Chapter 13 of 14

93%

CBSE Class 11 Mathematics★ 6-8 marks · CBSE Class 11 Mathematics Chapter 13

Statistics (Mean Deviation, Variance, Standard Deviation)

Data has a STORY — and statistics gives you the tools to tell it. This chapter covers measures of dispersion for DATA: range, mean deviation, variance, and standard deviation for both ungrouped and grouped data. CBSE Class 11 Mathematics Chapter 13.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

Ask AI about this

🔤Build your vocabulary from this chapterLexiGenome mines the key words, explains them in your language, and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Compute mean deviation about the mean and about the median for ungrouped data
2Calculate variance and standard deviation for ungrouped and grouped (discrete and continuous) data
3Use the computational formula σ² = Σxᵢ²/n − (x̄)² to find variance efficiently
4Compute the Coefficient of Variation (C.V.) and use it to compare two datasets
5Determine which dataset is more consistent given their means and standard deviations

💡

Why this chapter matters

Measures of dispersion are central to data analysis in mathematics, economics, and science. Variance and standard deviation directly extend to normal distributions in Class 12 statistics and are essential tools for understanding data consistency and reliability in real-world applications.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

🔗

Statistics (Class 9-10)

Mean, median, and mode from earlier classes are required as the central values around which dispersion is measured

Statistics

"The mean tells you WHERE. Variance tells you how FAR. Together, they describe EVERYTHING."

1. Chapter Overview

This is the DATA side of statistics (complementing the Economics Statistics chapters). Given a dataset, how SPREAD OUT is it? This chapter covers: range, mean deviation about mean and median, variance, standard deviation, and coefficient of variation — for both ungrouped and grouped (continuous) data.

2. Measures of Dispersion

Why Dispersion?

Two datasets can have the SAME MEAN but completely different SPREAD
Dispersion measures quantify VARIABILITY, CONSISTENCY, RELIABILITY

Measures Covered

Measure	Formula (Ungrouped Data)
Range	Max — Min
Mean Deviation about mean	Σ
Mean Deviation about median	Σ
Variance (σ²)	Σ(xᵢ — x̄)² / n
Standard Deviation (σ)	√[Σ(xᵢ — x̄)² / n]

3. Variance and Standard Deviation

Formulas

Ungrouped Data

Variance (σ²) = Σ(xᵢ — x̄)² / n
Standard Deviation (σ) = √(Variance)

Alternative (Computational) Formula

σ² = Σxᵢ²/n — (Σxᵢ/n)² = (mean of squares) — (square of mean)

Grouped (Discrete) Data

σ² = Σfᵢ(xᵢ — x̄)² / Σfᵢ

Grouped (Continuous) Data

Step deviation method using assumed mean and class interval — simplifies calculations with large numbers.

4. Coefficient of Variation (C.V.)

C.V. = (σ / x̄) × 100
A UNIT-FREE measure of relative dispersion
Allows COMPARISON of variability across datasets with different means or units
Lower C.V. = greater consistency (less variability)

5. Comparing Two Datasets

Equal means → compare STANDARD DEVIATIONS (lower = more consistent)
Different means → compare COEFFICIENTS OF VARIATION (C.V.)

6. Exam Focus

Mean deviation about mean and median — for ungrouped data
Variance and standard deviation — for ungrouped AND grouped (discrete and continuous)
Step deviation method for continuous grouped data
Coefficient of variation — formula, when used, interpretation
Comparing datasets using C.V.

7. Key Formulas

Formula	Ungrouped Data
Mean Deviation (about mean)	Σ
Variance (σ²)	Σ(xᵢ — x̄)² / n
Standard Deviation (σ)	√(σ²)
Coefficient of Variation	(σ / x̄) × 100

8. Conclusion

Statistics transforms raw numbers into MEANING:

MEAN DEVIATION: How far, on average, are the values from the centre?
VARIANCE and STANDARD DEVIATION: The STANDARD measures of spread. Standard deviation has the same units as the original data.
C.V.: The relative measure. For comparing apples and oranges — datasets with different means or units.

