Q.The means of two samples of sizes 60 and 120 respectively are 35.4 and 30.9 and the standard deviations 4 and 5. Obtain the standard deviation of the sample of size 180 obtained by combining the two samples.

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 combined set whenever the two groups have different means, because part of the combined spread comes from the groups being centred at different places, not just from spread within each group. The same formula can be used in reverse to find an unknown group's mean or standard deviation when the combined statistics and the other group's statistics are known.