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Chapter 13: Statistics

Class 11 • Mathematics • Chapter 13

Statistics

Measuring not just the average of a data set, but how spread out the numbers are.

CBSE / NCERT Aligned   School Revise

Chapter Roadmap

Measures of Dispersion • Range • Mean Deviation • Variance • Standard Deviation • Coefficient of Variation • Key Results • Applications
1

Why Statistics Matters

In earlier classes you learned to find the average of a data set, such as the mean, median and mode. But an average alone can be misleading. Two cricketers might both average 50 runs, yet one scores steadily around 50 every match while the other swings wildly between 5 and 95. To tell them apart we need to measure the spread of the data, and that is what this chapter is about.

Measures of dispersion describe how scattered the numbers are around their centre. You will meet the range, the mean deviation, the variance and the standard deviation, each one giving a sharper picture than the last. The standard deviation in particular is one of the most widely used numbers in all of statistics, from weather records to quality control in factories.

Key idea
The mean tells you the centre of a data set; a measure of dispersion tells you how spread out it is. A small standard deviation means the data is clustered and consistent; a large one means it is widely scattered.
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Key Terms You Must Know
Term Meaning Example
Dispersion How spread out a data set is around its centre. Two sets can share a mean but differ in spread
Range The difference between the largest and smallest values. For 3, 8, 12 the range is 9
Mean deviation The average of the distances of the values from a centre (mean or median). Always taken as a positive distance
Variance The average of the squared distances from the mean, written σ². σ² = (1/n) Σ(xₐ − mean)²
Standard deviation The square root of the variance, written σ. σ = √(variance)
Coefficient of variation The standard deviation as a percentage of the mean. CV = (σ / mean) × 100
Observation A single value xₐ in the data set. Each match score is an observation
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Core Concepts, Step by Step

1. Measures of Dispersion

A measure of dispersion is a single number that captures how scattered a data set is. If every value is close to the mean, the dispersion is small; if the values are spread far and wide, it is large. We study four such measures in this chapter: the range, the mean deviation, the variance and the standard deviation. They increase in usefulness, with the range being the quickest but crudest, and the standard deviation being the most informative.

2. The Range

The range is the simplest measure of spread: it is just the largest value minus the smallest value. For the data 4, 9, 2, 15, 7, the range is 15 − 2 = 13. The range is quick to find and easy to understand, but it has a serious weakness: it depends only on the two extreme values and ignores everything in between, so a single unusual reading can make it misleading.

3. Mean Deviation

A better measure looks at how far every value sits from a central value. The mean deviation is the average of these distances, always taken as positive. The mean deviation about the mean is (1/n) Σ |xₐ − mean|, and the mean deviation about the median is (1/n) Σ |xₐ − median|. We use the absolute value so that values above and below the centre do not cancel out. The mean deviation is least when taken about the median.

4. Variance

Instead of taking absolute values, the variance squares each distance from the mean, which also removes negative signs and is easier to handle algebraically. The variance is σ² = (1/n) Σ (xₐ − mean)². The table below shows the standard way to compute it: list the values, find each deviation from the mean, square the deviations, and average them. A handy shortcut, proved later, is σ² = (Σxₐ²)/n − (mean)².

Computing the variance of 2, 4, 6, 8, 10 (mean = 6)

xₐ xₐ − mean (xₐ − mean)²
2 −4 16
4 −2 4
6 0 0
8 2 4
10 4 16
Total 0 40

5. Standard Deviation

Because the variance is built from squared distances, its units are squared too (for example, squared rupees), which is awkward. Taking the square root fixes this and gives the standard deviation σ = √(variance). The standard deviation is measured in the same units as the data, which is why it is the most popular measure of spread. For the data in the table above, the variance is 40/5 = 8, so the standard deviation is √8 = 2√2, about 2.83.

6. Coefficient of Variation

To compare the spread of two data sets that have different means or different units, we use the coefficient of variation, CV = (σ / mean) × 100. Because it is a percentage, it lets us compare consistency fairly. The data set with the smaller coefficient of variation is the more consistent (less scattered) one. This is exactly how we decide which of two cricketers, or which of two machines, is the more reliable.
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Key Results & Proofs

Three results make variance calculations faster and explain how it behaves when the data is shifted or scaled.

Result 1: The Shortcut Formula for Variance

Statement. The variance can be written as σ² = (Σxₐ²)/n − (mean)².

