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CBSE · Class 11 · Mathematics

Statistics: Variance and Standard Deviation (Ch 13)

4 concepts · 32 practice questions with answers · Free to read

Variance and standard deviation

From daily life

Mean deviation uses absolute values, which are awkward in algebra. Squaring the gaps instead gives the variance, and its square root brings the spread back to the original unit.

Simple definition

The variance is the average of the squared gaps from the mean. The standard deviation (SD) is its square root, in the same unit as the data.

Formula: Variance = Σ(x − mean)² ÷ n; SD = √variance

Rule 1: Steps — Mean, gaps, squares, average, square root.

  • Easy: 4, 6, 8, 10, 12: mean 8; squared gaps 16, 4, 0, 4, 16; sum 40; variance 8; SD ≈ 2.83.
  • Medium: 3, 7, 11, 15: mean 9; squared gaps 36, 4, 4, 36; variance 20; SD ≈ 4.47.
  • Exam-type: Variance of 5, 5, 5, 5 is 0 because there is no spread.

Rule 2: Effect of changes — Adding shifts the data; multiplying stretches it.

  • Easy: Adding a number to every value: SD does not change.
  • Medium: Multiplying every value by k: SD is multiplied by |k|, variance by k².
  • Exam-type: 4, 6, 8, 10, 12 times 3 gives SD ≈ 8.49.
Remember

SD is never negative. SD = 0 only when all values are equal.

Common mistake

Forgetting to take the square root when SD is asked.

Memory clue

Square it, average it, root it.

Practice questions

Easy
  1. 1Find the variance and SD of 4, 6, 8, 10, 12.
  2. 2Find the variance of 5, 5, 5, 5.
  3. 3Find the variance and SD of 1, 2, 3, 4, 5.
Medium
  1. 4Find the variance and SD of 3, 7, 11, 15.
  2. 5The variance is 36. Find the SD.
  3. 6The SD is 5. Find the variance.
Challenging
  1. 7Each value of 4, 6, 8, 10, 12 is multiplied by 3. Find the new SD.
  2. 8The mean of 5 numbers is 6 and the sum of their squares is 190. Find the variance.
Show answers (try them yourself first)
  1. 18 and 2.83
  2. 20
  3. 32 and 1.41
  4. 420 and 4.47
  5. 56
  6. 625
  7. 78.49
  8. 8190/5 − 36 = 2

Shortcut formula for variance

From daily life

Subtracting the mean from every value is slow when the mean is not a whole number. A shortcut uses only the sum of squares and the mean.

Simple definition

Variance = (sum of x²) ÷ n − (mean)². It gives the same answer as the long method.

Formula: Variance = Σx² ÷ n − mean²

Rule 1: Using the shortcut — Find Σx² and the mean.

  • Easy: 2, 4, 6, 8: Σx² = 120; 120/4 − 5² = 5.
  • Medium: 1, 3, 5: Σx² = 35; 35/3 − 3² = 2.67.
  • Exam-type: n = 10, Σx = 50, Σx² = 290: mean 5; variance = 29 − 25 = 4; SD = 2.

Rule 2: Working backwards — Use the formula to find the sum of squares.

  • Easy: If n = 20, mean 12 and SD 3: variance 9; Σx²/20 = 9 + 144 = 153; Σx² = 3060.
  • Medium: Two sets 10, 20, 30 and 110, 120, 130 have the same SD (shift does not matter).
  • Exam-type: Variance can never be negative.
Remember

Square the mean after dividing; do not divide Σx² by n squared.

Common mistake

Using (Σx)² ÷ n instead of Σx² ÷ n.

Memory clue

Mean of squares minus square of mean.

Practice questions

Easy
  1. 1Use the shortcut to find the variance of 2, 4, 6, 8.
  2. 2Use the shortcut to find the variance of 1, 3, 5.
  3. 3Use the shortcut to find the variance of 6, 8, 10, 12, 14.
Medium
  1. 4For 10 values Σx = 50 and Σx² = 290. Find the variance and SD.
  2. 5For 8 values Σx = 40 and Σx² = 240. Find the variance.
  3. 6Find the variance of 7, 9, 11, 13.
Challenging
  1. 7For 20 values the mean is 12 and the SD is 3. Find Σx².
  2. 8Compare the SD of 10, 20, 30 with that of 110, 120, 130.
Show answers (try them yourself first)
  1. 15
  2. 22.67
  3. 38
  4. 44 and 2
  5. 530 − 25 = 5
  6. 65
  7. 73060
  8. 8Equal, both ≈ 8.16

Variance for grouped data

From daily life

A teacher has marks of 40 students in a frequency table. We want the SD without listing 40 numbers.

