# Calculating Means From Frequency Tables

Do you remember how to calculate the mean (or the average)? It's the sum of all the scores divided by the total number of scores:

$\text{mean} = \dfrac{\text{Sum of the Scores}}{\text{Number of Scores}}$

## Example

For example, to find the mean of the scores:

1,17,18 and 12
we
1. Find the sum of the scores (add up the scores): $1 + 17 + 18 + 12 = 48$
2. Divide by the number of scores ($4$) to give $48 \div 4 = 12$
to calculate the mean of $12$.

Sometimes it isn't so simple as being given a list of scores. Our scores (or other data) might be presented in a frequency table like the one below. The frequency table lists each of the scores, together with its frequency (the number of times it occurs)

Score Frequency
5 6
6 4
7 3
8 2
9 1
10 1
This table tells us that the score 5 occurred 6 times, the score 6 occurred 4 times, the score 7 occurred 3 times, and so on.

To find the average of these scores we could do it the hard way, by listing out all the scores, adding them together and dividing by the number of scores like this:

$\dfrac{5 + 5+ 5 + 5+ 5+ 5 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 9 + 10}{\text{number of scores}}$
or we could use multiplication to total the scores:
$5 \times 6 + 6 \times 4 + 7 \times 3 + 8 \times 2 + 9 \times 1 + 10 \times 1 = 112$
and add up the numbers in the frequency column to give the number of scores:
$6 + 4 + 3 + 2 + 1 + 1 = 17$
Finally, we can calculate the mean:
$\dfrac{112}{17} = 6.588\dots$

Let's try another example.

### Example

Christo is watching some old British comedies on TV, and laughing his head off. Sam thinks this is hilarious. He decides to work out the average number of times that Christo laughs at an episode. So, he draws up the following frequency table:

Times Christo Laughs in an Episode Frequency
6 10
7 12
8 6
9 4
10 2

Now he needs to work out the mean number of laughing episodes. The answer is

\begin{align*} \text{mean} &= \dfrac{6 \times 10 + 7 \times 12 + 8 \times 6 + 9 \times 4 + 10 \times 2}{10 + 12 + 6 + 4 + 2}\\ &= \dfrac{60 + 84 + 48 + 36 + 20}{34}\\ &\approx 7.29 \end{align*}
So, the mean number of times that Christo laughs in an episode is 7.29 (correct to two decimal places).

## Summation Notation

We can actually use notation to make our formulas more compact.

The symbol $\Sigma$ means to add. So, we can write down the sum of our frequencies as $\Sigma f$, where $f$ runs over all the frequencies.

In the above example, $\Sigma f$ would be:

$\Sigma f = 10 + 12 + 6 + 4 + 2 = 34$

We can also use this notation to add up the products of our scores and their frequencies like this:

$\Sigma fx = 10\times 6 + 12 \times 7 + 6 \times 8 + 4 \times 9 + 2\times 10 = 248$

Finally, we use the notation $\overline{x}$ to stand for the mean, so we have the formula:

$\overline{x} = \dfrac{\Sigma fx}{\Sigma f} = \dfrac{248}{34} \approx 7.29$

We can also add columns and rows to our table to keep track of our calculations as follows:

$x$ $f$ $fx$
6 10 60
7 12 84
8 6 48
9 4 36
10 2 20
TOTALS 34 248
And then, we can quickly calculate
$\text{Mean} = \dfrac{248}{34} \approx 7.29$
That's the most efficient way to do it, Sam!

### Description

This chapter series is on Data and is suitable for Year 10 or higher students, topics include

• Accuracy and Precision
• Calculating Means From Frequency Tables
• Correlation
• Cumulative Tables and Graphs
• Discrete and Continuous Data
• Finding the Mean
• Finding the Median
• FindingtheMode
• Formulas for Standard Deviation
• Grouped Frequency Distribution
• Normal Distribution
• Outliers
• Quartiles
• Quincunx
• Quincunx Explained
• Range (Statistics)
• Skewed Data
• Standard Deviation and Variance
• Standard Normal Table
• Univariate and Bivariate Data
• What is Data

### Audience

Year 10 or higher students, some chapters suitable for students in Year 8 or higher

### Learning Objectives

Learn about topics related to "Data"

Author: Subject Coach
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