The mean, median, mode and range; finding averages from frequency tables and from grouped data using the midpoint and an estimated mean; and comparing distributions using an average and the range.

What this dot point is asking

Edexcel expects you to find the four key measures (mean, median, mode and range), to calculate them from frequency tables, to estimate the mean of grouped data using midpoints, and to compare two distributions using an average and the range. Knowing which average suits which situation, and handling grouped data carefully, are the main skills.

The four measures

Each measure summarises data differently, and each has strengths.

The mean uses all the data but is affected by extreme values (outliers). The median is resistant to outliers, which is why it is preferred for skewed data such as house prices or incomes. The mode is the only average for qualitative data (the most popular colour, say). The range measures spread but, like the mean, is sensitive to a single extreme value.

Averages from a frequency table

When data is in a frequency table, multiply each value by its frequency to find the mean efficiently.

For a table of shoe sizes with frequencies, the mean shoe size is the sum of (size $\times$ frequency) divided by the total number of people, not the average of the sizes alone.

Estimating the mean of grouped data

Grouped data hides the exact values, so the mean can only be estimated using the midpoint of each class as a stand-in.

For grouped data you also identify the modal class (the class with the highest frequency) and the class containing the median, rather than a single value.

Comparing distributions

To compare two sets of data, use one measure of average (to compare typical values) and one measure of spread (to compare consistency). For example, "Team A has a higher mean score, so they tend to score more, but Team B has a smaller range, so they are more consistent." Edexcel rewards a comparison in context, not just two numbers.

Choosing the right average

Part of the skill is knowing which average to use. The mean is best when the data is fairly symmetric with no extreme values, because it uses every value. The median is better for skewed data or data with outliers, such as salaries, where a few very large values would drag the mean up and misrepresent the typical case. The mode is the right choice for categorical data, where "average" can only mean "most common", such as the most popular shoe size a shop should stock. A question may give a context and ask which average is most appropriate and why, so be ready to justify the choice, not just calculate it.

Try this

Q1. Find the median of $7, 3, 9, 4, 7, 2, 8$. [2 marks]

• Cue. Order them: $2, 3, 4, 7, 7, 8, 9$. The middle (4th) value is $7$.

Q2. Five numbers have a mean of $12$. Four of them are $10, 11, 13, 15$. Find the fifth. [2 marks]

• Cue. The total must be $5 \times 12 = 60$; the four given sum to $49$, so the fifth is $11$.