Consider a data set with m observations. If the data are sample data, you compute the sample standard deviation, s, whereas if the data are population data, you compute the population standard deviation, σ.
a. Derive a mathematical formula that gives σ in terms of s when both are computed for the same data set. (Hint: First note that, numerically, the values of . x and μ are identical. Consider the ratio of the defining formula for σ to the defining formula for s.)
b. Refer to the three data sets in Exercise 3.171. Verify that your formula in part (a) works for each of the three data sets.
c. Suppose that a data set consists of 15 observations. You compute the sample standard deviation of the data and obtain s = 38.6. Then you realize that the data are actually population data and that you should have obtained the population standard deviation instead. Use your formula from part (a) to obtain σ.