How to Find Mean of a Grouped Frequency Table

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If you have organized your observations in a frequency table, you can find the mean directly from the table without writing it out as a long list of numbers. Here you’ll learn how.

Rule

Mean: Grouped Data with Known Frequencies

You multiply every observational value $x_{i}$ by its frequency $f_{i}$ , add them together, then divide by the total number of observations:

Mean = \frac{x_{1} f_{1} + x_{2} f_{2} + \dots + x_{n} f_{n}}{Number of observations}

Here’s the rule with a more theoretical approach:

Rule

Mean: Grouped Data with Known Frequencies

Multiply every observational value $x_{i}$ by its frequency $f_{i}$ , then divide the sum of these values by the total number $N = \sum f_{i}$ of observations.

\overset{}{x} = \frac{x_{1} f_{1} + x_{2} f_{2} + \dots + x_{n} f_{n}}{N}

Example 1

Below is an example of a table of frequencies. A class of 20 students had a test in mathematics. Their grades were as follows:

Grade Frequency
0 1
1 2
2 3
3 7
4 7
Total 20

Find the mean grade.

\begin{matrix} \overset{}{x} = \frac{0 \cdot 1 + 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 7 + 4 \cdot 7}{20} \\ \overset{}{x} = \frac{57}{20} \\ \overset{}{x} = 2.85 \end{matrix}

The mean grade of the class was $2.85$ .

Mean: Grouped Data with Known Relative Frequency

If you have a table of frequencies but only know the relative frequencies, you can find the mean in the following way:

Theory

Mean: Grouped Data With Known Relative Frequency

Multiply each point with its relative frequency

r = \frac{f}{n}

, where

f

is the frequency and

n

is the total number of observations, and add all these values together.

\overset{}{x} = x_{1} r_{1} + x_{2} r_{2} + \dots + x_{n} r_{n}

Example 2

A school of 974 students has given you a table containing relative frequencies of the grades its students have achieved in Latin, and have asked you to find the mean Latin grade.

Grade Relative Frequency
$0$ $\frac{49}{974} = 0.050$
$1$ $\frac{78}{974} = 0.080$
$2$ $\frac{304}{974} = 0.312$
$3$ $\frac{306}{974} = 0.314$
$4$ $\frac{237}{974} = 0.243$
Total 1

\begin{array}{l} \overset{}{x} & = 0 \cdot \frac{49}{974} + 1 \cdot \frac{78}{974} + 2 \cdot \frac{304}{974} \\ + 3 \cdot \frac{306}{974} + 4 \cdot \frac{237}{974} \\ = 2.62 \end{array}

\begin{array}{l} \overset{}{x} & = 0 \cdot \frac{49}{974} + 1 \cdot \frac{78}{974} + 2 \cdot \frac{304}{974} + 3 \cdot \frac{306}{974} + 4 \cdot \frac{237}{974} \\ = 2.62 \end{array}

The school’s mean grade in Latin is

2.62

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What Is Arithmetic Mean?

What Is the Mean of Grouped Data?

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Grade	Relative Frequency

$0$	$\frac{49}{974} = 0.050$

$1$	$\frac{78}{974} = 0.080$

$2$	$\frac{304}{974} = 0.312$

$3$	$\frac{306}{974} = 0.314$

$4$	$\frac{237}{974} = 0.243$

Total	1