Construct and interpret frequency polygons, histograms with equal and unequal class widths using frequency density, and cumulative frequency curves, and read the median and quartiles from a cumulative frequency curve.

What this dot point is asking

Grouped continuous data needs its own displays. CCEA expects you to draw and read frequency polygons, histograms (including with unequal class widths using frequency density), and cumulative frequency curves, and to estimate the median and quartiles from a cumulative frequency curve. The histogram with unequal widths and the cumulative frequency curve are among the most demanding representation skills in the course, and both appear regularly at Higher tier.

Frequency polygons

A frequency polygon shows the shape of grouped data by plotting the frequency against the midpoint of each class and joining the points with straight lines.

It is especially useful for comparing two distributions on the same axes, because two polygons overlaid show which group is higher or more spread out without the clutter of two sets of bars. The points are plotted at class midpoints, not boundaries, and the lines are not extended beyond the data.

Histograms and frequency density

A histogram looks like a bar chart but is for continuous grouped data, so the bars touch. The crucial idea is that area, not height, represents frequency.

To find a missing frequency from a histogram, reverse the process: read the frequency density (height), multiply by the class width, and you have the frequency.

Cumulative frequency

Cumulative frequency is a running total of the frequencies, used to estimate the median and quartiles for grouped data.

To read the curve, use the cumulative frequency (vertical) axis: the median is at $\tfrac{n}{2}$, the lower quartile $Q_1$ at $\tfrac{n}{4}$, and the upper quartile $Q_3$ at $\tfrac{3n}{4}$. Read across to the curve and down to the value axis. The interquartile range is $Q_3 - Q_1$, the spread of the middle half of the data, and these are all estimates because the exact values within each class are unknown.

Why this matters

These displays are the gateway to the spread topic: the cumulative frequency curve feeds straight into quartiles, the interquartile range and box plots, while the histogram links to frequency density and the shape of a distribution, which underlies the normal distribution at Higher tier. Reading a missing frequency from a histogram, and reading quartiles from a curve, are reliable sources of marks once the area-equals-frequency idea is secure. Misreading a histogram as a bar chart is one of the most common and costly errors in the whole subject.