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..

Blood Platelet Count of Males (1000 cells /$\mathbf{\mu L}$ )	Frequency(f)
100-199	25
200-299	92
300-399	28
400-499	0
500-599	2

Short answer

The computed mean of blood counts is 255.6(1000 cells / $\mu L$).It is different from 255.1(1000 cells / $\mu L$), the actual values.

Step-by-step solution

Given information

The distribution of frequencies of the blood platelet count of males is as follows:

Blood Platelet Count of Males (1000 cells / $\mu L$)	Frequency(f)
100-199	25
200-299	92
300-399	28
400-499	0
500-599	2

The actual measure of the mean of the blood platelet count of males is 255.1(1000 cells/$\mu L$ ).

Determine the midpoints of each class

The formula for midpoints using the upper and lower class limits is:

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

The formula for the mean of frequency distribution is:

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

Therefore, the values are tabulated as follows:

Blood Platelet Count of Males (1000 cells /$\mu L$)	Frequency (f)	Lower limit	Upper limit	Midpoint (x)
100-199	25	100	199	149.5
200-299	92	200	299	249.5
300-399	28	300	399	349.5
400-499	0	400	499	449.5
500-599	2	500	599	549.5

The mean is computed using the totals as follows:

Blood Platelet Count of Males (1000 cells /$\mu L$)	Frequency (f)	Midpoint (x)	$f\times x$
100-199	25	149.5	3737.5
200-299	92	249.5	22954
300-399	28	349.5	9786
400-499	0	449.5	0
500-599	2	549.5	1099
Totals:	$\sum f=147$		$\sum f\times x=37576.5$

Substitute the values in the formula to obtain:

$$\begin{gathered} \overline{x}=\frac{37576.5}{147} \\ =255.6224 \end{gathered}$$

Thus, the mean blood platelet count of males is 255.6(1000 cells / $\mu L$).

Compare the mean value with the actual mean value

The actual mean value is 255.1(1000 cells /$\mu L$).

The tabulated or computed values are 255.6(1000 cells /$\mu L$), which is different from 255.1, the actual value.

Thus, theactual mean value is different from the one computed from the frequency distribution.