Question

Sample standard deviation from Site A is s_A = $34.20, and sample standard deviation for Site B is s_B = $31.96. The sample standard deviation of difference is s_D = $4.7. Explain how to sample standard deviation of difference is so much less than the sample standard deviation of prices from both sites.

Answer 1

The individual standard deviations gives the spread of the individual random variables. On the hand the standard deviaiton of difference s_d actually gives the standard deviation of the differences. If the two variables are highly positively correlated then as one variable increases, the other also increases and as it decreases, the other also decreases, this makes the corresponding differences very very low and highly close to each other thus decreasing the standard deviation of differences.

Therefore while the individual standard deviations gives a spread of the individual data samples, the spread of the difference is defined by the standard deviation of differences.