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 xi, where i = 1, 2... n. n is how many numbers there are. We have x1, x2, ... going up to xn.
  • Each value xi occurs 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.
  • Σfi is the sum of each fi in the column. Σfi = f1 + f2 + ... + fn.
  • Σfixi is the sum of each fixi in the column. Σfixi = f1x1 + f2x2 + ... + fnxn.
The formula for finding the mean, x̄ (said "x bar") is shown below:

Don't forget: The xi's and fi's stand in for numbers. In our example above, x1 = 5, f1 = 2, x2 = 6, f2 = 3 etc. Σfi = 11 and Σfixi = 77. We can calculate the mean, x̄:
x̄ = Σfixi / Σfi = 77 ÷ 11 = 7

Lesson Slides

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Interactive Widget

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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, xi is each number and fi is the frequency with which each number occurs. Σ meaning "sum of". Σfi is the sum of the frequencies and Σfixi 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:

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
It is possible to write this as:
{(3 × 1) + (1 × 2) + (3 × 3)} ÷ 7
The left-most number in each bracket is the frequency of each number (fi). This multiplies each number that appears in the set (xi). Hence each bracket is fi × xi, or fixi. Each fixi is being summed, which gives Σfixi. 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: Σfi. Putting it together gives:
x̄ = Σfixi / Σfi