Proof

Expand the squared deviation and use the fact that the average of the values is the mean.

Let the mean be m. Start from the definition σ² = (1/n) Σ(xₐ − m)².  –  definition of variance
Expand the square: (xₐ − m)² = xₐ² − 2m xₐ + m².  –  open the bracket
σ² = (1/n)Σxₐ² − 2m(1/n)Σxₐ + m² sum term by term
But (1/n)Σxₐ = m, so the middle term is −2m·m = −2m².  –  the mean appears
σ² = (Σxₐ²)/n − m² since −2m² + m² = −m²

This avoids working out each deviation separately, which is much faster for large data sets.
Result 2: Adding a Constant Leaves Variance Unchanged

Statement. If every observation is increased by the same constant k, the variance does not change.

Proof

Shifting every value by the same amount shifts the mean by the same amount too.

New values are xₐ + k, and the new mean is m + k.  –  everything shifts by k
New deviation = (xₐ + k) − (m + k) = xₐ − m.  –  the k cancels
The squared deviations are therefore exactly the same as before.  –  nothing has changed
new variance = old variance so the variance is unchanged

So adding marks of grace to every student leaves the spread of the marks exactly as it was.
Result 3: Multiplying by a Constant Scales the Variance

Statement. If every observation is multiplied by a constant k, the variance is multiplied by k² and the standard deviation by |k|.

Proof

Scaling every value by k scales each deviation by k, so each squared deviation by k².

New values are k xₐ, and the new mean is k m.  –  everything scales by k
New deviation = k xₐ − k m = k(xₐ − m).  –  factor out k
Squaring gives k²(xₐ − m)².  –  the k² comes out
new variance = k² × old variance average of the squared deviations
new σ = |k| × old σ take the square root

So changing units, say from metres to centimetres (k = 100), multiplies the standard deviation by 100.
5

Worked Examples
Example 1

Question: Find the range of the data 5, 8, 12, 3, 10.

▶ Show full working

The range is the largest value minus the smallest.

Largest value is 12, smallest is 3.  –  identify the extremes
range = 12 − 3 = 9 subtract

Answer: The range is 9.
Example 2

Question: Find the mean deviation about the mean for 4, 6, 8, 10, 12.

▶ Show full working

Find the mean, then average the absolute distances from it.

Mean = (4 + 6 + 8 + 10 + 12)/5 = 8.  –  find the mean
Absolute deviations: 4, 2, 0, 2, 4.  –  distances from 8
MD = (4 + 2 + 0 + 2 + 4)/5 average them
MD = 12/5 = 2.4 simplify

Answer: The mean deviation about the mean is 2.4.
Example 3

Question: Find the mean deviation about the median for 3, 6, 9, 12, 15.

▶ Show full working

The data is already in order, so the middle value is the median.

Median = middle value = 9.  –  find the median
Absolute deviations: 6, 3, 0, 3, 6.  –  distances from 9
MD = (6 + 3 + 0 + 3 + 6)/5 = 18/5 average them
MD = 3.6 simplify

Answer: The mean deviation about the median is 3.6.
Example 4

Question: Find the variance of 2, 4, 6, 8, 10.

▶ Show full working

Find the mean, then average the squared deviations.

Mean = 30/5 = 6.  –  find the mean
Squared deviations: 16, 4, 0, 4, 16.  –  square each distance from 6
variance = (16 + 4 + 0 + 4 + 16)/5 average them
variance = 40/5 = 8 simplify

Answer: The variance is 8.
Example 5

Question: Find the standard deviation of 2, 4, 6, 8, 10.

▶ Show full working

The standard deviation is the square root of the variance found above.

σ = √(variance) = √8 take the square root of 8
σ = 2√2 ≈ 2.83 simplify

Answer: The standard deviation is 2√2, about 2.83.
Example 6

Question: Find the variance of 1, 2, 3, 4, 5 using the shortcut formula.

▶ Show full working

Use σ² = (Σxₐ²)/n − (mean)².

Mean = 15/5 = 3; Σxₐ² = 1 + 4 + 9 + 16 + 25 = 55.  –  find the pieces
σ² = 55/5 − 3² apply the shortcut
σ² = 11 − 9 = 2 simplify

Answer: The variance is 2.
Example 7

Question: Find the mean of the data with values xₐ = 2, 4, 6 and frequencies fₐ = 1, 2, 1.

