Statistics – Class 11 Mathematics notes, important questions and MCQs

Chapter overview

Statistics summarises data. Measures of dispersion such as range, mean deviation, variance and standard deviation show how spread out data is.

Dispersion

• Range = maximum − minimum.
• Mean deviation about mean = Σ|xᵢ − x̄|/n (or about median).
• Variance σ² = Σ(xᵢ − x̄)²/n = Σxᵢ²/n − x̄²; standard deviation σ = √variance.

Comparison

• Coefficient of variation CV = (σ/x̄) × 100; higher CV means less consistent.
• Adding a constant to all values does not change σ; multiplying by k multiplies σ by |k|.
• For grouped data, use class mid-points with frequencies.

Key terms

Standard deviation

square root of variance, the typical distance from the mean

Coefficient of variation

σ/x̄ × 100, used to compare variability

Mean deviation

average of absolute deviations of observations from the mean or median

Formulas

• σ² = Σxᵢ²/n − x̄²
• CV = (σ/x̄) × 100

Important questions with answers

Quick revision

• SD unaffected by adding a constant.
• Variance = square of SD.
• Lower CV = more consistent.

Exam tips

• The series with the lower CV is more consistent.
• For grouped data use the step-deviation method uᵢ = (xᵢ − A)/h to simplify variance: σ² = h²[Σfᵢuᵢ²/N − (Σfᵢuᵢ/N)²].

Practice MCQs

1. Range of 3, 7, 2, 9, 5 is

   1. 6
   2. 9
   3. 7
   4. 5
   Show answer

   Answer: (c) 7. 9 − 2.

2. Standard deviation is the square root of

   1. mean
   2. variance
   3. range
   4. median
   Show answer

   Answer: (b) variance. σ = √σ².

3. Variance of 2, 4, 6, 8, 10 is

   1. 4
   2. 6
   3. 8
   4. 10
   Show answer

   Answer: (c) 8. Mean 6; deviations² sum 40/5.

4. If each value is increased by 5, the standard deviation

   1. increases by 5
   2. doubles
   3. stays the same
   4. becomes zero
   Show answer

   Answer: (c) stays the same. Shift does not change spread.

5. Mean deviation about the mean of 4, 7, 8, 9, 10, 12, 13, 17 is

   1. 2.5
   2. 4
   3. 3
   4. 5
   Show answer

   Answer: (c) 3. Mean 10; |dev| sum 24/8.

6. Mean 50, SD 10 gives CV

   1. 5%
   2. 10%
   3. 50%
   4. 20%
   Show answer

   Answer: (d) 20%. 10/50 × 100.

7. If each observation is multiplied by 3, the variance is multiplied by

   1. 3
   2. 9
   3. 6
   4. 1
   Show answer

   Answer: (b) 9. k².

8. The standard deviation of the first 10 natural numbers is about

   1. 3.03
   2. 2.87
   3. 8.25
   4. 5.5
   Show answer

   Answer: (b) 2.87. σ² = (n² − 1)/12 = 8.25.