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Question
Two groups of 20 and 30 students have means 50 and 60, and standard deviations 4 and 5 respectively. The variance of the combined group of 50 students is:
Solution Path
Combined mean = 56. Deviations: d1=−6d_1 = -6, d2=4d_2 = 4. Combined variance = [20(52)+30(41)]/50=45.4[20(52) + 30(41)]/50 = 45.4.
01Question Setup
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Two groups: n1=20n_1=20, xˉ1=50\bar{x}_1=50, σ1=4\sigma_1=4 and n2=30n_2=30, xˉ2=60\bar{x}_2=60, σ2=5\sigma_2=5. Find combined variance.
Find σ122\sigma_{12}^2
02Combined Mean
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Combined mean = (20×50+30×60)/50=2800/50=56(20 \times 50 + 30 \times 60)/50 = 2800/50 = 56. Then d1=−6d_1 = -6, d2=4d_2 = 4.
xˉ12=56\bar{x}_{12} = 56
03Combined Variance FormulaKEY INSIGHT
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Combined variance = [n1(σ12+d12)+n2(σ22+d22)]/(n1+n2)[n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)]/(n_1+n_2). The di2d_i^2 terms account for the gap between each group mean and the combined mean.
20(16+36)+30(25+16)50=227050\frac{20(16+36) + 30(25+16)}{50} = \frac{2270}{50}
04Final Answer
4/4
Combined variance = 2270/50=45.42270/50 = 45.4.
σ122=45.4\boxed{\sigma_{12}^2 = 45.4}
Concepts from this question2 concepts unlocked
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Combined Mean and Variance of Two Groups

STANDARD

When merging two groups, combined mean is the weighted average. Combined variance includes both the within-group variance and the between-group variance (dᵢ² terms).

σ122=n1(σ12+d12)+n2(σ22+d22)n1+n2\sigma_{12}^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}

A classic JEE question pattern: two groups with known means and SDs, find the combined variance. The dᵢ terms are the part students forget.

Combined varianceMerging two data setsGroup statistics
Practice (7 Qs) →

Variance Shortcut Formula

EASY

Variance = E(X²) - [E(X)]². Compute the mean of squares minus the square of the mean. Avoids computing individual deviations.

σ2=∑fixi2N−(∑fixiN)2\sigma^2 = \frac{\sum f_i x_i^2}{N} - \left(\frac{\sum f_i x_i}{N}\right)^2

Nearly every JEE variance question is faster with the shortcut. The direct formula with (xᵢ - mean)² takes longer and has more room for arithmetic errors.

Variance computationSD from raw dataGrouped data variance
Practice (10 Qs) →
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