## 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?## Step-by-Step:

# 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**.

**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

## Answer:

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**l**and less than an upper value_{i}**u**._{i}**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 (**l**=_{2}**u**,_{1}**l**=_{3}**u**)._{2} -
Each value occurs within each group with a frequency
**f**. We have_{i}**f**,_{1}**f**, ... going up to_{2}**f**._{n} -
**f**is the product of each_{i}x_{i}**x**with each_{i}**f**. We have_{i}**f**,_{1}x_{1}**f**, ... going up to_{2}x_{2}**f**._{n}x_{n} -
**m**is the midpoint of each group. It is halfway between the lower value_{i}**l**and the upper value_{i}**u**._{i}**m**= (_{i}**l**+_{i}**u**) ÷ 2._{i} -
**f**is the product of each_{i}m_{i}**f**with each_{i}**m**. We have_{i}**f**,_{1}m_{1}**f**, ... going up to_{2}m_{2}**f**._{n}m_{n} -
**Σf**is the sum of each_{i}**f**in the column._{i}**Σf**._{i}= f_{1}+ f_{2}+ ... + f_{n} -
**Σf**is the sum of each_{i}m_{i}**f**in the column._{i}m_{i}**Σf**._{i}m_{i}= f_{1}m_{1}+ f_{2}m_{2}+ ... + f_{n}m_{n}

**x̄**(said "x bar") is shown below:

**Don't forget:**The

**f**'s and

_{i}**m**'s stand in for numbers. In our example above,

_{i}**f**= 2,

_{1}**m**= 3,

_{1}**f**= 4,

_{2}**m**= 8 etc.

_{2}**Σf**= 9 and

_{i}**Σf**= 82. We can calculate the mean,

_{i}m_{i}**x̄**:

x̄ = Σf

_{i}m_{i}/ Σf_{i}= 82 ÷ 9 = 9.1## 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