Chapter 2, Problem 2UT

To determine

Compare the frequency histograms of men’s winning scores and women’s winning scores for different classes of 5, 7, and 10 and comment on general shape of the histograms.

Answer to Problem 2UT

The frequency histogram for the data on men’s and women’s winning scores with five classes is shown below:

The frequency histogram for the data on men’s and women’s winning scores with seven classes is shown below:

The frequency histogram for the data on men’s winning scores with ten classes is shown below:

The best choice for number of classes is seven.

Explanation of Solution

Calculation:

Class limits:

Class limits are the maximum and minimum values in the class interval.

Class Boundaries:

A class boundary is the midpoint between the upper limit of one class and the lower limit of the next class where the upper limit of the preceding class interval and the lower limit of the next class interval will be equal. The upper class boundary is calculated by adding 0.5 to the upper class limit and the lower class boundary is calculated by subtracting 0.5 from the lower class limit.

Frequency:

Frequency is the number of data points that fall under each class.

Men’s Winning Score with five classes:

From the given data set, the largest data point is 101 and the smallest data point is 50.

Class Width:

The class width is calculated as follows:

$\begin{array}{c}\text{Class width=}\frac{\left(\text{Largest data point}-\text{Smallest data point}\right)}{\text{Number of classes}}\\ \text{=}\frac{\left(101-50\right)}{5}\\ =10.2\\ \approx 11\end{array}$

The class width is 11. Hence, the lower class limit for the second class 61 is calculated by adding 11 to 50. Following this pattern, all the lower class limits are established. Then, the upper class limits are calculated.

The frequency distribution table is given below:

Class Limits	Class Boundaries	Frequency
50-60	49.5-60.5	2
61-71	60.5-71.5	13
72–82	71.5–82.5	8
83–93	82.5–93.5	5
94–104	93.5–104.5	4

Step-by-step procedure to draw the histogram using MINITAB software:

• Choose Graph > Bar Chart.
• From Bars represent, choose Values from a table.
• Under One column of values, choose Simple. Click OK.
• In Graph variables, enter the column of Frequency.
• In Categorical variables, enter the column of Winning Score Men.
• Click OK.

Thus, the histogram for men’s winning score with five classes is obtained.

Men’s Winning Score with seven classes:

From the given data set, the largest data point is 101 and the smallest data point is 50.

Class Width:

The class width is calculated as follows:

$\begin{array}{c}\text{Class width=}\frac{\left(\text{Largest data point}-\text{Smallest data point}\right)}{\text{Number of classes}}\\ \text{=}\frac{\left(101-50\right)}{7}\\ =7.28\\ \approx 8\end{array}$

The class width is 8. Hence, the lower class limit for the second class 58 is calculated by adding 8 to 50. Following this pattern, all the lower class limits are established. Then, the upper class limits are calculated.

The frequency distribution table is given below:

Class Limits	Class Boundaries	Frequency
50-57	49.5-57.5	1
58-65	57.5-65.5	3
66-73	65.5-73.5	13
74-81	73.5-81.5	5
82-89	81.5-89.5	6
90-97	89.5-97.5	2
98-106	97.5-106.5	2

Step-by-step procedure to draw the histogram using MINITAB software:

• Choose Graph > Bar Chart.
• From Bars represent, choose Values from a table.
• Under One column of values, choose Simple. Click OK.
• In Graph variables, enter the column of Frequency.
• In Categorical variables, enter the column of Winning Score Men.
• Click OK.

Thus, the histogram for men’s winning score with seven classes is obtained.

Men’s Winning Score with ten classes:

From the given data set, the largest data point is 101 and the smallest data point is 50.

Class Width:

The class width is calculated as follows:

$\begin{array}{c}\text{Class width=}\frac{\left(\text{Largest data point}-\text{Smallest data point}\right)}{\text{Number of classes}}\\ \text{=}\frac{\left(101-50\right)}{10}\\ =5.1\\ \approx 6\end{array}$

The class width is 6. Hence, the lower class limit for the second class 56 is calculated by adding 6 to 50. Following this pattern, all the lower class limits are established. Then, the upper class limits are calculated.

The frequency distribution table is given below:

Class Limits	Class Boundaries	Frequency
50-55	49.5-55.5	1
56-61	55.5-61.5	2
62-67	61.5-67.5	2
68-73	67.5-73.5	12
74-79	73.5-79.5	5
80-85	79.5-85.5	4
86-91	85.5-91.5	2
92-97	91.5-97.5	2
98-103	97.5-103.5	2
104-109	103.5-109.5	0

Step-by-step procedure to draw the histogram using MINITAB software:

• Choose Graph > Bar Chart.
• From Bars represent, choose Values from a table.
• Under One column of values, choose Simple. Click OK.
• In Graph variables, enter the column of Frequency.
• In Categorical variables, enter the column of Winning Score Men.
• Click OK.

Thus, the histogram for men’s winning score with ten classes is obtained.

Women’s Winning Score with five classes:

From the given data set, the largest data point is 101 and the smallest data point is 51.

