CBSE Class 11 Mathematics Chapter 13 Notes: Statistics

Chapter Introduction

Statistics is Chapter 13 of CBSE Class 11 Mathematics, falling under the Statistics and Probability unit. In Class 10, you learned about mean, median, and mode — measures of central tendency. This chapter goes further to study measures of dispersion: how spread out the data is. You will learn about range, mean deviation, variance, standard deviation, and coefficient of variation. This chapter typically carries 6-8 marks in board exams and the calculation-heavy nature means practice is essential.

The key insight: two distributions can have the same mean but very different spreads. Measures of dispersion tell you how much the data varies from the centre.

Key Definitions

Detailed Concepts

1. Measures of Dispersion — Overview

2. Mean Deviation for Ungrouped Data

About the mean (x-bar):

About the median (M):

Steps: 1. Calculate the mean (or median) of the data. 2. Find the deviation of each value from the mean (or median). 3. Take the absolute value of each deviation. 4. Sum all absolute deviations. 5. Divide by the number of observations.

Note: Mean deviation about the median is always less than or equal to mean deviation about the mean.

3. Mean Deviation for Grouped Data

Discrete frequency distribution:

Continuous frequency distribution:

Same formula, but x_i is replaced by the class mark of each class interval.

Steps: 1. Find class marks (midpoints of each class). 2. Calculate the mean x-bar.

4. Multiply by frequency f_i. 5. Sum and divide by N.

4. Variance and Standard Deviation — Ungrouped Data

Variance:

sigma^2 = (1/n) * Sigma (x_i - x-bar)^2

Standard Deviation:

sigma = sqrt(sigma^2) = sqrt[(1/n) * Sigma (x_i - x-bar)^2]

5. Variance and Standard Deviation — Grouped Data

Variance:

sigma^2 = (1/N) Sigma f_i (x_i - x-bar)^2

Standard Deviation:

sigma = sqrt[(1/N) Sigma f_i (x_i - x-bar)^2]

Where N = Sigma f_i and x_i is the class mark for continuous distributions.

6. Shortcut Method for Variance

When values are large, use an assumed mean A to simplify calculations:

Let d_i = x_i - A

*sigma^2 = (1/N) Sigma f_i d_i^2 - [(1/N) Sigma f_i d_i]^2*

Steps: 1. Choose A close to the actual mean (usually the middle class mark). 2. Compute d_i = x_i - A for each value/class mark. 3. Compute f_i d_i and f_i d_i^2. 4. Sum both columns. 5. Apply the formula.

Why it works: This formula is derived from expanding (x_i - x-bar)^2 and simplifying using the algebraic identity. The result does not depend on the choice of A.

7. Coefficient of Variation

CV = (sigma / x-bar) x 100

Purpose: To compare the variability of two distributions with different means or different units.

• If CV(A) < CV(B): Distribution A is more consistent (less variable).
• If CV(A) > CV(B): Distribution A is more variable (less consistent).

Direct vs Shortcut Method — Comparison

Important Numericals

Example 1: Mean deviation for ungrouped data

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

Mean = (4 + 6 + 8 + 10 + 12)/5 = 40/5 = 8

MD = (4 + 2 + 0 + 2 + 4)/5 = 12/5 = 2.4

Example 2: Variance and SD for ungrouped data

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

Mean = 50/5 = 10

Deviations squared: (6-10)^2 = 16, (8-10)^2 = 4, (10-10)^2 = 0, (12-10)^2 = 4, (14-10)^2 = 16

Variance = (16 + 4 + 0 + 4 + 16)/5 = 40/5 = 8

SD = sqrt(8) = 2*sqrt(2) = 2.83 (approx.)

Example 3: Grouped data — variance and SD

N = 35, Sigma f_i * x_i = 765

x-bar = 765/35 = 21.86 (approx.)

Sigma f_i * (x_i - x-bar)^2 = 1125 + 200 + 375 + 1575 = 3275

Variance = 3275/35 = 93.57 (approx.)

SD = sqrt(93.57) = 9.67 (approx.)

Example 4: Shortcut method

For data with class marks 5, 15, 25, 35 and frequencies 5, 8, 15, 7, take A = 25.

N = 35, Sigma f_i d_i = -110, Sigma f_i d_i^2 = 3500

sigma^2 = 3500/35 - (-110/35)^2 = 100 - (110/35)^2 = 100 - 9.88 = 90.12 (approx.)

Example 5: Coefficient of variation comparison

Team A scores: mean = 50, SD = 10. Team B scores: mean = 40, SD = 8.

CV(A) = (10/50) x 100 = 20%

CV(B) = (8/40) x 100 = 20%

Both teams have equal consistency since CV is the same.

Common Mistakes to Avoid

2. Confusing n and N: For ungrouped data, divide by n (number of observations). For grouped data, divide by N = Sigma f_i (total frequency), not by the number of classes.

