Presenting data in graphs

A histogram is read for shape before it is read for numbers, because four features of the shape carry information no frequency table shows at a glance. The first is centre: the class where the bars are tallest is where the bulk of the patients lie, and it is a visual estimate of the median and the mode. The second is spread: the horizontal distance the bars occupy says how much the patients differ from one another, so two wards with the same average weight can look completely different if one distribution is narrow and the other wide. The third is skewness. A distribution is symmetric when the left and right halves of the histogram are rough mirror images. It is skewed to the right, or positively skewed, when a thin tail of unusually large values extends to the right of the main peak; waiting times, length of stay and drug doses are almost always right-skewed, because most patients are quick and a few take a very long time. It is skewed to the left when the long tail runs towards the low values. Skewness matters because it changes which average to report: in a strongly right-skewed distribution the mean is dragged upward by the few extreme values while the median barely moves, so a report that gives only the mean will overstate the typical patient. The fourth feature is modality. A histogram with a single peak is unimodal; one with two clearly separate peaks is bimodal, and a second peak is a hypothesis about mixed populations — a bimodal body-mass histogram in a general clinic may be one peak of children and one of adults, so the clinic should report the two groups separately rather than one average that describes nobody. Finally, look for unusual observations and for empty classes. A bar standing alone far from the rest, or a class interval containing no patients at all, is a prompt to check the data: a weight recorded as 7 kg instead of 70 kg, a misplaced decimal, or a genuine clinical threshold such as an admission rule that keeps the least severe patients out of the ward. A histogram never proves an error; it tells the nurse where to look.

Before a histogram is drawn, its author makes a decision the reader never sees: how wide each class interval will be and where its boundaries fall. The tables in this section use ten-kilogram classes, 40–49, 50–59 and so on. Printed boundaries are shorthand, and the shorthand hides a rule. If weights are recorded to the nearest kilogram, the class printed as 40–49 really covers 39.5 kg up to but not including 49.5 kg, so a weight of 49.5 kg belongs to the next bar. Stating the convention matters because two colleagues who choose different conventions will put the same patient into different bars. The width itself changes the picture. The same 200 patients drawn in 20-year classes may show one broad mound, while the same patients drawn in 5-year classes may reveal two humps that the wide classes averaged away. Neither picture is a lie; the second simply has finer resolution, and the first conceals a real feature of the population. The safe habits are to use equal widths, to keep the number of classes between about five and twenty for a teaching figure, and to print the boundaries so that a reader can check. Equal widths also keep the rule simple, because with equal widths the height of each bar is proportional to the frequency. When widths are unequal — often useful for a long right tail, where the last class might be “100 kg and over” — height is no longer the message. The quantity that must then be proportional to frequency is the area of the bar, that is frequency divided by class width, a quantity called density. Redrawing unequal classes at equal height makes the widest class look as common as the narrowest, which is exactly the error the histogram was supposed to prevent. Two related displays appear in the notes and deserve a word. Joining the midpoints of the tops of the bars gives a frequency polygon, a single line that is easier to superimpose on a second distribution when two groups must be compared. Drawing a box with a line at the median and whiskers out to the extreme values gives a box-and-whisker plot, a compact figure that lets a reader compare the centre and spread of a dozen wards side by side where a dozen histograms would be unreadable.

A graph can be arithmetically perfect and still mislead, and the choices that do the misleading are made before the drawing starts. The first is the baseline. In a bar chart the length of the bar is the message, so the vertical axis must begin at zero: cutting it off at 80 % turns a change from 88 % to 92 % into a bar that looks twice as tall as its neighbour. The distortion is not a matter of opinion; it follows from the fact that the eye compares lengths. A line graph is different, because there the message is the slope, and the slope depends on the scale of the axis. A line graph may legitimately start above zero when the values being followed are all close to 100 %, such as hand-hygiene compliance, provided that the axis is labelled and the reader is told. The same series drawn on a zero-based axis can look almost flat, which is honest about the size of the change but hides the trend; the answer is to show the axis, not to hide it. The second choice is the form of the marks. The eye compares angles and areas far less accurately than it compares lengths, which is why a pie chart is a good display for two or three shares and a poor one for eight, and why a three-dimensional pie — the circle tilted into an ellipse — makes the sectors at the front look larger than the sectors at the back. A donut chart has exactly the same geometry as a pie with the middle removed; it does not add information. The third choice is the order of the categories. A bar chart of nominal categories may be sorted by frequency, because those categories have no natural order, but an ordinal variable such as triage level already has an order, and sorting it alphabetically throws that order away. The fourth is completeness: a graph is a disclosure of method as well as of result. It should state the number of observations, the unit on each axis and the period covered; a percentage without its denominator cannot be audited. These rules are not etiquette. In infection control, in a mortality review or in a quality-improvement presentation to a manager, the graph is often the only part of the report that is read, and a reader persuaded by a truncated axis will support the wrong change. Drawing honestly means asking, before the figure is saved, what a suspicious colleague would need in order to disagree with me.