Grouped data, cumulative frequency and box plots

Estimated mean of grouped data

Once data is grouped, the individual values are gone. Use the midpoint of each class as a stand-in:

The word estimated is not decoration. The exam asks "why is this only an estimate?" and the answer is: the exact values within each class are unknown, so every value is treated as if it sat at the midpoint.

Modal class

The class with the highest frequency — but only when all classes have the same width. With unequal widths, the modal class is the one with the highest frequency density.

Cumulative frequency

Plot the cumulative total against the upper class boundary — not the midpoint, not the lower boundary. Then read off, for $n$ values:

The IQR measures the spread of the middle half of the data, so unlike the range it is not disturbed by one extreme value.

Box-and-whisker plots

A box plot shows five numbers: minimum, $Q_{1}$, median, $Q_{3}$, maximum. Comparing two box plots means comparing a measure of centre and a measure of spread, each in a separate sentence — a comparison of only one of them will not get full marks.