What you'll learn
This topic covers the statistical measures — mean, median, mode and range — for both discrete and grouped data. In this guide you will learn how to calculate each average, how to find the range, how to estimate the mean from a grouped frequency table, and how to choose the most suitable average. These are core statistics skills used throughout GCSE.
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 common value.
Range — the difference between the largest and smallest values.
Modal class — the class with the highest frequency in grouped data.
Core concepts
Mean, median, mode and range
The mean is the sum of the values ÷ the number of values. The median is the middle value when ordered (average the middle two if there is an even number). The mode is the most frequent value, and the range (a measure of spread) is largest − smallest.
Finding the median position
For n values in order, the median is in position (n + 1) ÷ 2. With 11 values the median is the 6th; with 10 values it is the average of the 5th and 6th. Always order the data first.
Averages from a frequency table
For discrete data in a frequency table, the mean is Σ(value × frequency) ÷ Σfrequency. The mode is the value with the highest frequency, and the median is found by counting through the frequencies to the middle position.
Estimating the mean of grouped data
For grouped data, use the midpoint of each class as the value: mean ≈ Σ(midpoint × frequency) ÷ Σfrequency. This is an estimate, because the exact values are unknown. The modal class is the class with the highest frequency.
Choosing the best average
The mean uses all the data but is affected by outliers; the median is better for skewed data; the mode suits the most common value or categorical data. Pick the average that best represents the situation.
Worked examples
Example 1: Mean
Find the mean of 4, 7, 9, 10.
(4 + 7 + 9 + 10) ÷ 4 = 30 ÷ 4 = 7.5.
Example 2: Median
Find the median of 3, 8, 2, 7, 5.
Order: 2, 3, 5, 7, 8. Middle value = 5.
Example 3: Estimated mean (grouped)
Two classes: midpoint 5 (freq 4), midpoint 15 (freq 6). Estimate the mean.
(5×4 + 15×6) ÷ 10 = (20 + 90) ÷ 10 = 11.
Common mistakes and how to avoid them
Not ordering for the median. Always sort the data first.
Forgetting to multiply by frequency. Use value × frequency for the mean.
Using boundaries not midpoints. Grouped mean uses class midpoints.
Calling the grouped mean exact. It is an estimate.
Confusing modal class with frequency. The mode is the class, not its frequency.
Exam technique for Statistical Measures
Mean = sum ÷ count; remember × frequency in tables.
Order the data to find the median at position (n + 1) ÷ 2.
Use midpoints to estimate the mean of grouped data.
Identify the modal class as the most frequent class.
Choose the average that best fits the data.
Quick revision summary
The four measures are: mean (sum of values ÷ number of values), median (the middle value when ordered, at position (n + 1) ÷ 2, averaging the middle two if even), mode (the most frequent value), and range (largest − smallest, a measure of spread). For discrete frequency tables, the mean is Σ(value × frequency) ÷ Σfrequency. For grouped data, estimate the mean using the midpoint of each class — Σ(midpoint × frequency) ÷ Σfrequency — and identify the modal class as the most frequent class; this mean is only an estimate because exact values are unknown. Choose the mean for full data (but beware outliers), the median for skewed data, and the mode for the most common or categorical value. The common errors are not ordering for the median, forgetting to multiply by frequency, using boundaries instead of midpoints, and calling the grouped mean exact. Order data, multiply by frequency, use midpoints for grouped data, and pick the most suitable average.