Mean, Median, Mode and Range

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1. 1

   Mean: add everything, divide by the count

   4, 7, 7, 9, 13 → 40 ÷ 5 = 8

   The mean uses every value, which is its strength and its weakness. It represents the whole data set, and a single extreme value drags it away from where most of the data actually sits.

2. 2

   Median: order first, then take the middle

   4, 7, 7, 9, 13 → median 7

   Ordering the values is not optional and is the step most often skipped. With an even number of values, average the middle two — which can give a median that is not one of the original values.

3. 3

   Mode: the value that appears most

   4, 7, 7, 9, 13 → mode 7

   A data set can have two modes, several, or none at all if every value appears once. The mode is the only average that works on non-numerical data such as favourite colours.

4. 4

   Range: largest minus smallest

   13 − 4 = 9

   The range measures spread, not average, and questions asking for 'the average and the range' want two different kinds of thing. A small range means the data is tightly clustered whatever the average happens to be.

5. 5

   Mean from a frequency table: multiply, total, divide

   Σ(value × frequency) ÷ Σfrequency

   Multiply each value by how often it occurred, total those products, then divide by the total frequency. Dividing by the number of rows instead of the total frequency is the defining error of frequency-table questions.

Choosing between them

Each average answers a slightly different question. The mean is the fair share if everything were redistributed equally. The median is the typical value, unaffected by extremes. The mode is the most common outcome, and the only one that works for categories.

The choice matters most when the data is skewed. House prices and salaries are almost always reported as medians, because a small number of very large values pull the mean upward to a figure that describes almost nobody.

• Mean — total ÷ count. Uses all data, sensitive to outliers
• Median — middle value in order. Ignores extremes
• Mode — most frequent. Works on categories
• Range — largest − smallest. Measures spread

What one extreme value does

Take 2, 3, 3, 4, 88. The mean is 20 — larger than four of the five values, and a poor description of the data. The median is 3, which describes it well. Whenever a data set contains an outlier, the median is the more honest summary.

This is worth showing rather than stating, because exam questions increasingly ask students to justify a choice of average rather than just to compute one. A student who has seen a mean of 20 in a data set topping out at 4 for most values can explain why convincingly.

Frequency tables and grouped data

In a frequency table, dividing by the number of rows rather than the total frequency is the standard mistake. If ten people scored 3 and two scored 8, there are twelve values and not two, and the division must reflect that.

For grouped data, the individual values are unknown, so the midpoint of each class is used as an estimate. The result is an estimate of the mean, and questions expect that word — omitting it can cost the mark even when the arithmetic is right.

How we teach averages

We teach the four together and always ask which one suits the data, rather than teaching them as four separate calculations. Computing an average is easy; choosing the appropriate one is the skill that carries marks and the one that transfers outside the exam.

For frequency tables we require the total frequency to be written down as its own step before dividing. It is one extra number on the page and it prevents the most common error in the whole statistics strand.