Suppose that some measurements occur more than once and that the data x1, x2,..., xk are arranged in a frequency table as shown here:

The formulas for the mean and variance for grouped data are

Notice that if each value occurs once, these formulas reduce to those given in the text. Although these formulas for grouped data are primarily of value when you have a large number of measurements, demonstrate their use for the sample 1, 0, 0, 1, 3, 1, 3, 2, 3, 0, 0, 1, 1, 3, 2.
a. Calculate and s2 directly, using the formulas for ungrouped data.
b. The frequency table for the n = 15 measurements is as follows:

Observations Frequency i 2 where n f and 2X 1l f-4524 x 0123