Statistics · Data representation
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Bar charts & pie charts
Why a bar chart carries the frequency in its height while a pie chart carries only the share in its angle — so two identical-looking pies can stand for wildly different numbers — and how to convert between frequency and angle both ways.
Statistics · Data representation
Bar charts & pie charts
Why a bar chart carries the frequency in its height while a pie chart carries only the share in its angle — so two identical-looking pies can stand for wildly different numbers — and how to convert between frequency and angle both ways.
Why it works
A chart is a frequency table redrawn so that something you can see carries the number. Which measurement does the carrying — a height, a slice of a circle — and what that measurement is compared against, is the whole subject. Read the wrong measurement and a perfectly honest picture will hand you a false fact.A bar chart: the height is the frequency. The categories sit along the bottom, one bar each, all the same width, with a gap between them. The equal widths are not decoration: width means nothing here, so the bars are made identical in every way except the one way that does mean something — height. The gaps matter too. The horizontal axis of a bar chart is a list of labels, not a number line. There is nothing between "Tea" and "Coffee" the way there is something between cm and cm, so the bars must not touch.
That gap is also the line between a bar chart and a histogram. A histogram sits on a continuous number line, its bars touch, its classes may be different widths, and it is the area of each bar that carries the frequency. A bar chart carries the frequency in the height, full stop: there is no class width to divide by, and "frequency density" means nothing on a chart of favourite drinks.Two bars per category: a dual (comparative) bar chart. Put Year 7 and Year 8 side by side inside each category, with a key saying which is which, and you can answer "within this category, which year group is bigger?" and "how does the shape of the two years' choices differ?". The reason it works is that every bar still starts at zero, so any bar may be compared by eye with any other bar anywhere on the chart. (Below, only the first pair is labelled; in every pair Year 7 is the left-hand bar.)One bar per category, split: a composite (stacked) bar chart. Now the two year groups are stacked, so the full height of a bar is the total for that category. Below is exactly the same data as the dual chart above; Year 7 is the lower section of every bar, labelled on the first one.The totals (, , ) are now readable at a glance — that is what you gained. What you paid: only the bottom section still starts at zero. To find the size of a section higher up you must subtract — read the value where it ends, read the value where it starts, and take the difference. In the swimming bar the Year 8 section runs from to , so it is , not . And because the upper sections all float at different heights, your eye cannot compare them honestly. Choose composite bars when the totals are the point; choose dual bars when the comparison is the point.
A pie chart: the angle is the share. The whole circle stands for the whole data set, and once round is . Sharing out the circle therefore means sharing out in the same ratio as the frequencies: one item of data is worth degrees, so a category with frequency gets
That is not a rule to memorise. is simply the fraction of the data in that category, and you are taking that fraction of a full turn. Because those fractions add to , the angles must add to — the check that catches nearly every slip. Run it backwards and the same sentence still works: if you know the angle and the total, ; and if you know an angle and its frequency but not the total, that sector is of everything, so .
The trap, made concrete. Two cinemas each drew a pie chart of the films their customers watched one evening. On each chart the shaded quarter-circle at the top right — a sector — is comedy; the three unshaded sectors are the other kinds of film, and they play no part in what follows.It is very tempting to say "the same number of people watched a comedy at each cinema". Put the totals back in and watch it fall apart. Cinema A sold tickets, so its comedy sector is people. Cinema B sold , so its identical-looking sector is people. Same picture; against .
Neither chart lied. A pie chart deliberately throws the total away — that is exactly what makes it good at showing how one whole is divided and useless for showing how big anything is. So you may compare two pie charts' proportions freely ("comedy was a quarter of the tickets at both cinemas"), and you may compare amounts only after you have put the totals back in. If the totals are not given, the amounts simply cannot be found.
Choosing the chart. "How many, and which is biggest?" is a bar chart question. "How is this one whole divided up?" is a pie chart question. "Compare two groups category by category" wants dual bars. "How big is each total, and what is it made of?" wants composite bars. And every chart carries its own check: bar frequencies must add to the total, pie angles must add to .