If n1 = 10, n2 = 5 are the sizes of a male and female student group with mean ages x̄1 = 10, x̄2 = 4, respectively, with an equal standard deviation σ1 = σ2 = 1, the standard deviation of the combined series with size n1 + n2 and combined mean x̄ = 8 is equal to:
3
To find the standard deviation of the combined series, we need to calculate the combined variance first. Given:
- n1 = 10 (size of male group)
- n2 = 5 (size of female group)
- x̄1 = 10 (mean age of male group)
- x̄2 = 4 (mean age of female group)
- n = n1 + n2 = 15 (size of combined group)
- x̄ = 8 (combined mean age)
- σ1 = σ2 = 1 (standard deviation of each group)
Variance of each group (σ1² and σ2²) = 1² = 1
The formula for combined variance σ² for two groups is:
σ² = [(n1 * σ1²) + (n2 * σ2²) + (n1 * (x̄1 - x̄)²) + (n2 * (x̄2 - x̄)²)] / nSubstitute the values:
σ² = [(10 * 1) + (5 * 1) + (10 * (10 - 8)²) + (5 * (4 - 8)²)] / 15
= [10 + 5 + 10 * 4 + 5 * 16] / 15
= [15 + 40 + 80] / 15
= 135 / 15
= 9
Thus, the standard deviation σ is:
σ = √9 = 3
The standard deviation of the combined series is 3.
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