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 / )	Frequency(f)
0-99	1
100-199	51
200-299	90
300-399	10
400-499	0
500-599	0
600-699	1

Short answer

The mean value computed from the table is 224.1 (1000 cells / $\mu L$), which is different from the actual value of 224.3 (1000 cells /$\mu L$ ).

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)
0-99	1
100-199	51
200-299	90
300-399	10
400-499	0
500-599	0
600-699	1

The actual measure of the mean of the blood platelet count for males is 224.3(1000 cells/$\mu L$ ).

Determine the midpoints of each class

The midpoint x is computed using the class limits as follows:

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

Therefore, the mean is computed from the frequency distribution by the formula:

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

Therefore, the values are tabulated as follows:

Blood Platelet Count of Males (1000 cells / )	Frequency (f)	Lower limit	Upper limit	Midpoint (x)
0-99	1	0	99	49.5
100-199	51	100	199	149.5
200-299	90	200	299	249.5
300-399	10	300	399	349.5
400-499	0	400	499	449.5
500-599	0	500	599	549.5
600-699	1	600	699	649.5

The mean is computed using the totals as follows:

Blood Platelet Count of Males (1000 cells / $\mu L$)	Frequency (f)	Midpoint (x)	$f\times x$
0-99	1	49.5	49.5
100-199	51	149.5	7624.5
200-299	90	249.5	22455
300-399	10	349.5	3495
400-499	0	449.5	0
500-599	0	549.5	0
600-699	1	649.5	649.5
Totals:	$\sum f=153$		$\sum f\times x=34273.5$

Substituting the values in the formula, you get:

$$\begin{gathered} \overline{x}=\frac{34273.5}{153} \\ =224.0098 \end{gathered}$$

Thus, the mean blood platelet count of males is 224.1(1000 cells / $\mu L$).

Compare the mean value with the actual mean value

Compare it to the computed mean value. The actual mean value is almost the same as the computed mean of 224.1 (1000 cells / ).

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