'Without data, you're just another person with an opinion. With data and statistics, you're a person with evidence.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Mean Deviation about Mean

MD(x̄) = Σ|xᵢ − x̄| / n (ungrouped); MD(x̄) = Σfᵢ|xᵢ − x̄| / Σfᵢ (grouped)

Uses absolute values to prevent positive and negative deviations from cancelling

Mean Deviation about Median

MD(M) = Σ|xᵢ − M| / n (ungrouped)

Mean deviation about the median is always ≤ mean deviation about the mean

Variance (Ungrouped)

σ² = Σ(xᵢ − x̄)² / n

Variance is in squared units of the original data

Computational Formula for Variance

σ² = (Σxᵢ²/n) − (x̄)² = mean of squares − square of mean

Faster to compute when individual deviations are inconvenient to calculate

Standard Deviation

σ = √(σ²) = √[Σ(xᵢ − x̄)²/n]

Same units as original data; more interpretable than variance

Grouped Data Variance

σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ

xᵢ is the class mark (midpoint) for continuous data

Coefficient of Variation

C.V. = (σ / x̄) × 100

Unit-free; lower C.V. means more consistent (less relative variation); used to compare datasets with different means or units

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Computing Σ(xᵢ − x̄) instead of Σ(xᵢ − x̄)² for variance

✓ Σ(xᵢ − x̄) always equals 0 (positive and negative deviations cancel). Variance uses SQUARED deviations: σ² = Σ(xᵢ − x̄)²/n. For mean deviation, use ABSOLUTE values: Σ|xᵢ − x̄|/n.

WATCH OUT

✗ Dividing by (n−1) instead of n for variance

✓ In Class 11 NCERT, variance is σ² = Σ(xᵢ − x̄)²/n (divide by n, not n−1). The (n−1) denominator is used in inferential statistics (sample variance) which is beyond this syllabus.

WATCH OUT

✗ Comparing datasets using standard deviation when their means are different

✓ When two datasets have DIFFERENT means, compare using Coefficient of Variation (C.V. = σ/x̄ × 100). Standard deviation comparison is only valid when the means are equal.

WATCH OUT

✗ Using the wrong measure: range instead of standard deviation

✓ Range = Max−Min uses only two values and ignores all others. Standard deviation uses ALL data points and gives a more reliable measure of spread. Range is a quick estimate only.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 13.1

Exercise 13.1

Range and mean deviation about mean and median for ungrouped data

Questions

Ex 13.2

Exercise 13.2

Variance and standard deviation for ungrouped data, computational formula

Questions

Ex 13.3

Exercise 13.3

Variance and standard deviation for grouped (discrete and continuous) data

Questions

Ex 13.4

Exercise 13.4

Coefficient of variation — comparing two datasets

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Mean Deviation

Find the mean deviation about the mean for: 2, 4, 5, 7, 8.

▶ Show solution

Mean x̄ = (2+4+5+7+8)/5 = 26/5 = 5.2. Deviations: |2−5.2|=3.2, |4−5.2|=1.2, |5−5.2|=0.2, |7−5.2|=1.8, |8−5.2|=2.8. Sum = 9.2. MD = 9.2/5 = 1.84.

Q2MEDIUM· Variance and SD

Find the variance and standard deviation of: 6, 8, 10, 12, 14.

▶ Show solution

Mean x̄ = (6+8+10+12+14)/5 = 50/5 = 10. Deviations from mean: −4, −2, 0, 2, 4. Squared deviations: 16, 4, 0, 4, 16. Sum = 40. Variance σ² = 40/5 = 8. Standard deviation σ = √8 = 2√2 ≈ 2.83.

Q3HARD· Coefficient of Variation

Two series A and B have: Series A — mean=40, SD=10; Series B — mean=20, SD=6. Which series is more consistent?

▶ Show solution

C.V.(A) = (σ_A/x̄_A)×100 = (10/40)×100 = 25%. C.V.(B) = (σ_B/x̄_B)×100 = (6/20)×100 = 30%. Since C.V.(A) = 25% < C.V.(B) = 30%, Series A is MORE CONSISTENT (less relative variation) than Series B.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Measures of central tendency (mean, median) tell WHERE data is centred; measures of dispersion tell HOW SPREAD OUT it is
•Mean deviation = average of absolute deviations from the mean or median; always non-negative
•Variance σ² = Σ(xᵢ−x̄)²/n: average of squared deviations; always ≥ 0
•Standard deviation σ = √(variance): in the same units as original data; most commonly used measure of spread
•Computational shortcut: σ² = (Σxᵢ²/n) − x̄² avoids computing each (xᵢ−x̄) individually
•Coefficient of Variation CV = (σ/x̄)×100: unit-free, used to compare variability across different datasets
•Lower CV = more consistent; higher CV = more variable or unreliable
•For grouped continuous data: use class marks (midpoints) as xᵢ, weights are frequencies fᵢ

