Statistics

Frequency distributions- Grouped data

Frequently, business statistics deals with hundreds or even thousands of values in a set. In dealing with such a large amount of values, it is often easier to represent the data by dividing the values into equal-size groups known as classes, creating grouped data.

The number of values in each class is called the frequency, with the resulting chart called a frequency distribution or frequency table. The purpose of a frequency distribution is to organize large amounts of data in a more compact form without changing the essential information contained in those values.

Steps to construct a frequency distribution - grouped data

Step 1. Divide the data into equal-size classes.
Step 2. Record the frequency of values within each class.
Step 3. Write the frequency of each class in a column labeled "frequency (f)"

The following data (in thousands of dollars) represent the net annual income for a sample of taxpayers:

36 23 86 94 52 40 17 11 39 73
99 13 39 44 73 23 18 98 56 67
12 33 55 76 34 46 87 12 32 92

Graph this data set in a frequency table having 9 class intervals:

Class interval	Frequency
10-19	6
20-29	2
30-39	6
40-49	3
50-59	3
60-69	1
70-79	3
80-89	2
90-99	4

Number of classes

The type and number of classes for dividing the data are decided based on the available data. Classes are formed by specifying ranges that will be used to group the data. As a general guideline, between 5 and 20 classes are recommended. While choosing the number of classes ensure that the class width is equal for all class intevals, the classes should be non-overlapping and exhaustive.

Steps to divide the data into classes:

Step 1. Find the minimum and maximum value in the data set.
Step 2. Fix a number of equi-distant intervals which cover the entire range between the smallest and largest value without overlapping. These are termed as class intevals and their boundary points are known as class boundaries.
Step 3. The number of observations in the data that belong to a particular class interval is called class frequency.

Width of classes

The second step in constructing a frequency distribution for a quantitative data is to choose a width for the classes, which is usually the same for all classes.

Step 1. To determine an approximate class width, identify the largest and smallest data values.
Step 2. Once the desired number of classes has been specified, the class width is calculated by:

(It is rounded off to a more convenient round off value)

Cumulative frequency table for grouped data

For grouped data, the cumulative frequency is the total frequency of all data points less than or equal to the upper class boundary of the class interval under consideration.

Construct a cumulative frequency table for the given distribution:

Chest girth (cm)	75-79	80-84	85-89	90-94	95-99
Number of students	6	7	10	8	6

Solution:

Chest girth (cm)	Number of students	Cumulative frequency
75-79	6	6
80-84	7	13
85-89	10	23
90-94	8	31
95-99	6	37