TMUA Statistics: Averages, Spread and Data

Averages and spread

The mean is total divided by count, so a changed total or count changes the mean predictably. The range and the interquartile range ($Q_3-Q_1$) measure spread.

Histograms

On a histogram the area of a bar is the frequency, so frequency $=$ frequency density $\times$ class width.

Grouped data

To estimate the mean, multiply each class midpoint by its frequency, add, and divide by the total frequency.

Reading data

Mean, median and mode respond differently to extreme values: adding one very large value raises the mean but barely moves the median. Correlation shows a link, not a cause.

Interpolation (estimating inside the data range) is more reliable than extrapolation (estimating outside it). On a cumulative frequency graph, read the median at half the total frequency.

Worked examples

Common mistakes

• Reading the height of a histogram bar as the frequency.
• Using the class boundary instead of the midpoint for grouped data.
• Averaging two means without weighting by group size.
• Forgetting that an estimate from grouped data is not exact.

Test yourself on this topic with our free practice questions.