## The Lesson

We can find the mean of a set of numbers that are presented in a grouped frequency table.

## How to Find the Mean from a Grouped Frequency Table

Finding the mean from a grouped frequency table is slightly more complicated than finding the mean from a frequency table. This is because a grouped frequency table presents continuous data (whereas a frequency table presents discrete data). We only know which groups our data is in, not each of the values of our data. We use the midpoint of each group as our estimate of the values within each group (see Note).

## Question

The grouped frequency table below shows the test scores for a class of students. What is the mean test score? # 1

Add another column onto the table, labelled Midpoint. For each row of the table, find the midpoint of each group in the Score column. Add the lowest and higest number in each group and divide by 2.
• The midpoint of 1 - 5 = (1 + 5) ÷ 2 = 3.
• The midpoint of 6 - 10 = (6 + 10) ÷ 2 = 8.
• The midpoint of 11 - 15 = (11 + 15) ÷ 2 = 13.
• The midpoint of 16 - 20 = (16 + 20) ÷ 2 = 18.
Enter the answer in the Midpoint column. # 2

Add another column onto the table, labelled Frequency × Midpoint. For each row of the table, multiply the entry in the Frequency column with the entry in the Midpoint column. Enter the answer in the Frequency × Midpoint column. Note: The columns have been labelled (1), (2), (3) and (4). (4) = (2) × (3) indicates the entry in column (4) are the product of the entries in column (2) and (3).

# 3

Add another row at the bottom of the table, labelled Total. Add the numbers in the Frequency column, and write the total underneath in the Total row.
2 + 4 + 2 + 1 = 9 # 4

Add the numbers in the Frequency × Midpoint column, and write the total underneath in the Total row.
6 + 32 + 26 + 18 = 82 # 5

Divide the total of the Frequency × Midpoint column (82) by the total of the Frequency column (9).
82 ÷ 9 = 9.1

The mean of the test scores is 9.1. ## A Formula to Find the Mean from a Grouped Frequency Table

There is a formula to find the mean from a grouped frequency table. To use it, we must introduce some formal notation. • Our data is grouped. In each group, each value x is greater than a lower value li and less than an upper value ui. i is the number of each group, where i = 1, 2... n. n is how many groups there are. Because the values are continous, the upper value in one group becomes the lower value in the next group (l2 = u1, l3 = u2).
• Each value occurs within each group with a frequency fi. We have f1, f2, ... going up to fn.
• fixi is the product of each xi with each fi. We have f1x1, f2x2, ... going up to fnxn.
• mi is the midpoint of each group. It is halfway between the lower value li and the upper value ui. mi = (li + ui) ÷ 2.
• fimi is the product of each fi with each mi. We have f1m1, f2m2, ... going up to fnmn.
• Σfi is the sum of each fi in the column. Σfi = f1 + f2 + ... + fn.
• Σfimi is the sum of each fimi in the column. Σfimi = f1m1 + f2m2 + ... + fnmn.
The formula for finding the mean, (said "x bar") is shown below: Don't forget: The fi's and mi's stand in for numbers. In our example above, f1 = 2, m1 = 3, f2 = 4, m2 = 8 etc. Σfi = 9 and Σfimi = 82. We can calculate the mean, :
x̄ = Σfimi / Σfi = 82 ÷ 9 = 9.1

## Lesson Slides

The slider below gives another example of how to find the mean from a grouped frequency table. Open the slider in a new tab

## Interactive Widget

Here is an interactive widget to help you learn about finding the mean from a grouped frequency table.

## Why Use the Midpoint?

The grouped frequency table tells you how many values in our data are within each group. That means we don't know exactly what our original data is, only which groups it falls into. Let's look at the first row of the grouped frequency table used as an example in this lesson: We only know that there are 2 values in the range 1 - 5. We don't know exactly what those values are. They could be...
1, 2 3, 5 2.4, 4.7
... or any other two numbers between 1 and 5. The lowest the numbers can be are...
1, 1
... but this would likely be an underestimate the values. The highest the numbers can be are...
5, 5
... but this would likely be an overestimate the values. So we choose exactly halfway throught the group 1 - 5, which is 3. We assume that the 2 numbers in the 1 - 5 group are:
3, 3

## Grouped Frequency Tables Are for Continuous Data

A grouped frequency table is for continuous data. Continuous data can take any value (within a range). For example, it may take any value from 1 - 10: 1.5, 2.31, 3.05. This is unlike discrete data, which can only take certain values. For example: 1, 2, 3. It can't take values in between these values: it can't take 1.5.