Bhisham C. Gupta - Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

Example 2.6.1 (Ages of basketball fans) Find the mean of the grouped data that is the frequency distribution of a group of 40 basketball fans watching a basketball game (see Table 2.6.1).

Solution:Using formula ( 2.6.2), we have

Table 2.6.1 Age distribution of a group of 40 basketball fans watching a basketball game.

Class	Frequency ( )	Class midpoint ( )
	8	(10+20)/2 = 15	120
	10	(20+30)/2 = 25	250
	6	(30+40)/2 = 35	210
	11	(40+50)/2 = 45	495
	5	(50+60)/2 = 55	275

To compute the median of a grouped data set, follow the steps given below:

1 Step 1. Determine the rank of the median that is given by

2 Step 2. Locate the class containing the median and then proceed as follows:Add the frequencies of classes starting from class 1 and continue until the sum becomes greater than or equal to . Then, the class containing the median is identified.

3 Step 3. Once we identify the class containing the rank of the median, then the median is given by(2.6.3)

where

lower limit of the class containing the median

frequency of the class containing the median

class width of the class containing the median

Example 2.6.2(Median of a grouped data) Find the median of the grouped data in Example 2.6.1.

Solution:

1 Step 1. Rank of the median .

2 Step 2. Add the frequencies until the sum becomes greater than or equal to 20.5, that is,Stop at the class whose frequency is 6, so that the class containing the median is .

3 Step 3. Using (2.6.3), we have

To find the mode of a grouped data set is simple. This is because we need only to find the class with the highest frequency. The mode of the grouped data is equal to the midpoint of that class. But, if there are several classes with the same highest frequency, then there are several modes that are equal to the midpoints of such classes.

In Example 2.6.1, the mode is equal to the midpoint of the class , since it has the highest frequency, 11. Thus

The population and the sample variance of grouped data are computed by using the following formulas:

(2.6.4)

(2.6.5)

where and are as defined earlier in this section.

Example 2.6.3(Variance of a grouped data) Determine the variance of the grouped data in Example 2.6.1.

Solution:From the data in Table 2.1, we have

Substituting these values and in formula ( 2.6.5), we obtain

The population and the sample standard deviation are found by taking the square root of the corresponding variances. For example, the standard deviation for the grouped data in Example 2.6.1is

1 Use the frequency distribution table you prepared in Problem 4 of Section 2.3to do the following:Determine the mean, median, and mode of the grouped data.Determine the variance and the standard deviation of the grouped data.

2 Use the frequency distribution table you prepared in Problem 5 of Section 2.3, to do the following:Determine the mean, median, and mode of the grouped data.Determine the variance and the standard deviation of the grouped data.

3 Use the frequency distribution table you prepared in Problem 6 of Section 2.3to do the following:Determine the mean, median, and mode of the grouped data.Determine the variance and the standard deviation of the grouped data.

4 The following data give the systolic blood pressures of 30 US male adults whose ages are 30–40 years old:113122111119125113123122115115112117121116118116109109112116122109110115109115120122125111Determine the mean, median, and mode of these data.Determine the variance and the standard deviation of these data.Prepare a frequency distribution table for these data.Use the frequency distribution table of part (c) to determine the mean, median, and mode of the grouped data. Compare your results with those in part (a) and comment.Use the frequency distribution table of part (c) to determine the variance and the standard deviation of the grouped data. Compare your results with those in part (b) and comment.