Finding the mean, median, mode and range, averages from frequency tables, and the median and interquartile range from grouped data at Higher tier.

What this dot point is asking

AQA wants you to calculate the mean, median, mode and range, find averages from frequency tables, and at Higher tier estimate the mean and find the median and interquartile range from grouped data. Beyond the calculations, AQA wants you to compare two data sets using an average and a measure of spread together, which is where the interpretation marks lie.

The four basic measures

For the data $3, 5, 5, 8, 9$: the mean is $\dfrac{30}{5} = 6$, the median is the third value $5$, the mode is $5$, and the range is $9 - 3 = 6$. With an even number of values, the median is the mean of the two middle values. The mean uses every value (so it is affected by outliers), the median is resistant to outliers, and the mode is the only average usable for non-numerical data.

Averages from a frequency table

When data is given as values with frequencies, the mean is the sum of each value times its frequency, divided by the total frequency. Add a "value times frequency" column, total it, and divide by the total of the frequency column. The mode is the value with the highest frequency, and the median is found by counting through the cumulative frequencies to the middle position.

Grouped data at Higher tier

When data is grouped into class intervals, individual values are lost, so the mean can only be estimated using the midpoint of each class.

Interquartile range and comparison

The interquartile range (IQR) is the upper quartile minus the lower quartile, the spread of the middle half of the data. For ordered data the lower quartile is the value a quarter of the way through and the upper quartile three quarters through; the median is the halfway value. The IQR ignores the extreme values, so it is more reliable than the range when outliers are present. When comparing two data sets, quote one average (usually the median or mean) and one spread (range or IQR) and interpret both in context.

Choosing the right average

Part of the skill is knowing which average suits the data. The mean uses every value and is best for symmetric data with no extreme outliers, but a single very large value drags it up, making it misleading. The median ignores the actual size of extreme values and so represents skewed data, such as house prices or incomes, more fairly. The mode is the only average that works for categorical data (the most popular colour) and is useful when the most common value matters, such as the most frequent shoe size to stock. Exam questions often ask which average is most appropriate and why, so a one-line justification is worth rehearsing.

Finding the modal class and the effect of changes

For grouped data you cannot give a single mode; instead you state the modal class, the interval with the highest frequency. A subtle exam point is how changes to the data affect the averages: adding a constant to every value raises the mean, median and mode by that constant but leaves the range and IQR unchanged, while an extra outlier shifts the mean noticeably but the median barely at all. Reasoning about these effects, rather than just computing, is increasingly common in the reformed specification.