Mode or mean? Summarising data — Mathematics, 11–13
Categories can only have a mode; numbers can have a mode and a mean, and the question of the study decides which to use.
Two kinds of data, two toolboxes
Qualitative data are categories, such as favourite colour or favourite animal. Quantitative data are numbers that measure or count, such as siblings, shoe size or minutes of reading. The mode, the value or category that appears most often, can be found in both kinds. The mean can only be calculated for quantitative data: you cannot add blue to green and divide by two.
The question decides the measure
A summary measure is one value or category that describes a whole set of data. Which one to choose depends on the question of the study. To know “which colour was chosen most?”, the mode answers. To know “how many books did each pupil read, on average?”, the mean answers. A measure that answers a different question does not just fail to help: it can mislead.
Twelve shoe sizes
Twelve pupils have shoe sizes 35, 36, 36, 37, 36, 38, 37, 36, 35, 37, 36, 38. Mode: 36 appears 5 times (37 three times; 35 and 38 twice each). Mean: the sum is 437 and 437 ÷ 12 ≈ 36.4. A shop that wants to know which size to order most uses the mode, 36; 36.4 is not even a size. To answer “what is the average size of the class?”, use the mean.
A number is not always a quantity
Not everything written with numbers is quantitative data. The numbers on the shirts of a team, 3, 7, 7, 10 and 11, are just labels: you can add them and divide, 38 ÷ 5 = 7.6, but “the average shirt is 7.6” says nothing about the team. Before calculating a mean, ask yourself: does it make sense to add these values? If the numbers only identify things (shirts, codes, door numbers), the data are qualitative and only the mode can be found.
Which cake to bake, how long to wait
A bakery deciding which cake to bake most looks at the mode of the preferences: the flavour chosen by more customers. A school that wants to know how long pupils take to get there looks at the mean of the minutes. In the news, “the most common value” points to the mode and “on average” to the mean. Knowing which one you are being given helps you read the information.
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