Statistical Measures: Mean, Median, Mode and Range — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the three averages and the simplest measure of spread. By the end of this guide you should be able to calculate the mean, median, mode and range from a list of values, and know which average suits which situation.

You should also be able to find averages from a frequency table, estimate the mean from grouped data, work backwards from a mean to a missing value, and explain the effect of an outlier on each measure.

The organising idea is that an average is a single number standing in for a whole set, and the three averages stand in differently. The mean shares the total out equally, so every value influences it. The median takes the value in the middle, so only position matters and size does not. The mode reports what happens most often, so it ignores everything else entirely. None is more correct than the others — they answer different questions, and a question asking which is "most appropriate" is asking which of those three behaviours suits the data in front of you.

Key terms and definitions

Mean — the total of the values divided by how many there are.

Median — the middle value when the data is in order.

Mode — the most frequently occurring value.

Range — the largest value minus the smallest. A measure of spread, not an average.

Outlier — a value far away from the rest of the data.

Frequency — how many times a value occurs.

Modal class — the class with the highest frequency, used when data is grouped.

Bimodal — having two modes.

Core concepts

The three averages and the range

The mean is the sum of the values divided by the number of values. For 4, 7, 9 and 12, the total is 32 and there are 4 values, so the mean is 8.

The median is the middle value once the data is in order. Ordering first is not optional, and forgetting it is the commonest error in the whole topic.

The mode is the value occurring most often. A set can have more than one mode, or none at all if every value appears once.

The range is largest minus smallest. It measures spread, not average, and a question asking for "an average and a measure of spread" wants one of the first three plus this.

Finding the median position

For n values in order, the median sits at position (n + 1) ÷ 2.

With 11 values, (11 + 1) ÷ 2 = 6, so the median is the 6th value.

With 10 values, (10 + 1) ÷ 2 = 5.5, which means the median lies halfway between the 5th and 6th values — so add those two and halve.

Note this formula gives the position, not the value. Writing "the median is 5.5" when the position was 5.5 is a frequent slip.

Choosing the right average

The mean uses every value, which makes it a good summary when the data is evenly spread. That same property makes it vulnerable: a single extreme value drags it noticeably.

The median is unaffected by extremes, since it depends only on position. It is the right choice when the data is skewed or contains an outlier.

The mode is the only average available for categorical data such as favourite colour or shoe size sold, and it is what a shop means by its most popular item.

A question asking which is most appropriate wants a reason, usually about outliers: "the median, because the value of 200 is much larger than the others and would distort the mean".

The effect of an outlier

An outlier moves the mean and the range a great deal, and the median and mode hardly at all.

For the values 3, 4, 5, 6 and 82: the mean is 20, which is larger than four of the five values and describes none of them. The median is 5, sitting comfortably among the data.

The range is 79, driven entirely by the outlier, whereas removing it would leave a range of 3.

This asymmetry is exactly why the median-and-interquartile-range pairing exists for skewed data.

Averages from a frequency table

Where a table lists values with their frequencies, the same definitions apply but the arithmetic changes.

For the mean, multiply each value by its frequency, add those products, then divide by the total frequency. Adding a "value × frequency" column to the table makes this visible and earns method marks.

For the median, find the total frequency, use (n + 1) ÷ 2 to get the position, then count through the frequencies until you reach it. A running total column makes the counting reliable.

For the mode, read off the value with the highest frequency — not the highest value, and not the highest frequency itself. If the frequencies are 3, 9 and 5 for values 10, 20 and 30, the mode is 20, not 9.

For the range, subtract the smallest value present from the largest, ignoring the frequencies entirely.

Estimating the mean from grouped data

Once data is grouped into classes the original values are lost, so the mean can only be estimated.

Use the midpoint of each class to stand for every value in it. Multiply each midpoint by its frequency, add the products, and divide by the total frequency.

Divide by the total frequency, not by the number of classes — there were far more data values than classes.

With grouped data the mode becomes the modal class, given as an interval rather than a single number, and the range can only be estimated too, from the extremes of the outer classes.

Working backwards from a mean

Many questions give a mean and ask for a missing value, which works because the mean fixes the total.

If five numbers have a mean of 12, the total must be 5 × 12 = 60. Knowing four of them, the fifth is 60 minus their sum.

