Collection, Tabulation & Presentation of Grouped Data – SS1 Mathematics Lesson Note (Third Term)

Grouped data are data in which two or more values are put together in one class, shown as a class interval in the first column of a frequency table. Data are grouped when there are many values and an ordinary frequency table would show no pattern. Each class has limits, boundaries, a class mark and a class width.

Behavioural objectives

By the end of the lesson, students should be able to:

• explain what grouped data are and why data are grouped;
• organise raw data into a grouped frequency table with tally and frequency;
• state the class limits of a class interval;
• find the class boundaries, class width and class mark (mid-value) of a class interval;
• identify the class interval with the highest frequency.

What is grouped data?

Data are said to be grouped if two or more values are put together as one under one cell. In this case, the variable column (the first column) is known as the class interval. Other terms are associated with grouped data and they are listed below.

Class interval

Examples: 1-10, 11-20, 21-30, …

Class boundaries

Class boundaries are the possible extra length created for the class interval. Each boundary is 0.5 below the lower limit or 0.5 above the upper limit:

0.5-10.5, 10.5-20.5, 20.5-30.5, …

Class mark (mid mark), x

The class mark is found by adding the two ends of the class interval and dividing by 2:

(1 + 10) ÷ 2, (11 + 20) ÷ 2, (21 + 30) ÷ 2, …

Therefore, the class marks are 5.5, 15.5, 25.5, …

Class width

The class width is also known as the class size. It is the difference between the upper class boundary and the lower class boundary.

Why do we group data?

When a data set has a large number of values, it is cumbersome to prepare its frequency table. For example, below is a list of scores out of 60 obtained by SS 3 students in a test:

30, 12, 58, 23, 25, 14, 8, 20, 5, 35
27, 38, 53, 32, 36, 15, 14, 37, 13, 50
31, 19, 34, 51, 25, 30, 39, 10, 42, 33
55, 16, 45, 18, 56

The list has 35 scores. If the data are organised in a frequency table, the table will show 32 different scores. Each of them occurs once, except 14, 25 and 30, which occur twice.

The frequency table and the bar chart of this data would not be very useful because the result would show no pattern. To overcome this problem, we can organise the data into groups or classes.

How do we form a grouped frequency distribution?

Before we group the data, we consider the range first, which is 5 to 58. With this range, the data can be grouped into class intervals such as 1-10, 11-20, 21-30, 31-40, 41-50 and 51-60.

When data are divided into groups, the result is called a grouped frequency distribution. The groups or classes into which the data are arranged are called class intervals. The first class interval is 1-10, the second class interval is 11-20, and so on. Since each class interval covers 10 possible marks, we say that the class width is 10 marks. The frequency distribution table for this data is shown below.

What is grouped discrete data?

In grouped discrete data, the data are usually whole numbers. For this reason, the class intervals do not overlap, because each mark can appear in only one interval. It is wrong to use intervals such as 1-10, 10-20 and 20-30, because 10 appears in both the 1st and the 2nd class intervals, and 20 appears in both the 2nd and the 3rd class intervals.

However, when we group discrete data, we are treating it as though it were continuous.

What is grouped continuous data?

When dealing with continuous data, the variable is measured on a continuous scale. It is important to know where to place values that appear to be between groups or classes. For example, the frequency distribution below shows the weight of 50 students to the nearest kg.

This data is continuous, so we need to find the class boundaries (or the class mid-value) and the width of the class intervals.

Class limits

The end numbers of each class interval are known as the class limits of that interval. In the table above, the 1st class is 40-44. These figures give the class interval. The end numbers 40 and 44 are the class limits: 40 is the lower class limit and 44 is the upper class limit.

Similarly, for the 2nd class, the class interval is 45-49. 45 is the lower class limit and 49 is the upper class limit.

Class boundaries

When data are given to the nearest unit, the class interval 40-44 theoretically includes all weights from 39.5 kg to 44.5 kg. We say that the 1st class interval has class boundaries of 39.5 kg and 44.5 kg. 39.5 kg is the lower class boundary and 44.5 kg is the upper class boundary.

Each class boundary can be found by adding the upper limit of one class to the lower limit of the next class and dividing the result by 2.

For the 2nd class:

• Lower class boundary = (44 + 45) ÷ 2 = 44.5 kg
• Upper class boundary = (49 + 50) ÷ 2 = 49.5 kg

For the 3rd class:

• Lower class boundary = 49.5 kg
• Upper class boundary = (54 + 55) ÷ 2 = 54.5 kg

and so on.

Notice that, in this case, for each class interval:

• To obtain the lower class boundary, subtract 0.5 from the lower class limit.
• To obtain the upper class boundary, add 0.5 to the upper class limit.

Class width of a class interval

The class width is also called the class size.

Class width = upper class boundary – lower class boundary

For the 1st interval, class width = 44.5 – 39.5 = 5

For the 2nd interval, class width = 49.5 – 44.5 = 5

Class mid-value (class mark)

The mid-value of a class is known as the class mark. For a given class interval, the class mid-value is exactly halfway between the lower limit and the upper limit.

Class mid-value = (lower class boundary + upper class boundary) ÷ 2

For example, the class mid-value of the 1st interval = (39.5 + 44.5) ÷ 2 = 42

The class mid-value of the 2nd interval = (44.5 + 49.5) ÷ 2 = 47, and so on.

Evaluation

Question 1. In a particular company, the amount of money, to the nearest naira, spent by workers on transportation to work daily was recorded as follows:

30, 60, 120, 200, 80, 90, 74, 240, 236, 125, 40, 75, 110, 120
220, 130, 180, 60, 90, 112, 150, 210, 245, 135, 140, 80, 100, 125
215, 240, 50, 60, 180, 190, 180, 148, 120, 88, 138, 195, 248, 130
140, 150, 154, 208, 225, 65, 145

a) Construct a grouped frequency distribution of this data, taking equal intervals 0-49, 50-99, …
b) Find the class boundaries and the class marks of each class interval.
c) Use the frequency distribution to find the class interval with the highest frequency.
d) State the width of each class interval.

Question 2. The weights of a group of students, to the nearest kg, are given below:

65, 70, 60, 46, 51, 55, 59, 63, 68, 53, 47, 53, 72, 53, 67, 62, 64, 70, 57, 56
73, 56, 48, 51, 58, 63, 65, 62, 49, 64, 53, 59, 63, 50, 48, 72, 67, 56, 61, 64

With the class intervals 45-49, 50-54, 55-59 and so on, show the class boundaries, class marks, tally and frequencies, in that order.

Assignment

The weights, to the nearest kg, of a group of people are shown in the table below.

Use the table to answer questions 1 to 4.

1. Copy and complete the table.
2. What is the modal class?
   A. 51-60  B. 61-70  C. 71-80  D. 31-40
3. Find the class widths of the last two class intervals.
   A. 90.5 – 80.5 = 10  B. 50.5 – 40.5 = 10  C. 41 – 40 = 1  D. 70 – 51 = 19
   A. 90.5 – 70.5 = 20  B. 80.5 – 70.5 = 10  C. 100.5 – 90.5 = 10  D. 50.5 – 20.5 = 30
4. Estimate the mode of the frequency distribution.