Cumulative frequency, histograms and box plots
These three graphical displays are tested heavily in CCEA Paper 2 (calculator). Each requires you to draw, read, and interpret the display.
Cumulative frequency
Cumulative frequency (CF) is the running total of frequencies up to and including each class.
Plotting: plot the cumulative frequency against the upper class boundary (not the midpoint). Draw a smooth S-shaped curve through the plotted points. Always start from (lower bound of first class, 0).
Reading the graph:
- Median: the value at CF = n/2.
- Lower quartile (Q1): the value at CF = n/4.
- Upper quartile (Q3): the value at CF = 3n/4.
- IQR = Q3 − Q1.
- Percentile: the value at CF = (percentile/100) × n.
For the percentage above/below a given value: read off the CF at that value, then calculate the percentage.
Box plots (box-and-whisker diagrams)
A box plot displays the five-number summary:
- Minimum value (left whisker).
- Lower quartile (Q1) (left edge of box).
- Median (line inside box).
- Upper quartile (Q3) (right edge of box).
- Maximum value (right whisker).
The box covers the IQR (middle 50% of data). The whiskers extend to the min and max.
CCEA Higher: identify and mark outliers (values more than 1.5 × IQR from Q1 or Q3) if required.
Histograms
A histogram shows the distribution of continuous grouped data. Unlike a bar chart:
- The frequency density is plotted on the y-axis (not frequency).
- Frequency density = frequency ÷ class width.
- The area of each bar = frequency (so bars of different widths can be compared fairly).
- There are no gaps between bars.
Reading a histogram:
- To find frequency from a bar: frequency = frequency density × class width.
- To find the number of values above/below a boundary: sum the relevant frequencies.
Comparing distributions: use the same axis scale for histograms from two groups, and comment on shape, centre, and spread.
CCEA examiner context
CCEA Paper 2 typically provides a frequency table, asks you to complete the cumulative frequency table, draw the curve, draw a box plot, and compare two distributions. Histograms appear less often but are Higher tier. Always use a ruler for box plots.
⚠Common mistakes
- Plotting CF at midpoints instead of upper class boundaries.
- Not starting the cumulative frequency curve at zero.
- Reading median at n, not n/2 (always halve the total frequency).
- Frequency density on histogram: plotting frequency (not density) when class widths differ.
- Box plot with wrong quartile positions: whiskers should be at min/max, not at Q1 ± 1.5 IQR (unless outliers are marked).
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