Which average should I use and when?

Use the mean when data is roughly symmetrical and you want every value to count, the median when there are outliers because it resists them, and the mode for qualitative data or the most common value. The mean is the only average affected strongly by extreme values.

How do I estimate the mean from grouped data?

Use the midpoint of each class as the value, multiply each midpoint by its frequency, sum these, then divide by the total frequency. It is only an estimate because the exact values within each class are unknown.

What is the difference between the range and the interquartile range?

The range is the maximum minus the minimum and is distorted by outliers. The interquartile range is the upper quartile minus the lower quartile, measuring the spread of the middle half, so it ignores extreme values and is preferred for skewed data.

How do I calculate standard deviation?

Find the mean, then use the square root of the mean of the squared values minus the mean squared, that is the square root of $\frac{\sum x^2}{n} - \bar{x}^2$. From a frequency table, weight each value and its square by the frequency.

How should I compare two data sets?

Compare one average to contrast the typical value and one measure of spread to contrast consistency, and write both in context. Never compare two averages with no spread, or two spreads with no average.

EnglandStatistics

AQA GCSE Statistics Summarising data: a complete overview of averages, spread and standard deviation

A deep-dive AQA GCSE Statistics guide to Summarising data. Covers measures of central tendency, measures of spread, quartiles and the interquartile range, standard deviation, and comparing distributions, with the calculations and exam patterns AQA repeats.

Generated by Claude Opus 4.815 min read8382Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this module demands
Measures of central tendency
Measures of spread
Quartiles and the interquartile range
Standard deviation
Comparing distributions
How this module is examined
Check your knowledge

What this module demands

Summarising data is the calculation heart of the course. AQA tests the four averages, the measures of spread, standard deviation and, above all, comparing two data sets correctly in context. Most marks come from accurate calculation followed by a clear interpretation.

This guide walks through the five topics in specification order, then sets out the exam patterns AQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Measures of central tendency

The module opens with measures of central tendency: the mean, median and mode, averages from frequency tables, the estimated mean from grouped data, and the weighted mean. The mean from a frequency table, $\frac{\sum fx}{\sum f}$ , is a recurring calculation.

Measures of spread

Measures of spread covers the range, interquartile range and percentiles, the effect of outliers, and choosing a measure. The interquartile range is preferred when outliers are present because it uses only the middle half.

Quartiles and the interquartile range

Quartiles and the IQR covers finding quartiles from a list and from cumulative frequency, the interquartile range, percentiles and identifying outliers with the $1.5 \times \text{IQR}$ rule.

Standard deviation

Standard deviation covers variance and standard deviation, calculating it from a list and a frequency table, and interpreting it as spread about the mean. It uses every value, so it is more sensitive to outliers than the interquartile range.

Comparing distributions

Comparing distributions covers comparing two data sets with one average and one measure of spread, describing skewness, and writing the comparison in context. This is the skill examiners reward most heavily.

How this module is examined

A typical AQA profile for this module:

Averages. The four averages and the estimated mean from grouped data.
Spread. The range, interquartile range and percentiles, and the effect of outliers.
Standard deviation. Calculating it from a list or table and interpreting it.
Comparison. Comparing two distributions with an average and a spread, in context.

Check your knowledge

A mix of recall and calculation questions covering this module. Attempt them under timed conditions, then check against the solutions.

Find the mean of $5, 7, 9, 11$ . (2 marks)
Find the estimated mean for classes $0 \le t < 4$ and $4 \le t < 8$ with frequencies $3$ and $7$ . (3 marks)
Find the range of $3, 9, 4, 12, 7$ . (1 mark)
A data set has $Q_1 = 14$ and $Q_3 = 26$ . Find the interquartile range. (1 mark)
Find the standard deviation of $2, 4, 6, 8$ . (3 marks)
A distribution has mean $40$ and median $34$ . Describe its skew. (1 mark)
Which average is least affected by an outlier? (1 mark)
State the two things you must compare when comparing two data sets. (2 marks)

Solutions

Step 1: Q1 - mean of a list

The mean is the sum of all values divided by how many values there are:

\frac{5 + 7 + 9 + 11}{4} = \frac{32}{4} = 8

Step 2: Q2 - estimated mean from grouped data

Use each class midpoint as a representative value, multiply by frequency, sum the results, then divide by the total frequency:

Midpoints are $2$ and $6$ ; $\sum fx = 2 \times 3 + 6 \times 7 = 48$ ; estimated mean $= \dfrac{48}{10} = 4.8$ .

Step 3: Q3 - range

The range is the maximum value minus the minimum value, giving the total spread:

12 - 3 = 9

Step 4: Q4 - interquartile range

The interquartile range is $Q_3 - Q_1$ , measuring the spread of the middle half and ignoring extreme values:

26 - 14 = 12

Step 5: Q5 - standard deviation

First find the mean, then use $\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}$ . The mean of $2, 4, 6, 8$ is $5$ , and:

\sum x^2 = 4 + 16 + 36 + 64 = 120, \quad \sigma = \sqrt{\frac{120}{4} - 25} = \sqrt{5} \approx 2.24

Step 6: Q6 - skewness from mean and median

When the mean is greater than the median, the longer tail pulls the mean upward, indicating a positive (right) skew.

Answer: positively skewed, because the mean is greater than the median.

Step 7: Q7 - average least affected by outliers

Outliers shift the mean significantly but the median only moves by one rank position, so it is far more resistant to extreme values.

Answer: the median (or the mode).

Step 8: Q8 - comparing two distributions

A full comparison must state a measure of the typical value (an average) and a measure of how spread out the data is, both set in the context of the question:

One average for the typical value and one measure of spread for consistency, in context.

Sources & how we know this

AQA GCSE Statistics (8382) specification — AQA (2017)