The variance σ² of the data
is _________.

Solution

Maths sir · 3 steps

Find the mean first, then add up the frequency-weighted squared distances from it.

1. Find the mean. N = Σ fᵢ = 3+2+3+2+6+3+3 = 22 and Σ fᵢ xᵢ = 0+2+15+12+60+36+51 = 176. So x̄ = 176/22 = 8.

   xᵢ	fᵢ	(xᵢ - 8)²	fᵢ (xᵢ - 8)²
   0	3	64	192
   1	2	49	98
   5	3	9	27
   6	2	4	8
   10	6	4	24
   12	3	16	48
   17	3	81	243
   Total	22	not added	640

   σ² = (Σ fᵢ (xᵢ - x̄)²)/N = 640/22 ≈ 29.09

2. Check. The shortcut gives the same result. Σ fᵢ xᵢ² = 2048, so σ² = 2048/22 - 8² = 93.09 - 64 = 29.09.
3. So. To the nearest integer, the variance is 29.

The common slip: 30. Dividing by N - 1 = 21 instead of N = 22 gives 640/21 ≈ 30.5, which rounds to 30. That is the sample variance. The variance of the data divides by N.

The idea to keep. For a frequency table, multiply every squared deviation by its frequency and divide by N = Σ fᵢ. A whole-number mean makes the deviation method quick.

Answer from the NTA final answer key. Solution by PrepWiser.