Class Features of Grouped Data

When data is grouped into class intervals, each class has special features. These features have exact names, and CSEC exam questions often ask you to identify or calculate them.

In this lesson, you will learn how to find the class limits, class boundaries, class width, and class midpoint. These ideas are important when working with grouped frequency tables, histograms, frequency polygons, and grouped mean calculations.

What you will learn

• Identify lower and upper class limits
• Find lower and upper class boundaries
• Find the class width
• Find the class midpoint
• Understand why class boundaries are used in grouped data

Key Ideas

What is a Class Interval?

A class interval is a group or range of values used in a grouped frequency table.

For example, the table below shows grouped marks:

Mark	Frequency
10 - 19	4
20 - 29	7
30 - 39	5

The classes are 10 - 19, 20 - 29, and 30 - 39. Each class interval contains a group of values.

Class Limits

The class limits are the values written at the beginning and end of a class interval.

There are two class limits:

• The lower class limit is the smallest value stated in the class.
• The upper class limit is the largest value stated in the class.

For example, in the class 20 - 29:

Class Interval	Lower Class Limit	Upper Class Limit
20 - 29	20	29

Example 1: For the class 30 - 39, state the lower and upper class limits.

1. The first value written is 30.
2. The last value written is 39.
3. Answer: Lower class limit = 30, upper class limit = 39.

Class Boundaries

The class boundaries are the true limits of a class interval. They are used to remove gaps between classes.

For example, look at these classes:

Class	Next Class
10 - 19	20 - 29

There appears to be a gap between 19 and 20. The class boundaries remove that gap by using halfway values.

For data measured to the nearest whole number:

For the class 10 - 19:

• Lower boundary = 10 - 0.5 = 9.5
• Upper boundary = 19 + 0.5 = 19.5

So the class boundaries are 9.5 and 19.5.

Why Class Boundaries Are Important

Class boundaries are important because they show the true continuous range of a class. This is especially useful when drawing a histogram, where bars must touch because the data is continuous.

Class Interval	Class Boundaries
10 - 19	9.5 - 19.5
20 - 29	19.5 - 29.5
30 - 39	29.5 - 39.5

Notice that the upper boundary of one class is the same as the lower boundary of the next class. This means there are no gaps.

Class Width

The class width is the size of the class interval. It tells us how wide each class is.

To find the class width, subtract the lower class boundary from the upper class boundary.

For the class 20 - 29:

• Lower boundary = 19.5
• Upper boundary = 29.5
• Class width = 29.5 - 19.5 = 10

Example 2: For the class 60 - 64, find the class width.

1. Lower boundary = 60 - 0.5 = 59.5.
2. Upper boundary = 64 + 0.5 = 64.5.
3. Class width = 64.5 - 59.5 = 5.
4. Answer: 5

Class Midpoint

The class midpoint is the middle value of a class interval. It is also called the class mark.

The midpoint is used when estimating the mean from grouped data and when drawing frequency polygons.

To find the class midpoint, add the lower and upper class limits, then divide by 2.

For the class 40 - 49:

• Lower class limit = 40
• Upper class limit = 49
• Midpoint = (40 + 49) ÷ 2 = 89 ÷ 2 = 44.5

Example 3: For the class 50 - 54, find the class midpoint and width.

1. Midpoint = (50 + 54) ÷ 2.
2. Midpoint = 104 ÷ 2 = 52.
3. Boundaries are 49.5 and 54.5.
4. Width = 54.5 - 49.5 = 5.
5. Answer: Midpoint = 52, class width = 5.

Complete Worked Example

Example 4: For the class 20 - 29, state the class limits, boundaries, width, and midpoint.

Feature	Answer	Reason
Lower class limit	20	First value stated in the class
Upper class limit	29	Last value stated in the class
Lower class boundary	19.5	20 - 0.5
Upper class boundary	29.5	29 + 0.5
Class width	10	29.5 - 19.5
Class midpoint	24.5	(20 + 29) ÷ 2

Summary Table

Class Inter­val	Class Limits	Class Bound­aries	Class Width	Class Mid­point
10 - 19	10 and 19	9.5 and 19.5	10	14.5
20 - 29	20 and 29	19.5 and 29.5	10	24.5
30 - 39	30 and 39	29.5 and 39.5	10	34.5
50 - 54	50 and 54	49.5 and 54.5	5	52

Common Mistakes

• Confusing class limits with class boundaries.
• Using the class limits instead of the boundaries to find class width.
• Forgetting to subtract 0.5 from the lower limit and add 0.5 to the upper limit.
• Finding the midpoint by subtracting instead of averaging.
• Thinking the class width of 20 - 29 is 9. It is actually 10 when using boundaries.
• Forgetting that class midpoint is also called the class mark.

Exam-style multiple choice

Practice Questions

1. For the class 30 - 39, state the lower and upper class limits.
2. For the class 30 - 39, find the class boundaries.
3. For the class 40 - 49, find the class midpoint.
4. For the class 60 - 64, find the class width.
5. For the class 70 - 79, state the class limits, boundaries, width, and midpoint.
6. For the class 100 - 109, find the lower boundary and upper boundary.
7. For the class 15 - 19, find the class midpoint.
8. Explain the difference between class limits and class boundaries.

Answers

1. Lower class limit = 30, upper class limit = 39
2. 29.5 and 39.5
3. 44.5
4. 5
5. Limits: 70 and 79; boundaries: 69.5 and 79.5; width: 10; midpoint: 74.5
6. 99.5 and 109.5
7. 17
8. Class limits are the values written in the class interval. Class boundaries are the true limits found by going half a unit below and above the stated limits.