Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Celebrity Net Worth Listed below are the highest amounts of net worth (in millions of dollars) of celebrities. The celebrities are Tom Cruise, Will Smith, Robert De Niro, Drew Carey, George Clooney, John Travolta, Samuel L. Jackson, Larry King, Demi Moore, and Bruce Willis. Are the measures of variation typical for all celebrities?

250 200 185 165 160 160 150 150 150 150

Short answer

The range of the net worths is equal to $100.0 million.

The sample variance of the net worths is equal to 1034.4${\left(\text{million}\text{dollars}\right)}^2$.

The sample standard deviation of the net worth is equal to $32.2 million.

These computed values are not typical for all the celebrities because they have been computed from a sample consisting of only the highest amounts of the net worths of a few celebrities.

Step-by-step solution

Given information

A sample consisting of the highest amounts of the net worths of 10 celebrities (n) is given.

Computation of the measures of variation

The range of a sample is computed by deducting the minimum value from the maximum value.

$$\begin{gathered} \text{Range}=\text{Largest}\text{Value}-\text{Lowest}\text{Value} \\ =250-150 \\ =100.0\text{million}\text{dollars} \end{gathered}$$

Therefore, the range of net worths is equal to $100.0 million.

Thevariance$\left(s^2\right)$ and thestandard deviation (s)assess the amount of dispersion present in a set of values.

The sample variance is calculated as shown below:

$s^2=\frac{{\sum }_{1=1}^n{\left(x_i-\overline{x}\right)}^2}{n-1}$

Here,

x represents the observations, and

$\overline{x}$is the sample mean.

The sample mean is calculated as

.$$\begin{gathered} \overline{x}=\frac{{\sum }_{1=1}^n{x}_i}{n} \\ =\frac{250+200+...+150}{10} \\ =\frac{1720}{10} \\ \approx 172 \end{gathered}$$

Thus, the sample mean is $172.0 million.

Now, the variance is calculated as

$$\begin{gathered} s^2=\frac{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}{n-1} \\ =\frac{{\left(250-172\right)}^2+{\left(200-172\right)}^2+...+{\left(150-172\right)}^2}{10-1} \\ =\frac{9310}{9} \\ =1034.4 \end{gathered}$$

.

Therefore, the variance of the net worths is equal to $1034.4${\left(\text{million}\text{dollars}\right)}^2$.

The sample standard deviation is calculated as shown below

$$\begin{gathered} s=\sqrt{s^2} \\ =\sqrt{1034.4} \\ \approx 32.2 \end{gathered}$$

Therefore, the standard deviation of net worths is equal to $32.2 million.

 Step 3: Interpretation 

As the sample comprises only those celebrities who have the highest amounts of net worth, the measures of variation are not typical for all the celebrities as they do not represent the entire group of celebrities.