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	1	Mean deviation calculation for small ungrouped datasets
Long Answer	4-6	1	Variance and standard deviation (grouped or ungrouped), Coefficient of Variation comparison

Prep strategy

Construct a computation table with columns xᵢ, xᵢ−x̄, (xᵢ−x̄)², fᵢ·(xᵢ−x̄)² for systematic calculation — it prevents arithmetic errors
Practise the computational formula σ² = (Σxᵢ²/n) − (x̄)² — it is faster in exams than computing each deviation separately
For C.V. problems: state the formula, compute both C.V. values, then explicitly state which is lower and therefore more consistent

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Finance and Investment Risk

The standard deviation of stock returns measures investment risk. Higher SD = more volatile (risky) investment. Portfolio diversification aims to reduce the combined SD of returns.

Quality Control in Manufacturing

In Six Sigma manufacturing, processes are considered under control when measurements stay within ±3σ of the mean. Standard deviation is the key metric for process consistency.

Medical Research and Clinical Trials

Drug efficacy studies report mean improvement ± standard deviation, allowing doctors to judge whether the treatment effect is consistent across patients or highly variable.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Always compute the mean first and verify it before proceeding — all dispersion formulas depend on an accurate mean
Use a table format with columns for xᵢ, fᵢ, fᵢxᵢ, xᵢ², fᵢxᵢ² — this systematic approach is expected in board exam solutions
For C.V. comparison questions: compute both values numerically, then state 'since C.V.(A) < C.V.(B), A is more consistent'
Don't forget to take the square root for standard deviation — stopping at variance is a common error that costs the final mark

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Effect of change of scale and origin: if yᵢ = (xᵢ−a)/h, then σ_y = σ_x/h — shifting data doesn't change SD but scaling changes it proportionally
Variance of combined data: if two groups have means x̄₁, x̄₂ and variances σ₁², σ₂² and sizes n₁, n₂, the combined variance formula incorporates both within-group and between-group variation

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainHigh

JEE AdvancedMedium

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Standard deviation is mathematically more tractable — the squared function is differentiable and algebraically convenient, unlike the absolute value. SD is also the basis for the normal distribution and all of inferential statistics.

Use CV when comparing the variability of two datasets with DIFFERENT means or DIFFERENT units. For example, comparing consistency of exam scores (mean 60) vs. heights (mean 170 cm) — SD would not be comparable, but CV normalises by the mean.

The mean x̄ is the balance point of the data. Deviations above the mean are exactly cancelled by deviations below it. This is why variance uses SQUARED deviations — they are all non-negative and capture the actual magnitude of spread.

Practice more, ask doubts, find a tutor

CBSE Class 11 mock tests

CBSE Class 11 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Statistics (Mean Deviation, Variance, Standard Deviation) usually read these next.

Sets

The language of all mathematics begins here — with SETS. This foundational chapter covers set notation, types of sets (empty, finite, infinite, equal, subsets), Venn diagrams, and set operations (union, intersection, difference, complement). CBSE Class 11 Mathematics Chapter 1.

Relations and Functions

How are things CONNECTED? Relations and functions formalise the idea of association — from the Cartesian product of sets to the special case where every input has EXACTLY ONE output. This chapter covers domain, range, and types of functions. CBSE Class 11 Mathematics Chapter 2.

Trigonometric Functions

Sine, cosine, tangent — the ratios that relate angles to sides and UNDERLIE everything from physics to engineering. This chapter covers degree and radian measure, trigonometric functions of any angle, graphs, and key identities. CBSE Class 11 Mathematics Chapter 3.

Complex Numbers and Quadratic Equations

What is √(-1)? The answer — i — opens up an ENTIRE NEW NUMBER SYSTEM. This chapter introduces complex numbers (a + ib), their algebra, the Argand plane, and how they solve every quadratic equation. CBSE Class 11 Mathematics Chapter 4.