Question

Outliers. In Exercise 3.54 on pages 104-105, 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.

(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.

Short answer

Part (a) approx. 6.82 and 1.988

Part (b) 24 and 7

Part (c) Higher variance in second set of data.

Step-by-step solution

Part (a) Step 1. Given information.

The given data sets are:

Part (a) Step 2. Find the sample standard deviation for each data set.

Mean of first data:

$$\begin{gathered} \overline{x}=\frac{0+12+14+15+............+14+15+17}{14} \\ =\frac{189}{14} \\ =13.5 \\ \sigma =\sqrt{\frac{{\left(0-13.5\right)}^2+{\left(12-13.5\right)}^2+....+{\left(15-13.5\right)}^2+{\left(17-13.5\right)}^2}{14-1}} \\ \approx 6.8247 \end{gathered}$$

Mean of second data:

$$\begin{gathered} \overline{x}=\frac{10+14+15+....+14+15+16}{10} \\ =\frac{142}{10} \\ =14.2 \\ \sigma =\sqrt{\frac{{\left(10-14.2\right)}^2+{\left(14-14.2\right)}^2+.......+{\left(15-14.2\right)}^2+16-{\left(16-14.2\right)}^2}{10-1}} \\ \approx 1.988 \end{gathered}$$

Part (b) Step 1. Find the range for each data set.

Range$=\mathrm{Maximum}\mathrm{value}-\mathrm{Minimum}\mathrm{value}$

For data set I:

$24-0=24$

For data set II:

$17-10=7$

Part (c) Step 1. Compare the data.

The first data set's measures of variation are higher than the second data set's.

As a result, the first data set's measures of variance are higher than the second data set's.