3. Variance vs standard deviation: Variance is in squared units. Always take the square root to get SD. Do not report variance when the question asks for SD, or vice versa.

4. Class mark calculation: Class mark = (upper + lower)/2. A common error is using the class boundaries incorrectly, especially with continuous classes.

5. Shortcut method sign errors: d_i = x_i - A can be negative. When computing d_i^2, square the negative value correctly. The formula uses d_i^2 (always positive) and (Sigma d_i)^2 (square of the sum, which may have a different sign).

6. CV comparison direction: Smaller CV means more consistent. Students sometimes state the opposite.

7. Mean deviation about mean vs median: These are two different measures. Read the question carefully to determine which central value to use.

Exam Tips

• For 1-mark questions, know: definitions of variance, SD, CV; formula for CV.
• For 3-mark questions, practice: mean deviation for ungrouped data, variance/SD for small datasets.
• For 5-mark questions, practice: variance and SD for grouped data (with frequency table), shortcut method, CV comparison.
• In calculation questions, create a table with columns: x_i, f_i, f_ix_i, (x_i - x-bar), f_i(x_i - x-bar)^2. This is essential for organized work and partial marks.
• Always show the mean calculation first before computing deviations.
• For shortcut method, clearly state the assumed mean A you have chosen.
• When comparing two distributions using CV, clearly state which is more consistent and why.

Quick Revision Notes

NCERT Exercise Problem Patterns

Derivations and Key Proofs

Derivation 1: Shortcut method for variance

Let d_i = x_i - A (deviation from assumed mean A).

x_i - x-bar = (x_i - A) - (x-bar - A) = d_i - d-bar

(x_i - x-bar)^2 = (d_i - d-bar)^2 = d_i^2 - 2d_id-bar + d-bar^2

Sigma f_i(x_i - x-bar)^2 = Sigma f_id_i^2 - 2d-barSigma f_id_i + d-bar^2*Sigma f_i

Since d-bar = (1/N)Sigma f_id_i:

= Sigma f_id_i^2 - 2Nd-bar^2 + Nd-bar^2 = Sigma f_id_i^2 - Nd-bar^2

sigma^2 = (1/N)Sigma f_id_i^2 - d-bar^2 = (1/N)Sigma f_id_i^2 - [(1/N)Sigma f_id_i]^2

Derivation 2: Why MD about median is less than MD about mean

For any set of data, the sum of absolute deviations is minimized about the median. This is a property of the median — it is the point that minimizes the sum of absolute deviations, just as the mean minimizes the sum of squared deviations.

Additional Solved Examples

Example 6: Mean deviation about median

Find the mean deviation about the median for: 3, 9, 5, 3, 12, 10, 18, 5.

Arranged: 3, 3, 5, 5, 9, 10, 12, 18. Median = (5 + 9)/2 = 7

Sum = 4 + 4 + 2 + 2 + 2 + 3 + 5 + 11 = 33

MD = 33/8 = 4.125

Example 7: Shortcut method step by step

Data: 4, 7, 10, 13, 16. Take A = 10.

Sigma d_i = 0, Sigma d_i^2 = 90

sigma^2 = 90/5 - (0/5)^2 = 18 - 0 = 18

SD = sqrt(18) = 3*sqrt(2) = 4.24 (approx.)

Example 8: CV for consistency comparison

Batsman A: mean = 60, SD = 12. Batsman B: mean = 48, SD = 8.

CV(A) = (12/60) x 100 = 20%

CV(B) = (8/48) x 100 = 16.67%

Batsman B is more consistent (smaller CV), even though A scores more on average.

Practice Checklist

• Calculate mean deviation (about mean and median) for 3 ungrouped datasets
• Compute variance and SD for 2 grouped data distributions
• Use the shortcut method for at least 2 problems
• Compare consistency of 2 distributions using CV
• Practice creating frequency tables with all required columns
• Verify shortcut method results match direct method results

Quick Revision Notes

1. Range: R = Max - Min (simplest but least informative)

3. Mean deviation about median: Replace x-bar with Median M in the above formula 4. Variance (ungrouped): sigma^2 = (1/n) Sigma (x_i - x-bar)^2 5. Variance (grouped): sigma^2 = (1/N) Sigma f_i(x_i - x-bar)^2 6. Standard deviation: sigma = sqrt(variance) 7. Shortcut method: sigma^2 = (1/N) Sigma f_i d_i^2 - [(1/N) Sigma f_i d_i]^2, where d_i = x_i - A 8. Coefficient of variation: CV = (sigma/x-bar) x 100 9. CV comparison: Smaller CV = more consistent; larger CV = more variable 10. Class mark: (upper + lower)/2 11. N = Sigma f_i (total frequency, not number of classes) 12. Mean deviation about median is always <= mean deviation about mean