How to Find the Mean from a Grouped Frequency Table
Finding the Mean from a Grouped Frequency Table
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?
StepbyStep:
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
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 (l_{2} = u_{1}, l_{3} = u_{2}).

Each value occurs within each group with a frequency f_{i}. We have f_{1}, f_{2}, ... going up to f_{n}.

f_{i}x_{i} is the product of each x_{i} with each f_{i}. We have f_{1}x_{1}, f_{2}x_{2}, ... going up to f_{n}x_{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_{i}m_{i} is the product of each f_{i} with each m_{i}. We have f_{1}m_{1}, f_{2}m_{2}, ... going up to f_{n}m_{n}.

Σf_{i} is the sum of each f_{i} in the column. Σf_{i} = f_{1} + f_{2} + ... + f_{n}.

Σf_{i}m_{i} is the sum of each f_{i}m_{i} in the column. Σf_{i}m_{i} = f_{1}m_{1} + f_{2}m_{2} + ... + f_{n}m_{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, f_{1} = 2, m_{1} = 3, f_{2} = 4, m_{2} = 8 etc.
Σf_{i} = 9 and Σf_{i}m_{i} = 82. We can calculate the mean, x̄:
x̄ = Σf_{i}m_{i} / Σf_{i} = 82 ÷ 9 = 9.1