13 Research methods, statistics, critical review and evidence-based practice13.8 Descriptive statistics
13.8.9.2 Graphical presentation
8 min read6 figures
Written by Resident Doctor Committee · Published
Bar charts and histograms
The gap between bars is structural: separated bars mean separate categories, and touching bars mean a continuous scale.
Graph | Data | Bars | Vertical axis |
|---|---|---|---|
Bar chart | Discrete categories, nominal or ordinal | Separated, over named categories | One summary value per category, such as a count, proportion or mean |
Histogram | One continuous variable | Touching, over contiguous class intervals | Frequency density |
A bar chart compares groups and often carries error bars.
A bar chart differs from a box plot: the bar gives one summary value per category, while the box plot shows each group's spread through its median, quartiles and whiskers.
A headline figure that pools adjacent categories, such as agree plus strongly agree, is not the proportion in either one; read each from its own segment.
A histogram shows the frequency distribution, so a skew or a second peak (a bimodal distribution) shows at a glance.
Frequency density is frequency divided by class width, so the area of a bar, not its height, carries the frequency.
With equal class widths, height is proportional to frequency; with unequal widths, plotting raw counts as heights exaggerates the wide bins.
For 378 admissions by length of stay:
Class interval | Admissions | Frequency density |
|---|---|---|
0 to under 7 days | 140 | 140 / 7 = 20 per day |
7 to under 14 days | 70 | 70 / 7 = 10 per day |
14 to 56 days | 168 | 168 / 42 = 4 per day |
The widest bin holds the most admissions but has the lowest density. Broad intervals hide detail, and narrow ones exaggerate noise.
A frequency polygon joins the midpoints of the tops of a histogram's bars (class frequencies plotted at class midpoints) with straight lines, closed down to the horizontal axis.
Other displays of one variable
A pie chart shows shares of a whole for one categorical variable: each sector's angle and area match its share, and the slices add to 100%.
Its categories must be mutually exclusive and exhaustive, so a pie of substances used fails when patients use several.
Similar angles, such as 22% and 25%, are harder to compare than lengths on a common baseline, and the denominator must be given, since one share may be 40 or 4,000 patients.
A stem-and-leaf plot lists raw digits in rows, each leaf digit being one observation, so every individual value can be read back.
A population pyramid shows the age and sex structure of a population. It is a pair of back-to-back horizontal bar charts of counts per age band, one per sex, either side of a shared age axis.
A Kaplan-Meier curve falls in steps from 1.0 over time and shows the proportion still event-free at each time point.
Each event steps the curve down.
Censored participants are marked on the curve and kept in until they leave.
Box and whisker plots
A box plot draws the five-number summary: minimum, lower quartile, median, upper quartile and maximum. All five are rank-based, so the plot does not show the mean, mode, variance or standard deviation.
Part | What it marks |
|---|---|
Line inside the box | Median |
Box edges | Lower quartile (25th centile, first quartile) and upper quartile (75th centile, third quartile) |
Box length | Interquartile range (IQR), holding the middle 50% of observations |
Whiskers | The extreme observations within 1.5 times the IQR of each box edge |
Separate points | Outliers beyond the whiskers |
On a horizontal plot, three quarters of the data lie below the right-hand edge of the box.
Outliers are plotted as separate points, not at the whisker end, and each whisker ends at a real observation rather than at the calculated limit.
With no outliers plotted, each whisker holds about a quarter of the observations.
Some plots also mark points beyond 3 IQR, and other conventions run whiskers to the minimum and maximum or to set centiles, so check the caption.
Skew shows as an off-centre median and unequal whiskers, and is named after the longer tail.
A box plot suits a skewed continuous variable and shows several groups side by side, but it hides sample size, clusters and multiple peaks.
Shape, axes and error bars
Shape | Appearance | Reading |
|---|---|---|
Right skew | Long upper tail, as in length of stay | The mean often exceeds the median; median and IQR describe a typical stay, while the mean may matter for resources |
Bimodal | Two peaks | May reflect subgroups, but does not prove two populations |
Floor or ceiling effect | Scores bunch at a scale limit | Limits measurable variation and change |
Isolated distant point | One value far from the rest | Check for entry error (3,000 mg for 300); keep genuine extremes unless exclusion is justified |
Bar lengths should usually start at zero: an axis restricted to 21 to 25% exaggerates the gap between 22% and 24%.