▶ Show full working

Multiply each value by its frequency, add, and divide by the total frequency.

Σfₐxₐ = (1)(2) + (2)(4) + (1)(6) = 16.  –  sum of f times x
Total frequency N = 1 + 2 + 1 = 4.  –  add the frequencies
mean = 16/4 = 4 divide

Answer: The mean is 4.
Example 8

Question: A data set of 10 values has Σxₐ = 40 and Σxₐ² = 180. Find the variance and standard deviation.

▶ Show full working

Find the mean first, then use the shortcut formula.

Mean = 40/10 = 4.  –  find the mean
σ² = 180/10 − 4² shortcut formula
σ² = 18 − 16 = 2 simplify
σ = √2 square root of the variance

Answer: Variance 2 and standard deviation √2.
Example 9

Question: The variance of a data set is 25. Find its standard deviation.

▶ Show full working

The standard deviation is the square root of the variance.

σ = √25 take the square root
σ = 5 simplify

Answer: The standard deviation is 5.
Example 10

Question: If every observation of a data set is increased by 5, what happens to its variance?

▶ Show full working

Use the result that adding a constant shifts the mean but not the spread.

Each value and the mean both increase by 5.  –  everything shifts together
Each deviation (value − mean) is unchanged.  –  the 5 cancels
So the variance stays the same.  –  conclusion

Answer: The variance is unchanged.
Example 11

Question: The standard deviation of a data set is 4. If every observation is multiplied by 3, find the new standard deviation.

▶ Show full working

Multiplying every value by k multiplies the standard deviation by |k|.

new σ = |k| × old σ scaling rule
new σ = 3 × 4 = 12 with k = 3

Answer: The new standard deviation is 12.
Example 12

Question: Two batsmen both average 50 runs. One has a standard deviation of 5 runs, the other 8 runs. Who is more consistent?

▶ Show full working

Compare their coefficients of variation; the smaller one is more consistent.

CV (first) = (5/50) × 100 = 10% coefficient of variation
CV (second) = (8/50) × 100 = 16% coefficient of variation
The first batsman has the smaller CV.  –  compare

Answer: The first batsman is more consistent.
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Where You Meet This in Real Life

Quality control

Factories measure the standard deviation of product sizes to make sure items stay within tolerance; a small spread means reliable manufacturing.

Weather and climate

The spread of daily temperatures or rainfall, not just the average, tells us how changeable a place’s weather is.

Sport

Selectors compare the consistency of players using the coefficient of variation, rewarding steady performers over erratic ones.

Finance

The risk of an investment is measured by the standard deviation of its returns: a higher spread means a riskier, less predictable investment.

Research and surveys

Scientists report the standard deviation alongside the mean so that readers know how reliable and tightly grouped the results are.
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Practice Sets A to D

Practice Set A – Basics

A1. Find the range of 10, 15, 8, 20, 5.

▶ Reveal full working

Subtract the smallest from the largest.

Largest 20, smallest 5.
range = 20 − 5 = 15 subtract

Answer: 15.

A2. Find the mean of 2, 4, 6, 8.

▶ Reveal full working

Add the values and divide by how many there are.

mean = (2 + 4 + 6 + 8)/4 = 20/4 add and divide
mean = 5 simplify

Answer: 5.

A3. Find the variance of 1, 2, 3.

▶ Reveal full working

Mean is 2; average the squared deviations.

Squared deviations from 2: 1, 0, 1.
variance = (1 + 0 + 1)/3 = 2/3 average them

Answer: 2/3.

A4. If the variance of a data set is 16, find its standard deviation.

▶ Reveal full working

Take the square root.

σ = √16 = 4 square root

Answer: 4.

Practice Set B – Conceptual

B1. Why do we need a measure of dispersion when we already have the mean?

▶ Reveal full working

Think of two data sets with the same mean.

Two data sets can share a mean but be spread very differently.
Dispersion tells us how scattered the values are around that mean.

Answer: Because the mean alone does not show how spread out the data is.

B2. What is the range of a data set?

▶ Reveal full working

It uses only two values.

It is the largest value minus the smallest value.

Answer: The largest value minus the smallest value.

B3. Which is more affected by a single extreme value, the range or the standard deviation?

▶ Reveal full working

The range uses only the two ends.

