Distribution plots, and the dynamite plot

What this page is about

The commonest way a continuous outcome is drawn in papers and slides looks like this: one bar per group, the height is the mean, and a short whisker sticks out of the top. The picture has a nickname — the dynamite plot, because it looks like a stick of dynamite with a fuse.

The problem is not that it is ugly. The problem is that it draws two numbers per group: one bar height and one whisker length. You started with tens or hundreds of observations, and the reader ends up with two summaries and no way of telling how many people they were computed from.

This page draws one dataset four ways, and then works out exactly what the first of those four threw away. The data are MASS::birthwt: 189 mothers, with birth weight in grams as the outcome. The grouping variable is the same one the ANOVA example uses on t-tests and analysis of variance, so the two pages describe the same strata of the same dataset.

One dataset, four ways to draw it

What each of the four hands the reader, and what it withholds:

Look at the vertical axis of panel A. For bar lengths to be comparable, the axis has to start at zero, so the data end up squeezed against the top of the plot. The other three are under no such constraint: their axes can sit on the data, which buys a great deal of resolution on the same piece of paper.

What the dynamite plot hides in this dataset

The table above states principles. Principles do not convince anyone, so this section computes the numbers.

One: the group sizes differ several-fold, and the bar chart shows none of it

The largest group is 3.69 times the size of the smallest, while the three SDs are almost identical — largest over smallest is 1.14. Yet the three SE whiskers differ in length by a factor of 1.69. The reason is that SE is the SD divided by the square root of n, so the length of the whisker is mostly reporting sample size, not spread.

This dataset supplies an unusually clean example. The Black group has the smallest SD of the three (638.7 g), and the longest SE whisker of the three (125.3 g), because it contains only 26 mothers.

Two: the means look far apart, the individual babies overlap heavily

The first row is the representative one. The White and Black groups differ in mean birth weight by 383 grams, which on a bar chart is two visibly different heights. But draw one baby at random from each group, and the probability that the White one is heavier is only 0.654 — better than a coin flip, nowhere near two separate populations. And 30.2% of the White babies weigh something that falls inside the Black interquartile range.

That quantity — the probability of superiority — earns its place because it is free of the original units and because every reader already knows what 0.5 means. It is not a test, and it does not replace the inference on ANOVA and its post hoc comparisons. It answers a different question: how large is this group difference when it is applied to one patient.

Three: shape — these data are actually rather clean, and saying so is more useful

At this point a textbook usually adds that the bar chart also hid the skew, the bimodality and the outliers. In this dataset that sentence is mostly untrue, and reporting it honestly beats reciting the principle:

The largest absolute skewness of the three groups is only 0.42, which is close to symmetric, and of all 189 babies the 1.5 IQR rule flags 2. As for bimodality: at the default bandwidth of density(), the worst group has 3 local maxima and looks as though it contains a second cluster; widen the bandwidth by half and all three groups drop to 1. The extra peaks are a product of the smoothing parameter, not evidence of subpopulations.

So in this dataset the damage a dynamite plot does comes from unequal group sizes and from overlap, not from hidden skew or hidden modes. What happens when the shapes genuinely do differ is section five, on data constructed for the purpose.

Three error bars, identical in appearance, several-fold apart in length

The phrase “that error bar” keeps recurring above because the plot itself does not say which one it is. The same data support three:

• SD (standard deviation) describes how spread out the individuals are. It does not shrink with sample size — recruit a hundred more people and the SD is about the same.
• SE (standard error) describes how precisely the mean has been estimated. It is the SD divided by the square root of n, so it shortens as the sample grows.
• 95% CI is the SE multiplied by a t quantile, roughly two SEs. It answers where the mean plausibly lies.

In the White group the SD bar is 9.8 times as long as the SE bar. The same data and the same mean, and the reader sees a picture that differs by nearly an order of magnitude depending on which one the author picked. The SE is the shortest of the three, so choosing it makes the groups look as separated as they can possibly look — which is part of why it is chosen so often.

There is a related misreading: whether two error bars overlap is not a substitute for a test. Two 95% CIs can overlap slightly while the test of the difference in means still reaches statistical significance, and two SE bars can fail to overlap without it reaching significance. What settles a group comparison is the confidence interval of the difference, not whether two separate intervals happen to touch. t-tests and analysis of variance covers this properly.

One bar chart, three completely different datasets

Section three said it plainly: the three birthwt groups are fairly symmetric with almost no outliers. So what does happen when the shapes really do differ? The three samples below are constructed on purpose and are not the result of any study: each has 40 observations, a mean of 3000 and an SD of 700, so the SE is 110.7 for all three and the 95% CI half-length is 223.9 for all three. Their bar chart is one picture drawn three times.

The first four columns are identical and the last four are nothing alike. Sample 2 splits into two clusters and its mean falls in the gap between them, where nobody is — a position no observation comes near, and the height of the bar. Sample 3 has a skewness of 1.95, and its 4 extreme values drag the mean from a median of 2831.8 up to 3000.

