Chapter 2, Problem 2.2.12RE

Construct a histogram, frequency polygon, and ogive, using relative frequencies for the data in Exercise 6.

To determine

To construct: The histogram, frequency polygon and ogive using relative frequency of the data.

Answer to Problem 2.2.12RE

Output using EXCEL software is given below:

Histogram:

Frequency polygon:

Ogive:

Explanation of Solution

Given info: The given data shown below:

90	83	80	73	70	51	68	79	70	71
72	74	67	54	81	66	62	63	68	57
66	96	78	55	60	66	57	71	60	85
76	98	77	88	78	81	64	66	77	93
70

Calculation:

Step-by-step procedure to construct frequency distribution table is as follows:

The number of classes is 6.

Here, the minimum number is 51 and maximum number is 98. Thus, the range is, $98-51=47$

Class width is obtained by dividing the range by the number of classes.

Thus,

$\begin{array}{c}\text{Class}\text{width}=\frac{\text{Range}}{\text{Number}\text{of}\text{calsses}}\\ =\frac{47}{6}\\ =7.8\\ \approx 8\left(\text{rounded}\text{up}\text{for}\text{convenient}\right)\end{array}$

Since, we have to make 6 classes for given data we take class width 9 for convenient.

Lower class limits:

• The class limits are obtained by taking the minimum value or less than minimum value as lower limit for the first class. For second class, the lower limit is obtained by adding the class width to the previous class lower limit. That is, $51+9=60$. Similarly, the lower limits for the remaining classes can be obtained.

Upper class limits:

• For the first class, the upper limit is one less than the lower limit of the second class that is, $60-0.5=59.5$. The other upper limits are obtained by adding the upper limit of the preceding class and the class width.

Thus, the lower class and upper class limits are as follows:

Lower limit	Upper limit
50.5	$60-0.5=59.5$
$50.5+9=59.5$	$59.5+9=68.5$
$59.5+9=68.5$	$68.5+9=77.5$
$68.5+9=77.5$	$77.5+9=86.5$
$77.5+9=86.5$	$86.5+9=95.5$
$86.5+9=95.5$	$95.5+9=104.5$

• Make a tally mark for each entry in the corresponding class and continue for all number of children in the data.
• The number of tally marks in each class represents the frequency, f of that class.

Thus, the frequency distribution table the distribution of declaration of independence is as follows,

Class	Tally	Frequency $f$
50.5-59.5	$\cancel{\left|\right|\left|\right|}$	5
59.5-68.5	$\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}\left|\right|$	12
68.5-77.5	$\cancel{\left|\right|\left|\right|}\cancel{\left|\right|\left|\right|}|$	11
77.5-86.5	$\cancel{\left|\right|\left|\right|}\left|\right||$	8
86.5-95.5	$\left|\right||$	3
95.5-104.5	$\left|\right|$	2

Relative frequency:

The portion of the data that falls in the corresponding class is relative frequency of that class.

$\begin{array}{c}\text{Relative}\text{frequency}=\frac{\text{Class}\text{frequency}}{\text{Sample}\text{size}}\\ =\frac{f}{n}\end{array}$

Thus, the relative frequency for each class is tabulated below,

Class	Frequency $f$	Relative frequency
50.5-59.5	5	$\frac{5}{41}=0.122$
59.5-68.5	12	$\frac{12}{41}=0.293$
68.5-77.5	11	$\frac{11}{41}=0.268$
77.5-86.5	8	$\frac{8}{41}=0.195$
86.5-95.5	3	$\frac{3}{41}=0.073$
95.5-104.5	2	$\frac{2}{41}=0.049$
Total	41

Histogram:

Step-by-step procedure to construct the histogram by using EXCEL software is given below:

• Enter the data in column A of new worksheet, one number per cell.
• Enter the boundaries in column B.
• Enter the relative frequency in column C.
• From the toolbar, select the Data tab, the select data analysis.
• In data analysis, select HISTOGRAM and click ok.
• In HISTOGRAM dialog box, type C1:C6 in the input range.
• Click ok and histogram shown below,

The shape of the distribution is positive skewed.

Frequency polygon:

• Enter midpoint and  relative frequency in column A and column B respectively.
• Select the insert tab from the toolbar and line chart option.
• Select 2-D line chart.

Ogive:

Software procedure:

• Enter upper class boundaries and cumulative frequency in column A and column B respectively.
• Select the insert tab from the toolbar and line chart option.
• Select 2-D line chart