Mean Calculations

Definition of Mean

The mean is a measure of average, often called the average. It is found by adding all the data values together and then dividing by the number of values. The mean gives a single value that represents the central point of a data set.

For example, if you have the numbers 4, 7, and 9, the mean is:

Sum = 4 + 7 + 9 = 20

Number of values = 3

Mean = $\frac{20}{3} \approx 6.67$

Calculating Mean from Raw Data

When given a list of individual data points (raw data), the steps to calculate the mean are:

• Add all the data points together to find the total sum.
• Count the total number of data points.
• Divide the sum by the number of data points.

This method is straightforward and works for any set of numbers.

For instance, if the marks scored by 5 students in a test are 12, 15, 18, 10, and 20, then:

Sum = 12 + 15 + 18 + 10 + 20 = 75

Number of values = 5

Mean = $\frac{75}{5} = 15$

Mean from Frequency Tables

Sometimes data is presented in a frequency table, where each value is paired with how often it occurs (its frequency). To find the mean from such a table:

• Multiply each data value by its frequency to find the total contribution of that value.
• Add all these products to get the sum of all data values.
• Add all the frequencies to find the total number of data points.
• Divide the sum of the products by the total frequency.

This method accounts for repeated values efficiently.

For example, consider the following frequency table of test scores:

Score	Frequency
5	2
7	3
9	5

Calculate the mean:

Sum of products = $5 \times 2 + 7 \times 3 + 9 \times 5 = 10 + 21 + 45 = 76$

Total frequency = $2 + 3 + 5 = 10$

Mean = $\frac{76}{10} = 7.6$

Interpretation and Use of the Mean

The mean is a useful measure because it summarises a data set with a single value that represents the central tendency 013 where the data tends to cluster.

However, the mean can be affected by extreme values (outliers). For example, if one value is much larger or smaller than the others, it can pull the mean towards it, making it less representative of most data points.

For example, consider the data set: 4, 5, 6, 7, 50. The mean is:

Sum = $4 + 5 + 6 + 7 + 50 = 72$

Number of values = 5

Mean = $\frac{72}{5} = 14.4$

Here, the value 50 is an outlier and raises the mean above most of the data points.

Despite this, the mean is very useful for comparing different data sets to see which has a higher or lower average value.