Simple definition

For values x with frequencies f and total N: variance = Σf(x − mean)² ÷ N, or Σf·x² ÷ N − mean². For classes use the class mark as x.

Formula: Variance = Σf(x − mean)² ÷ N

Rule 1: Discrete — Weight each squared gap by f.

  • Easy: values 1, 2, 3 with frequencies 2, 3, 2: N = 7, mean 2, variance = 0.57.
  • Medium: values 2, 4, 6, 8 with frequencies 1, 3, 3, 1: N = 8, mean 5, variance = 3.
  • Exam-type: values 10, 20, 30 with frequencies 2, 4, 2: mean 20, variance = 50, SD ≈ 7.07.

Rule 2: Classes — Use class marks.

  • Easy: classes 0–10, 10–20, 20–30 with frequencies 4, 8, 4: class marks 5, 15, 25; mean 15; variance = 50.
  • Medium: classes 0–20, 20–40, 40–60, 60–80 with frequencies 2, 4, 4, 2: class marks 10, 30, 50, 70; mean 40; variance ≈ 366.67; SD ≈ 19.15.
  • Exam-type: Divide by N (sum of frequencies), not by the number of rows.
Remember

Class mark = (lower limit + upper limit) ÷ 2.

Common mistake

Using the class limits instead of the class marks.

Memory clue

Class mark, then weight.

Practice questions

Easy
  1. 1Find the variance for values 1, 2, 3 with frequencies 2, 3, 2.
  2. 2Find the variance for values 2, 4, 6, 8 with frequencies 1, 3, 3, 1.
  3. 3Find the variance and SD for values 10, 20, 30 with frequencies 2, 4, 2.
Medium
  1. 4Find the variance for classes 0–10, 10–20, 20–30 with frequencies 4, 8, 4.
  2. 5Find the variance and SD for classes 0–20, 20–40, 40–60, 60–80 with frequencies 2, 4, 4, 2.
  3. 6Find the variance and SD for values 5, 15 with frequencies 1, 1.
Challenging
  1. 7Find the variance for values 3, 6, 9, 12 with frequencies 1, 2, 2, 1.
  2. 8Find the variance for classes 10–20, 20–30, 30–40 with frequencies 5, 10, 5.
Show answers (try them yourself first)
  1. 10.57
  2. 23
  3. 350 and 7.07
  4. 450
  5. 5366.67 and 19.15
  6. 625 and 5
  7. 78.25
  8. 850

Coefficient of variation

From daily life

Rohan's test marks average 80 with SD 8; Sana's average 50 with SD 8. Same SD, but a gap of 8 marks matters more at 50 than at 80. So we compare spread relative to the mean.

Simple definition

The coefficient of variation (CV) = SD ÷ mean × 100, a percentage. The set with the smaller CV is more consistent (less variable).

Formula: CV = (SD ÷ mean) × 100

Rule 1: Compute CV — Divide the SD by the mean, multiply by 100.

  • Easy: Mean 50, SD 5: CV = 10%.
  • Medium: Mean 40, SD 6: CV = 15%.
  • Exam-type: 4, 6, 8, 10, 12: mean 8, SD ≈ 2.83; CV ≈ 35.36%.

Rule 2: Compare two sets — Lower CV means more consistent.

  • Easy: A: mean 20, SD 4 (CV 20%). B: mean 30, SD 5 (CV 16.67%). B is more consistent.
  • Medium: Same SD but different means: the one with the larger mean is more consistent.
  • Exam-type: CV has no unit.
Remember

Compare consistency by CV, not by SD alone, when the means differ.

Common mistake

Choosing the set with the smaller SD without checking the means.

Memory clue

Smaller CV, steadier set.

Practice questions

Easy
  1. 1Find the CV for mean 50 and SD 5.
  2. 2Find the CV for mean 40 and SD 6.
  3. 3A: mean 20, SD 4. B: mean 30, SD 5. Which is more consistent?
Medium
  1. 4Find the CV of 4, 6, 8, 10, 12.
  2. 5The CV is 25% and the mean is 40. Find the SD.
  3. 6The SD is 12 and the CV is 20%. Find the mean.
Challenging
  1. 7Series A: mean 100, SD 15. Series B: mean 60, SD 12. Which is more consistent?
  2. 8Each value of 4, 6, 8, 10, 12 is increased by 10. Find the new CV.
Show answers (try them yourself first)
  1. 110%
  2. 215%
  3. 3B (16.67% against 20%)
  4. 435.36%
  5. 510
  6. 660
  7. 7A (15% against 20%)
  8. 815.71%