Q.The mean height of 200 students is 65 inches. The mean heights of boys and girls are 70 inches and 62 inches respectively and the standard deviations are 8 and 10 respectively. Find the number of boys and the combined S.D.

When two groups of data are pooled into one combined data set, the mean and standard deviation of the combined set can be found directly from the size, mean and standard deviation of each group, without returning to the individual observations. If group 1 has n₁ items with mean x̄₁ and standard deviation σ₁, and group 2 has n₂ items with mean x̄₂ and standard deviation σ₂, the combined mean is the size-weighted average x̄c = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂). The combined standard deviation additionally accounts for the fact that each group's own mean is offset from the combined mean; writing d₁ = x̄₁ − x̄c and d₂ = x̄₂ − x̄c, the combined standard deviation is σc = √{[n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)]/(n₁ + n₂)}. The d² terms are essential — simply averaging the two variances would understate the true spread of the combi …