Question

In Exercises 29–32, find the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (Exercise 29) 36.2 years; (Exercise 30) 44.1 years; (Exercise 31) 224.3; (Exercise 32) 255.1..

Age (year) of Best Actress when Oscar was won	Frequency
20–29	29
30–39	34
40–49	14
50–59	3
60–69	5
70–79	1
80–89	1

Short answer

The mean value is computed as 36.2 years, which is equal to the actual value.

Step-by-step solution

Given information

The frequency distribution is given as follows:

Age (year) of Best Actress when Oscar was won	Frequency
20–29	29
30–39	34
40–49	14
50–59	3
60–69	5
70–79	1
80–89	1

The actual measure of mean is 36.2 years.

Determine the midpoints of each class

The mean for the frequency distribution is obtained using the formula

$\overline{x}=\frac{\sum \left(f\times x\right)}{\sum f}$, where fdenotes the frequencies and xrepresents the midpoints of the classes.

The midpoints x are computed for a class interval as follows:

$x=\frac{\text{Upper}\text{limit}+\text{Lower}\text{limit}}{2}$

Therefore, the values are tabulated as follows:

Age (in years) of Best Actress when Oscar was Won	Frequency (f)	Lower limit	Upper limit	Midpoint (x)
20–29	29	20	29	24.5
30–39	34	30	39	34.5
40–49	14	40	49	44.5
50–59	3	50	59	54.5
60–69	5	60	69	64.5
70–79	1	70	79	74.5
80–89	1	80	89	84.5

The mean for the values is computed using the formula

The computation for the sum totals is done as follows:

Age (in years) of Best Actress when Oscar was Won	Frequency (f)	Midpoint (x)	$f\times x$
20–29	29	24.5	710.5
30–39	34	34.5	1173
40–49	14	44.5	623
50–59	3	54.5	163.5
60–69	5	64.5	322.5
70–79	1	74.5	74.5
80–89	1	84.5	84.5
Totals:	$\sum f=87$		$\sum f\times x=3151.5$

Substituting the values in the formula, you get:

$$\begin{gathered} \overline{x}=\frac{3151.5}{87} \\ =36.2241 \end{gathered}$$

Thus, the mean age of the best actress when Oscar was won is 36.2 years.

Compare the mean value with the actual mean value

The actual mean value is the same as the one computed from the frequency distribution; that is, 36.2 years.