Calculate the mean, median and mode of a set of data, find the range, and choose an appropriate average

What this dot point is asking

SEAB wants you to calculate the three averages - mean, median and mode - and the range of a data set, and to comment on which average best represents the data. Each average summarises a data set with a single number, but they measure slightly different things, so knowing the difference matters.

The answer

The mean

The mean is the everyday "average": add up all the values and divide by how many there are.

For $3, 5, 7$ the mean is $\dfrac{15}{3} = 5$. The mean uses every value, so it is affected by very large or very small ones.

The median

The median is the middle value when the data is arranged in order. If there is an even number of values, the median is the mean of the two middle ones.

For $2, 4, 6, 8, 10$ the median is $6$. The median is not pulled around by extreme values, so it can be a fairer "typical" value.

The mode

The mode is the value that appears most often. A data set can have one mode, more than one mode, or no mode if all values appear equally. The mode is the only average that works for non-numerical data, such as the most common colour.

The range

The range measures spread, not average:

A small range means the data is close together; a large range means it is spread out.

Averages from a frequency table

When data is given in a frequency table, the same ideas apply. The mode is the value with the highest frequency. The median is found by counting through the frequencies to the middle position. The mean is the total of (value times frequency) divided by the total frequency. Reading averages from a table is common once data has been organised.

Choosing an average

The mean is best when the data is fairly even. The median is better when there are a few extreme values that would distort the mean. The mode is best for the most popular category, and is the only average that works for non-numerical data such as favourite colour. Questions often ask you to justify the choice, so be ready to give a short reason.

Examples in context

Example 1. Comparing test scores. Two classes can have the same mean score but very different ranges. The class with the smaller range had more consistent results, while the larger range shows a wider spread of ability. Reporting the mean and the range together gives a fuller picture than the mean alone.

Example 2. The effect of an outlier. In the salaries $20, 22, 24, 26, 200$ (in thousands), the single large value pulls the mean up to $58.4$, which represents nobody well. The median of $24$ is a far more typical figure. This is exactly when the median is preferred over the mean.

Try this

• Cue. Find the mean of $2, 4, 6, 8$. Sum is $20$, count is $4$, so mean $= 5$.
• Cue. Find the median of $3, 9, 1, 7, 5$. Order to $1, 3, 5, 7, 9$; the middle is $5$.
• Cue. Find the mode and range of $6, 2, 6, 9, 6, 2$. The mode is $6$ and the range is $9 - 2 = 7$.