## The Lesson

We can find the mean of a set of numbers that are presented in a frequency table.## How to Find the Mean from a Frequency Table

Finding the mean from a frequency table is easy.## Question

The frequency table below shows the test scores for a class of students.## Step-by-Step:

# 1

Add another column onto the table, labelled

**Score × Number**. For each row of the table, multiply the entry in the**Score**column with the entry in the**Frequency**column. Enter the answer in the**Scores × Frequency**column.**Note:**The columns have been labelled**(1)**,**(2)**and**(3)**.**(3) = (1) × (2)**indicates the entry in column**(3)**are the product of the entries in column**(1)**and**(2)**.# 2

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 + 3 + 2 + 2 + 1 + 1 = 11

# 3

Add the numbers in the

**Scores × Frequency**column, and write the total underneath in the**Total**row.
10 + 18 + 14 + 16 + 9 + 10 = 77

# 4

Divide the total of the

**Scores × Frequency**column (77) by the total of the**Frequency**column (11).
77 ÷ 11 = 7

## Answer:

The mean of the test scores is 7.## A Formula to Find the Mean from a Frequency Table

There is a formula to find the mean from a frequency table. To use it, we must introduce some formal notation.-
Let each value be
**x**, where_{i}**i**= 1, 2...**n**.**n**is how many numbers there are. We have**x**,_{1}**x**, ... going up to_{2}**x**._{n} -
Each value
**x**occurs with a frequency_{i}**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} -
**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}x_{i}**f**in the column._{i}x_{i}**Σf**._{i}x_{i}= f_{1}x_{1}+ f_{2}x_{2}+ ... + f_{n}x_{n}

**x̄**(said "x bar") is shown below:

**Don't forget:**The

**x**'s and

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

_{i}**x**= 5,

_{1}**f**= 2,

_{1}**x**= 6,

_{2}**f**= 3 etc.

_{2}**Σf**= 11 and

_{i}**Σf**= 77. We can calculate the mean,

_{i}x_{i}**x̄**:

x̄ = Σf

_{i}x_{i}/ Σf_{i}= 77 ÷ 11 = 7## Interactive Widget

Here is an interactive widget to help you learn about finding the mean from a frequency table.## What Is a Frequency Table?

A frequency table shows how often (how*frequently*) each number appears in a list of numbers.

## What Is the Mean?

The mean is an average of a set of numbers. The mean is found by adding all the numbers together and dividing by how many numbers there are.## What's in a Name?

"Mean" comes from the Old French "meien", which comes from the Latin "medianus", meaning "middle".## Why Does the Formula Work?

The formula for finding the mean from a frequency table is shown below:In this formula,

**x**is each number and

_{i}**f**is the frequency with which each number occurs.

_{i}**Σ**meaning "sum of".

**Σf**is the sum of the frequencies and

_{i}**Σf**is the sum of each value multiplied by its frequency. Why does the formula work? The mean is found by adding all the numbers in a set together and then dividing by how many numbers there are in the set. For example, consider the set of numbers given below:

_{i}x_{i}The mean is found by adding up the seven numbers, and then dividing by seven (how many numbers there are):

**{1 + 1 + 1 + 2 + 3 + 3 + 3} ÷ 7**

**{(3 × 1) + (1 × 2) + (3 × 3)} ÷ 7**

**f**). This multiplies each number that appears in the set (

_{i}**x**). Hence each bracket is

_{i}**f**×

_{i}**x**, or

_{i}**f**. Each

_{i}x_{i}**f**is being summed, which gives

_{i}x_{i}**Σf**. Finally it is divided by how many numbers there are in total, which is the frequency of all the numbers in the set. This is the sum of the frequencies of each number in the set:

_{i}x_{i}**Σf**. Putting it together gives:

_{i}
x̄ = Σf

_{i}x_{i}/ Σf_{i}