Averages and range

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The mean and the range

An average is one number that represents a whole set of data. The mean is the one most people mean by “average”: add up all the values and divide by how many there are. If a waiter's tips over five days were £42, £38, £55, £47 and £33, the total is £215 and the mean is £215 ÷ 5 = £43.

The range measures how spread out the data is: largest value minus smallest value. For the tips, £55 − £33 = £22. A small range means the values are consistent; a large range means they vary a lot. The range is a single number — write £22, not “£33 to £55”.

Median and mode (Level 2)

The median is the middle value when the data is in order. Always put the numbers in order first — this is the step most often missed. With an odd number of values there is one middle value. With an even number there are two, and the median is halfway between them: for 7, 9, 10, 12, 14, 15, the middle pair is 10 and 12, so the median is 11.

The mode is the value that appears most often. It is the only average that works for non-numerical data, such as the most popular sandwich. A set can have more than one mode, or none at all.

Each average has a purpose. The mean uses every value but can be pulled by one extreme value — one very high salary lifts the mean pay. The median is not affected by extremes, which is why house prices and pay are often reported as medians.

Averages from tables and grouped data (Level 2)

When data is in a frequency table, multiply each value by its frequency, add these products, and divide by the total frequency. If 4 households have 0 children, 7 have 1, 6 have 2 and 3 have 3, the total number of children is 0 + 7 + 12 + 9 = 28 across 20 households, a mean of 1.4.

With grouped data, such as delivery times of 0–10, 10–20 minutes and so on, you do not know the exact values, so use the midpoint of each group as an estimate. Multiply each midpoint by its frequency, add, and divide by the total. The answer is an estimated mean, and you should say so.

Comparing two data sets

A fair comparison uses one average and one measure of spread, and puts them in context. For example: “Team A had a higher mean score (24 against 22), so on average they did better. Team A's range was smaller (6 against 15), so their scores were more consistent.” Level 1 usually asks for the mean and range; Level 2 adds the median, mode, grouped data and written comparisons.

Watch out

Common mistakes

• Finding the median without putting the data in order first.
• Dividing by the number of groups instead of the total frequency in a table.
• Giving the range as two numbers (“33 to 55”) instead of the difference.
• Comparing only averages and ignoring spread, or not linking the comparison to the context.

Real life

Where you'll use this at work or home

• Working out average weekly earnings when hours or tips vary.
• Comparing the performance of two shifts, shops or teams.
• Understanding news reports about average pay or house prices.
• Tracking fuel use, energy readings or sales and spotting unusual weeks.