Finding the Mean from a Grouped Frequency Table
(KS3, Year 7)

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).

Step-by-Step:

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_i and less than an upper value 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 (l₂ = u₁, l₃ = u₂).
• Each value occurs within each group with a frequency f_i. We have f₁, f₂, ... going up to f_n.
• f_ix_i is the product of each x_i with each f_i. We have f₁x₁, f₂x₂, ... going up to f_nx_n.
• m_i is the midpoint of each group. It is halfway between the lower value l_i and the upper value u_i. m_i = (l_i + u_i) ÷ 2.
• f_im_i is the product of each f_i with each m_i. We have f₁m₁, f₂m₂, ... going up to f_nm_n.
• Σf_i is the sum of each f_i in the column. Σf_i = f₁ + f₂ + ... + f_n.
• Σf_im_i is the sum of each f_im_i in the column. Σf_im_i = f₁m₁ + f₂m₂ + ... + f_nm_n.

The formula for finding the mean, x̄ (said "x bar") is shown below:

Don't forget: The f_i's and m_i's stand in for numbers. In our example above, f₁ = 2, m₁ = 3, f₂ = 4, m₂ = 8 etc. Σf_i = 9 and Σf_im_i = 82. We can calculate the mean, x̄:

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... ... or any other two numbers between 1 and 5. The lowest the numbers can be are... ... but this would likely be an underestimate the values. The highest the numbers can be are... ... 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:

Top Tip

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.

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