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 Actor when Oscar was won	Frequency
20–29	1
30–39	28
40–49	36
50–59	15
60–69	6
70–79	1

Short answer

The mean value is computed as 44.5 years, which is not equal to the actual value.

Step-by-step solution

Given information

The frequency distribution for ages is as follows:

Age (in years) of Best Actor when Oscar was won	Frequency
20–29	1
30–39	28
40–49	36
50–59	15
60–69	6
70–79	1

The actual mean using the original list is 44.1 years.

Determine the midpoints of each class

The formula for the mean of frequency distribution is:

$\overline{x}=\frac{\sum \left(f\times x\right)}{\sum f}....\left(1\right)$

Here,

f represents the frequencies.

x denotes the midpoints of the classes.

The formula for determining the midpoints:

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

Compute midpoints for each class interval as follows:

Age (in years) of Best Actor when Oscar was won	Frequency (f)	Lower limit	Upper limit	Midpoint (x)
20–29	1	20	29	24.5
30–39	28	30	39	34.5
40–49	36	40	49	44.5
50–59	15	50	59	54.5
60–69	6	60	69	64.5
70–79	1	70	79	74.5

The computation for the sum totals is done as follows:

Age (in years) of Best Actor when Oscar was won	Frequency (f)	Midpoint (x)	$f\times x$
20–29	1	24.5	24.5
30–39	28	34.5	966
40–49	36	44.5	1602
50–59	15	54.5	817.5
60–69	6	64.5	387
70–79	1	74.5	74.5
Totals:	$\sum f=87$		$\sum f\times x=3871.5$

Substituting the values in the equation (1), you get:

$$\begin{gathered} \overline{x}=\frac{3871.5}{87} \\ =44.5 \end{gathered}$$

Thus, the mean age is computed as 44.5 years.

Compare the mean value with the actual mean value

The actual mean value is known to be 44.1 years.

The actual mean value is different from the one tabulated from the frequency distribution; that is, 44.5 years.