The same idea handles combined groups. If one group of 10 has a mean of 6 and another of 15 has a mean of 8, their totals are 60 and 120, so the combined mean is 180 ÷ 25 = 7.2.

Averaging 6 and 8 to get 7 is wrong, because the groups are different sizes — the larger group pulls the combined mean towards its own.

Comparing two sets of data

A fair comparison gives one statement about an average and one about the spread, both in context.

"The second class scored higher on average, with a median of 62 compared with 55, and their marks were more consistent, with a range of 20 compared with 34."

Comparing only averages leaves out half the picture, since two sets can share an average while differing completely in consistency.

Worked examples

Example 1: All four measures from a list

Find the mean, median, mode and range of 7, 3, 9, 3, 8, 6.

The mean: the total is 7 + 3 + 9 + 3 + 8 + 6 = 36, and there are 6 values, so the mean is 36 ÷ 6 = 6.

The median: order the data first, giving 3, 3, 6, 7, 8, 9. With 6 values the position is (6 + 1) ÷ 2 = 3.5, so average the 3rd and 4th: (6 + 7) ÷ 2 = 6.5.

The mode: 3 appears twice and everything else once, so the mode is 3.

The range: 9 − 3 = 6.

Note the median is not a value from the list, which is normal with an even number of values.

Example 2: Mean from a frequency table

Twenty families were asked how many children they have. Two have 0, five have 1, eight have 2, four have 3 and one has 5. Find the mean.

Multiply each value by its frequency: 0 × 2 = 0, 1 × 5 = 5, 2 × 8 = 16, 3 × 4 = 12, 5 × 1 = 5.

Add the products: 0 + 5 + 16 + 12 + 5 = 38.

The total frequency is 2 + 5 + 8 + 4 + 1 = 20.

The mean is 38 ÷ 20 = 1.9 children.

An answer that is not a whole number is perfectly acceptable here, since a mean need not be a possible individual value.

Example 3: Working backwards

The mean of six numbers is 14. Five of them are 9, 11, 15, 18 and 12. Find the sixth.

The mean fixes the total: 6 × 14 = 84.

The five known values add to 9 + 11 + 15 + 18 + 12 = 65.

The sixth value is 84 − 65 = 19.

Check: adding 19 to 65 gives 84, and 84 ÷ 6 = 14. ✓

Common mistakes and how to avoid them

Not ordering the data before finding the median. Order it every time, without exception.

Confusing the median's position with its value. (n + 1) ÷ 2 gives the position.

Giving the frequency as the mode. The mode is the value that occurs most often, not how often it occurs.

Dividing by the number of classes when estimating a mean. Divide by the total frequency.

Averaging two means of different-sized groups. Work with the totals instead.

Calling the range an average. It measures spread.

Using the mean when there is an obvious outlier. Choose the median and say why.

Exam technique for "Statistical Measures"

Write the ordered list out before finding a median or a range. It takes seconds and prevents the topic's most frequent error.

Add columns to a frequency table — "value × frequency" and a running total — rather than working in your head. Both carry method marks.

State which average you are using and why whenever a question asks what is most appropriate.

Show the total when working backwards from a mean, since that step is where the mark sits.

For comparisons, write two sentences with figures: one about the average, one about the spread.

Check a mean is plausible: it must lie between the smallest and largest values.

Quick revision summary

An average stands in for a whole set, and the three do it differently: the mean shares the total out so every value counts; the median takes the middle so only position counts; the mode reports what is commonest and ignores the rest.

Mean = total ÷ how many. Median = middle value, after ordering, at position (n + 1) ÷ 2. Mode = most frequent value. Range = largest − smallest, and it measures spread, not average.

An outlier drags the mean and the range a long way; the median barely moves. Choose the median for skewed data and say why.

From a frequency table: mean = total of (value × frequency) ÷ total frequency; median by counting through a running total; mode = value with the highest frequency, not the frequency itself.

From grouped data the mean is only an estimate, using class midpoints and dividing by the total frequency. The mode becomes the modal class.

Working backwards, a mean fixes the total: five numbers averaging 12 must total 60. Combine two groups through their totals, never by averaging their means.

Compare two sets with one average and one spread, quoted with figures and in context.