Three-dimensional perspective distorts sizes and makes values harder to read.
Error bar | Shows |
|---|---|
Standard deviation (SD) | Spread of the observations |
Standard error (SE) | Sampling precision of an estimate |
95% confidence interval | Uncertainty interval, often about the estimate plus or minus 1.96 SE |
A bar showing only a mean with an upward error bar hides the distribution. Descriptive counts need no error bars.
Scatter plots and regression
A scatter plot shows paired values of two continuous variables, one point per unit of observation, with the explanatory variable on the x-axis.
Adding a fitted regression line does not change the graph type.
Rising points with an upward-sloping line show the outcome increasing as the x-axis variable increases; nothing beyond the two plotted variables can be read from the plot.
Placing a variable on the x-axis does not establish causation.
An outlier departs from the trend of the others, with a residual (vertical distance from the fitted line) far larger than any other.
The point furthest from the line shows variation the predictor does not explain: for service spend against deprivation, this is what a postcode lottery means.
One extreme point can change a correlation greatly, and a strong inverted U can give a linear correlation near zero.
In Wilkinson and Pickett's graph of rich countries, the country with the lowest ratio of richest-fifth to poorest-fifth income is the most equal.
Logistic regression and nomograms
A logistic regression plot shows a binary outcome predicted from a continuous or ordinal variable as an S-shaped probability curve bounded by 0 and 1.
Its vertical axis is the estimated probability of the outcome coded 1, the dependent variable.
The shaded band around a fitted regression curve is the confidence interval of the estimated outcome at each value of the predictor.
Fagan's nomogram has three parallel scales: pre-test probability, likelihood ratio and post-test probability. A straight line through the first two reads off the post-test probability.
Forest plots
A forest plot has one row per study or comparison.
Element | Meaning |
|---|---|
Square | Point estimate, with area proportional to the study's weight in the pooled estimate |
Horizontal line | Confidence interval |
Arrowhead on the line | The interval runs past the edge of the plotted axis |
Vertical line | Line of no effect (an odds ratio of 1) |
Diamond at the foot | Pooled estimate, when one is given |
Square size reflects weight, not the length of the interval or the size of the effect.
The most precise estimate has the narrowest interval, usually from the largest trial or the most data.
A study is individually significant when its interval lies wholly clear of the line of no effect, judged separately from the pooled diamond.
On network meta-analysis plots against a comparator:
Plot | Best treatment |
|---|---|
Efficacy (response odds ratio) | The odds ratio furthest from the line of no effect |
Tolerability (dropout odds ratio) | The lowest odds ratio, with its interval wholly below 1 |
A treatment that is both effective and acceptable must be significantly in its favour on both plots.
Funnel, L'Abbé and Galbraith plots
Funnel plots in meta-analysis
A meta-analysis funnel plot places each study as one point by effect size against precision, with standard error on an inverted vertical axis; sloping limits give the funnel shape.
The largest, most precise studies sit at the apex and small imprecise ones at the base.
It is used to look for publication bias: a gap among small, imprecise studies with unfavourable results suggests, but does not prove, that trials are missing.
Funnel plots in audit
An audit funnel plot compares each unit's performance on a quality indicator with a benchmark, plotted against unit size.
Its control limits narrow as size grows, flagging units that diverge by more than chance.
Points beyond the outer 3 SD limit (about 99.8%) trigger an alarm; points between the 2 SD and 3 SD limits trigger only an alert.
L'Abbé and Galbraith plots
Plot | Axes | Each trial | Reading |
|---|---|---|---|
L'Abbé | Treated-arm event proportion against control-arm event proportion | A circle sized by sample size | Above the line of equality, the intervention arm has the higher event rate |
Galbraith (radial) | Standardised effect (z statistic) against precision | One point | The line through the origin has a slope equal to the pooled effect; trials outside bands two units either side suggest heterogeneity |
On a L'Abbé plot, a trial above the line favours the intervention when the event is a good outcome, and circle size does not decide direction.
Validity
Convergent validity is shown when a new measure correlates highly with an established measure of the same construct.
Discriminant validity is low correlation with measures of unrelated constructs.
External validity is how far findings hold for people, settings and times beyond the sample, so a well-conducted trial with a narrow, unrepresentative sample is limited in external but not internal validity.