step-deviation) method are covered for grouped data. The Miscellaneous Exercise closes the chapter with reverse-engineering problems: recovering missing observations from a known mean and variance, working out how scaling every observation by a constant affects the mean and standard deviation, and correcting the mean and standard deviation when one or more observations were wrongly recorded.

How the Chapter Builds

One Idea Leads to the Next

1

Measures of Dispersion & Range

Why a mean alone doesn't tell the whole story, and the simplest measure of spread. Exercise 13.1.

→
2

Mean Deviation

About the mean and the median, for ungrouped, discrete, and continuous data. Exercise 13.1.

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3

Variance & Standard Deviation

Squaring the deviations instead of taking absolute values, plus the shortcut step-deviation method. Exercise 13.2.

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4

Applications

Missing observations, scaling effects, and correcting wrongly recorded data. Misc. Exercise.

Quick Reference

Important Formulas — Chapter 13

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Statistics.

Mean Deviation (§13.3–13.4)

Mean deviation (ungrouped)

\text{M.D.}(\bar{x})=\dfrac{1}{n}\sum|x_i-\bar{x}|

Replace x̄ with the median M to get the mean deviation about the median instead.

Mean deviation (grouped data)

\text{M.D.}(\bar{x})=\dfrac{1}{\text{N}}\sum f_i|x_i-\bar{x}|

N is the sum of all frequencies; for continuous data, xi is the mid-point of each class.

Range

\text{Range}=\text{Maximum}-\text{Minimum}

The simplest, fastest measure of spread — but it only uses two of the data points and ignores the rest.

Variance and Standard Deviation (§13.5)

Variance and standard deviation

\sigma^2=\dfrac{1}{n}\sum(x_i-\bar{x})^2\qquad\sigma=\sqrt{\sigma^2}

Squaring the deviations avoids the positive/negative cancellation problem, without needing absolute values.

Variance for grouped data

\sigma^2=\dfrac{1}{\text{N}}\sum f_i(x_i-\bar{x})^2

Same idea as ungrouped data, weighted by each value's frequency.

Shortcut (step-deviation) method

\sigma^2=\dfrac{h^2}{\text{N}^2}\Big[\text{N}\sum f_iy_i^2-\big(\sum f_iy_i\big)^2\Big]

Using an assumed mean A and yi = (xi − A)/h keeps the arithmetic small even when the data values are large.

Applications (Miscellaneous Exercise)

Recovering Σx and Σx²

\sum x_i=n\bar{x}\qquad\sum x_i^2=n(\sigma^2+\bar{x}^2)

The key first step for almost every reverse-engineering problem in this chapter — find these two sums before doing anything else.

Effect of scaling by a constant

y_i=ax_i\ \Rightarrow\ \bar{y}=a\bar{x},\ \sigma_y^2=a^2\sigma_x^2

Multiplying every observation by a scales the mean by a and the variance by a².

Effect of adding a constant

y_i=x_i+a\ \Rightarrow\ \bar{y}=\bar{x}+a,\ \sigma_y^2=\sigma_x^2

Adding the same number to every observation shifts the mean but leaves the variance completely unchanged.

Decision Guide

Which Statistics Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is askingUse thisWhy
Find mean deviation about the mean or the medianM.D. formula with absolute deviationsCompute the mean/median first, then average the absolute value of every deviation from it (§13.3–13.4).
Data is given as a continuous frequency distributionUse class mid-pointsTreat each class as concentrated at its mid-point, then proceed exactly as for discrete data (§13.4).
Find variance or standard deviationSquare the deviations from the meanσ² = mean of the squared deviations; σ is its square root (§13.5).
The data values are large or awkward to work with directlyShortcut (step-deviation) methodPick an assumed mean A near the centre of the data and work with yi = (xi − A)/h instead (§13.5).
Some observations are missing but the mean/variance is knownRecover Σx and Σx² firstUse Σx = nx̄ and Σx² = n(σ² + x̄²), then solve the resulting equations for the missing values (Misc. Exercise).
Every observation is multiplied or added to by a constantApply the scaling/shifting ruleMultiplying scales the mean by the same factor and variance by its square; adding shifts the mean but not the variance (Misc. Exercise).
An observation was recorded wrongly and needs correctingAdjust Σx and Σx², then recomputeRemove the wrong value (and add the correct one, if replacing) from both sums before finding the new mean and SD (Misc. Exercise).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across both exercises.

  • Mixing up mean deviation about the mean and about the median — the question specifies which central value to use; substituting the wrong one gives a completely different (and wrong) numerical answer.
  • Getting the median formula wrong for even vs. odd n — for an odd number of observations the median is the middle one; for an even number it's the average of the two middle ones, and continuous data needs the full median-class formula.
  • Forgetting to use class mid-points for continuous data — every class in a grouped frequency table is treated as if all its observations sit exactly at the mid-point; skipping this step invalidates the whole calculation.
  • Dropping the absolute value in mean deviation — without the modulus, positive and negative deviations cancel out and the mean deviation always comes out as (close to) zero, which defeats its purpose.
  • Forgetting the h² factor in the step-deviation formula — the shortcut method's variance formula must be multiplied by h² at the end to undo the earlier division by h; skipping this gives a variance that's off by a factor of h².
  • Not converting to a continuous distribution before finding a median — some questions give class intervals like "16-20, 21-25" that must first be converted to continuous form (subtracting/adding 0.5) before the median formula can be applied correctly.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 13 — Choose an Exercise

13.1

Exercise 13.1

Mean Deviation — about the mean and the median, for ungrouped, discrete, and continuous data · 12 questions

Solve Exercise 13.1 →
13.2

Exercise 13.2

Variance and Standard Deviation — direct and shortcut step-deviation methods · 10 questions

Solve Exercise 13.2 →
M

Miscellaneous Exercise

Missing observations, scaling effects, and correcting wrongly recorded data · 6 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Statistics — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 13, Statistics.

How many exercises are there in Chapter 13, Statistics?
There are two main exercises — 13.1 (Mean Deviation, 12 questions) and 13.2 (Variance and Standard Deviation, 10 questions) — plus a Miscellaneous Exercise of 6 questions, totalling 28 questions.
What is mean deviation?
Mean deviation about a value 'a' is the average of the absolute deviations of every observation from 'a'. It is most commonly calculated about the mean or the median, and it measures how spread out a data set is — unlike the range, it takes every single observation into account, not just the extremes.
What is the difference between variance and standard deviation?
Variance is the mean of the squared deviations of the observations from their mean, denoted σ². Standard deviation, denoted σ, is simply the positive square root of the variance. Standard deviation is preferred as the practical measure of spread because it is expressed in the same units as the original data, whereas variance is in squared units.
Where can I find the official NCERT textbook for this chapter?
Statistics is Chapter 13 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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