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.

Behavioural objectives

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

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

What is grouped data?

Grouped data are data in which two or more values are put together under one cell of a frequency table.

• Class interval: the first column of the table, which holds the groups, for example 1-10, 11-20, 21-30.
• Frequency: the number of values that fall in a class interval.
• Use: data are grouped when there are many values and an ordinary frequency table would show no pattern.

What are the terms used with grouped data?

Five terms go with grouped data: class interval, class limits, class boundaries, class width and class mark.

• Boundaries of the next classes: 10.5-20.5, 20.5-30.5, and so on.
• Class marks of the next classes: (11 + 20) ÷ 2 = 15.5 and (21 + 30) ÷ 2 = 25.5, and so on.

Why do we group data?

We group data because a large data set is cumbersome to put in an ordinary frequency table.

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

• Number of scores: the list has 35 scores.
• Ordinary frequency table: it would show 32 different scores.
• Repeated scores: 14, 25 and 30 occur twice. Every other score occurs once.
• Problem: the table and its bar chart would show no pattern.
• Solution: organise the scores into groups or classes.

How do we form a grouped frequency distribution?

A grouped frequency distribution is a table that shows how many values fall in each class interval.

1. Find the lowest and the highest value. Here the range is 5 to 58.
2. Choose equal class intervals that cover the range, such as 1-10, 11-20, 21-30, 31-40, 41-50 and 51-60.
3. Make sure the intervals do not overlap, so each value fits in only one class.
4. Go through the data once. Make one tally mark in the correct class for each value.
5. Count the tally marks in each class to get the frequency.
6. Add the frequencies. The total must equal the number of values (35 here).

• Class width: each class covers 10 possible marks, so the class width is 10 marks.
• Tally marks: a stroke through four bars (shown as /) makes a group of five.

• Highest frequency: 9 students, in the classes 11-20 and 31-40.

What are the types of grouped data?

There are two types of grouped data: grouped discrete data and grouped continuous data.

What is grouped discrete data?

Grouped discrete data are whole-number values arranged in class intervals that do not overlap.

• Reason for no overlap: each mark can appear in only one interval.
• Wrong intervals: 1-10, 10-20 and 20-30, because 10 is in the 1st and 2nd classes, and 20 is in the 2nd and 3rd.
• Right intervals: 1-10, 11-20 and 21-30.
• Note: when we group discrete data, we treat it as though it were continuous.

What is grouped continuous data?

Grouped continuous data are values measured on a continuous scale, such as weight, height and time.

• Care needed: we must know where to place values that appear to lie between classes.
• Method: use class boundaries, which have no gaps between classes.

The table shows the weight of 50 students to the nearest kg.

• Highest frequency: 15 students, in the class 50-54.

How do we find class limits, boundaries, width and mark?

We find them from the class interval, using the weight table above.

What are class limits?

Class limits are the end numbers of a class interval.

• 1st class (40-44): the lower class limit is 40 and the upper class limit is 44.
• 2nd class (45-49): the lower class limit is 45 and the upper class limit is 49.

What are class boundaries?

Class boundaries are the exact ends of a class when data are given to the nearest unit.

• 1st class: 40-44 includes all weights from 39.5 kg to 44.5 kg.
• Lower class boundary: 39.5 kg. Upper class boundary: 44.5 kg.

There are two ways to find a boundary.

1. Method 1: add the upper limit of one class to the lower limit of the next class, then divide by 2.
2. Method 2: subtract 0.5 from the lower class limit, and add 0.5 to the upper class limit.

• 2nd class, lower boundary: (44 + 45) ÷ 2 = 44.5 kg.
• 2nd class, upper boundary: (49 + 50) ÷ 2 = 49.5 kg.
• 3rd class, lower boundary: 49.5 kg.
• 3rd class, upper boundary: (54 + 55) ÷ 2 = 54.5 kg.

What is class width?

Class width, also called class size, is the upper class boundary minus the lower class boundary.

• Formula: class width = upper class boundary – lower class boundary.
• 1st interval: 44.5 – 39.5 = 5.
• 2nd interval: 49.5 – 44.5 = 5.

What is the class mark (class mid-value)?

The class mark is the value exactly halfway between the two ends of a class.

• Formula: class mid-value = (lower class boundary + upper class boundary) ÷ 2.
• 1st interval: (39.5 + 44.5) ÷ 2 = 42.
• 2nd interval: (44.5 + 49.5) ÷ 2 = 47.
• Shortcut: (lower limit + upper limit) ÷ 2 gives the same answer, for example (40 + 44) ÷ 2 = 42.

What is the full working for the weight table?

Evaluation

1. Define grouped data and give two reasons why data are grouped.
2. Why is it wrong to use the class intervals 1-10, 10-20 and 20-30 for whole-number data?
3. For the class interval 61-70, find the class limits, class boundaries, class width and class mark.
4. 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, and so on.
   
   (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.
5. 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.

1. Copy and complete the table.
2. State the class interval with the highest frequency (the modal class).
3. Find the class widths of the last two class intervals.