The range depends only on the highest and lowest values.
So one extreme reading changes the range completely.

Answer: The range, since it depends only on the extremes.

B4. What does a small standard deviation tell you about a data set?

▶ Reveal full working

Relate spread to closeness to the mean.

A small standard deviation means the values lie close to the mean.
So the data is consistent and tightly clustered.

Answer: The data is closely clustered around the mean (consistent).

Practice Set C – Application / Numerical

C1. Find the mean deviation about the mean for 2, 4, 6, 8, 10.

▶ Reveal full working

Mean is 6; average the absolute deviations.

Absolute deviations from 6: 4, 2, 0, 2, 4.
MD = (4 + 2 + 0 + 2 + 4)/5 = 12/5 average
MD = 2.4 simplify

Answer: 2.4.

C2. Find the variance of 4, 8, 12, 16, 20.

▶ Reveal full working

Mean is 12; average the squared deviations.

Squared deviations: 64, 16, 0, 16, 64.
variance = (64 + 16 + 0 + 16 + 64)/5 = 160/5 average
variance = 32 simplify

Answer: 32.

C3. Find the standard deviation of 4, 8, 12, 16, 20.

▶ Reveal full working

Take the square root of the variance found above.

σ = √32 = 4√2 square root of 32

Answer: 4√2 (about 5.66).

C4. Find the variance of 2, 3, 5, 7, 8 using the shortcut formula.

▶ Reveal full working

Use σ² = (Σxₐ²)/n − (mean)².

Mean = 25/5 = 5; Σxₐ² = 4 + 9 + 25 + 49 + 64 = 151.
σ² = 151/5 − 25 shortcut formula
σ² = 30.2 − 25 = 5.2 simplify

Answer: 5.2.

Practice Set D – HOTS / Word Problems

D1. The mean of 5 observations is 6 and the variance is 4. If each observation is multiplied by 2, find the new mean and new variance.

▶ Reveal full working

Multiplying by k multiplies the mean by k and the variance by k².

new mean = 2 × 6 = 12 mean scales by 2
new variance = 2² × 4 = 16 variance scales by k²

Answer: New mean 12, new variance 16.

D2. A data set has mean 25 and standard deviation 5. Find its coefficient of variation.

▶ Reveal full working

Use CV = (σ / mean) × 100.

CV = (5/25) × 100 substitute
CV = 20% simplify

Answer: 20%.

D3. Find the mean deviation about the median for 5, 7, 9, 11, 13, 15.

▶ Reveal full working

With six values, the median is the average of the two middle ones.

Median = (9 + 11)/2 = 10.  –  median of an even count
Absolute deviations from 10: 5, 3, 1, 1, 3, 5.
MD = (5 + 3 + 1 + 1 + 3 + 5)/6 = 18/6 average
MD = 3 simplify

Answer: 3.

D4. The variance of 6 numbers is 8. If each number is increased by 4, find the new variance.

▶ Reveal full working

Use the result that adding a constant does not change the variance.

Each value and the mean rise by 4 together.
Every deviation is unchanged, so the variance is unchanged.
new variance = 8 same as before

Answer: 8 (unchanged).

Chapter Summary

Everything in One Glance

Dispersion

A measure of how spread out the data is; the mean alone is not enough.

  

Range

Largest value minus smallest value; quick but uses only the extremes.
 

Mean Deviation

Average of the absolute distances from the mean or the median.

  

Variance

σ² = (1/n)Σ(xₐ − mean)², or the shortcut (Σxₐ²)/n − (mean)².
 

Standard Deviation

σ = √(variance); measured in the same units as the data.

  

Shifting and Scaling

Adding a constant leaves variance unchanged; multiplying by k scales variance by k² and σ by |k|.
 
8

Are You Exam-Ready?

8-Point Exam Quick-Check

1 Find the range of 12, 7, 19, 4, 15.
 
2 Find the mean deviation about the mean for 3, 5, 7.
 
3 Find the variance of 1, 3, 5, 7, 9.
 
4 If a variance is 49, what is the standard deviation?
 
5 Write the shortcut formula for the variance.
 
6 What happens to the variance if every value is increased by 10?
 
7 What happens to the standard deviation if every value is doubled?
 
8 Two data sets have CV 12% and 18%; which is more consistent?
 

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Class 11 Mathematics Chapter 13: Statistics, Complete Notes and Practice

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