Class Width:

The class width is calculated as follows:

$\begin{array}{c}\text{Class width=}\frac{\left(\text{Largest data point}-\text{Smallest data point}\right)}{\text{Number of classes}}\\ \text{=}\frac{\left(101-51\right)}{5}\\ =10\end{array}$

The class width is 10. Hence, the lower class limit for the second class 61 is calculated by adding 10 to 51. Following this pattern, all the lower class limits are established. Then, the upper class limits are calculated.

The frequency distribution table is given below:

Class Limits	Class Boundaries	Frequency
51-60	50.5-60.5	1
61-70	60.5-70.5	5
71–80	70.5–80.5	12
81–90	80.5–90.5	8
91–101	90.5–101.5	6

Step-by-step procedure to draw the histogram using MINITAB software:

• Choose Graph > Bar Chart.
• From Bars represent, choose Values from a table.
• Under One column of values, choose Simple. Click OK.
• In Graph variables, enter the column of Frequency.
• In Categorical variables, enter the column of Winning Score Women.
• Click OK.

Thus, the histogram for women’s winning score with five classes is obtained.

Women’s Winning Score with seven classes:

From the given data set, the largest data point is 101 and the smallest data point is 51.

Class Width:

The class width is calculated as follows:

$\begin{array}{c}\text{Class width=}\frac{\left(\text{Largest data point}-\text{Smallest data point}\right)}{\text{Number of classes}}\\ \text{=}\frac{\left(101-51\right)}{7}\\ =7.14\\ \approx 7\end{array}$

The class width is 8. Hence, the lower class limit for the second class 59 is calculated by adding 8 to 51. Following this pattern, all the lower class limits are established. Then, the upper class limits are calculated.

The frequency distribution table is given below:

Class Limits	Class Boundaries	Frequency
51-58	50.5-58.5	1
59-66	58.5-66.5	1
67–74	66.5–74.5	6
75–82	74.5–82.5	11
83–90	82.5–90.5	7
91-98	90.5-98.5	3
99-107	98.5-107.5	3

Step-by-step procedure to draw the histogram using MINITAB software:

• Choose Graph > Bar Chart.
• From Bars represent, choose Values from a table.
• Under One column of values, choose Simple. Click OK.
• In Graph variables, enter the column of Frequency.
• In Categorical variables, enter the column of Winning Score Women.
• Click OK.

Thus, the histogram for women’s winning score with seven classes is obtained.

Women’s Winning Score with ten classes:

From the given data set, the largest data point is 101 and the smallest data point is 51.

Class Width:

The class width is calculated as follows:

$\begin{array}{c}\text{Class width=}\frac{\left(\text{Largest data point}-\text{Smallest data point}\right)}{\text{Number of classes}}\\ \text{=}\frac{\left(101-51\right)}{10}\\ =5\end{array}$

The class width is 5. Hence, the lower class limit for the second class 56 is calculated by adding 5 to 51. Following this pattern, all the lower class limits are established. Then, the upper class limits are calculated.

The frequency distribution table is given below:

Class Limits	Class Boundaries	Frequency
51-55	50.5-55.5	1
56-60	55.5-60.5	0
61–65	60.5–65.5	0
66–70	65.5–70.5	5
71–75	70.5–75.5	4
76-80	75.5-80.5	8
81-85	80.5-85.5	6
86-90	85.5-90.5	2
91-95	90.5-95.5	0
96-101	95.5-101.5	6

Step-by-step procedure to draw the histogram using MINITAB software:

• Choose Graph > Bar Chart.
• From Bars represent, choose Values from a table.
• Under One column of values, choose Simple. Click OK.
• In Graph variables, enter the column of Frequency.
• In Categorical variables, enter the column of Winning Score Women.
• Click OK.

Thus, the histogram for women’s winning score with ten classes is obtained.

Comparison of men’s and women’s winning score:

Five classes:

From the histogram on men’s and women’s winning scores with five classes, the following can be observed:

• The data values of men’s winning scores fall within 50 and 101, and the data values of women’s winning scores range between 51 and 101.
• The shape of distribution of men’s winning scores is skewed to the right and there are no unusual observations in the data as not even one data point is far from the overall bulk of data.
• The shape of distribution of women’s winning scores is approximately mound-shaped and there are no outliers.

Seven classes:

From the histogram on men’s and women’s winning scores with seven classes, the following can be observed:

• The data values of men’s winning scores fall within 50 and 101, and the data values of women’s winning scores range between 51 and 101.
• The shape of distribution of men’s winning scores is almost skewed to the right and there are no unusual observations in the data as not even one data point is far from the overall bulk of data.
• The shape of distribution of women’s winning scores is approximately mound-shaped and there are no outliers.

Ten classes:

From the histogram on men’s and women’s winning scores with seven classes, the following can be observed:

• The data values of men’s winning scores fall within 50 and 101 and the data values of women’s winning scores range between 51 and 101.
• The shape of distribution of men’s winning scores is slightly skewed to the right and there are no unusual observations in the data as not even one data point is far from the overall bulk of data. There is only one peak in the distribution.
• The shape of distribution of women’s winning scores is skewed to the left and there is an unusual observation in the data as there are few observations that fall away from the overall bulk of data.