Outliers. In Exercise 3.54 on pages 126–127, we discussed outliers, or observations that fall well outside the overall pattern of the data. The following table contains two data sets. Data Set II was obtained by removing the outliers from Data Set I.

Data Set I Data Set II 0 12 14 15 23 10 14 15 17 0 14 15 16 24 12 14 15 10 14 15 17 14 15 16

a. Compute the sample standard deviation of each of the two data sets.

b. Compute the range of each of the two data sets.

c. What effect do outliers have on variation? Explain your answer.

Grouped-Data Formulas. When data are grouped in a frequency distribution, we use the following formulas to obtain the sample mean and sample standard deviation.

Grouped-Data Formulas x¯ = xi fi n and s =

(xi − x¯)

2 fi n − 1 , where xi denotes either class mark or midpoint, fi denotes class frequency, and n (= fi) denotes sample size. The sample standard deviation can also be obtained by using the computing formula s =

x 2 i fi − (xi fi)2/n n − 1 .

In general, these formulas yield only approximations to the actual sample mean and sample standard deviation