Find the mean, median, mode and range of a data set, estimate the mean from grouped data, find the modal class, and use averages and range to compare two distributions.

What this dot point is asking

Averages summarise a data set with a single typical value, and spread measures how varied the data is. In the CCEA Handling Data strand you must find the mean, median, mode and range, estimate the mean from grouped data, identify the modal class, and use an average together with a measure of spread to compare two distributions. The comparison questions carry the AO2 and AO3 reasoning marks, so a good answer states both an average and a spread and interprets them in context.

The three averages and the range

Each average summarises the data differently, and each has its uses.

The mean uses every value but is pulled by extreme values (outliers). The median is unaffected by outliers, which makes it the better average for skewed data. The mode is the only average that works for non-numerical data.

Finding the median carefully

To find the median you must first order the data. For an odd number of values the median is the single middle one; for an even number it is the mean of the two middle values. With $n$ values, the median is at position $\tfrac{n + 1}{2}$, which helps locate it in a long list or a frequency table.

Averages from grouped data

When data is grouped into classes, you do not know the exact values, so you estimate the mean using the midpoint of each class as a representative value.

Comparing distributions

A full comparison uses two things: an average and a measure of spread. Compare the means or medians to say which group is higher on average, then compare the ranges (or interquartile ranges) to say which group is more consistent. The interpretation must be in context, for example "Class A scored higher on average and was also more consistent because its range was smaller."

Choosing which average to compare matters. If one data set contains an obvious outlier, comparing medians is fairer than comparing means, because the mean of that set is distorted. The mode is the right choice only when you care about the most frequent value, such as the most common shoe size a shop should stock. A reliable structure for a comparison answer is two sentences: one about the average ("on average, A is higher/lower than B") and one about the spread ("A is more/less consistent than B because its range or IQR is smaller/larger"). Marks are lost when a candidate quotes the numbers but never says what they mean for the two groups being compared, so always finish with the interpretation.

Why this matters

Averages and spread are the everyday language of statistics, used in reports, league tables and research, and the comparison questions are exactly the AO2 and AO3 reasoning CCEA rewards. The grouped-mean method links to histograms and frequency tables, and the idea of spread leads on to the interquartile range, cumulative frequency and box plots. Choosing the most suitable average for the data is a judgement the exam tests.