Run it yourself

library(MASS)
data(birthwt, package = "MASS")
bw <- birthwt
bw$race_f <- factor(bw$race, levels = 1:3, labels = c("White", "Black", "Other"))
sp <- split(bw$bwt, bw$race_f)

# 1. Bar chart with SE whiskers. Here for comparison, not as a recommendation.
m  <- sapply(sp, mean)
se <- sapply(sp, function(x) sd(x) / sqrt(length(x)))
mids <- barplot(m, ylim = c(0, 3600), ylab = "Birth weight (g)")
arrows(mids, m, mids, m + se, angle = 90, code = 2, length = 0.05)

# 2. Box plot. One line.
boxplot(bwt ~ race_f, data = bw, ylab = "Birth weight (g)")

# 3. Violin plot: density() + polygon(), no extra package needed.
violin <- function(x, at, width = 0.35, col = "#dfe6ee") {
  d <- density(x, adjust = 1.2)
  k <- d$x >= min(x) & d$x <= max(x)      # do not draw beyond the data
  dy <- d$y[k] / max(d$y) * width
  polygon(c(at + dy, rev(at - dy)), c(d$x[k], rev(d$x[k])), col = col)
}
plot(NA, xlim = c(0.5, 3.5), ylim = range(bw$bwt), xaxt = "n",
     xlab = "", ylab = "Birth weight (g)")
axis(1, at = 1:3, labels = levels(bw$race_f))
for (i in 1:3) violin(sp[[i]], i)

# 4. Raincloud: half a violin, a narrow box, and every observation.
plot(NA, xlim = c(0.5, 3.5), ylim = range(bw$bwt), xaxt = "n",
     xlab = "", ylab = "Birth weight (g)")
axis(1, at = 1:3, labels = levels(bw$race_f))
set.seed(1)                               # without a fixed seed the points move
for (i in 1:3) {
  x <- sp[[i]]
  d <- density(x, adjust = 1.2)
  k <- d$x >= min(x) & d$x <= max(x)
  dy <- d$y[k] / max(d$y) * 0.34
  polygon(c(rep(i, sum(k)), rev(i + dy)), c(d$x[k], rev(d$x[k])),
          col = "#dfe6ee", border = "#4d6a8c")
  boxplot(x, at = i - 0.02, add = TRUE, boxwex = 0.09,
          axes = FALSE, outline = FALSE)
  points(jitter(rep(i - 0.22, length(x)), amount = 0.09), x,
         pch = 16, cex = 0.6, col = adjustcolor("#4d6a8c", 0.4))
  text(i - 0.22, min(bw$bwt), paste0("n = ", length(x)), cex = 0.8)
}

import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.api as sm

bw = sm.datasets.get_rdataset("birthwt", "MASS").data
bw["race_f"] = bw["race"].map({1: "White", 2: "Black", 3: "Other"})

# Box plot and violin plot, one line each
sns.boxplot(data=bw, x="race_f", y="bwt")
sns.violinplot(data=bw, x="race_f", y="bwt", inner=None, cut=0)

# A raincloud has to be stacked by hand: half violin, narrow box, jittered points
fig, ax = plt.subplots(figsize=(6, 4))
sns.violinplot(data=bw, x="race_f", y="bwt", inner=None, cut=0,
               split=True, hue=1, legend=False, ax=ax)
sns.boxplot(data=bw, x="race_f", y="bwt", width=0.08, showfliers=False,
            boxprops={"zorder": 3}, ax=ax)
sns.stripplot(data=bw, x="race_f", y="bwt", size=3, alpha=0.45,
              jitter=0.18, dodge=False, ax=ax)
ax.set_ylabel("Birth weight (g)")

# Group sizes have to be added yourself; seaborn will not print them
for i, (lab, grp) in enumerate(bw.groupby("race_f", sort=False)):
    ax.text(i, bw["bwt"].min(), f"n = {len(grp)}", ha="center", fontsize=8)

Journals have started asking for the individual data points

This is no longer a matter of taste; it has reached submission guidelines.

Weissgerber and colleagues surveyed papers in leading physiology journals for PLOS Biology (2015) and found that the great majority presented continuous data as bar charts with error bars, often at sample sizes so small that the two summaries meant very little. Their central argument is the figure in section five above: one bar chart is compatible with many datasets that look nothing like each other.

What has changed since:

• The PLOS Biology submission guidelines recommend presenting continuous data so that the reader can evaluate the distribution — scatter plots, box plots, histograms — and require the full distribution to be shown, rather than summary statistics alone, for small samples (9 or fewer independent observations per group).
• eLife requires figure legends to state the sample size and the definition of the error bars, along with the statistical tests and the replicates. That is the previous section’s problem written into a rule.
• A growing number of journals discourage bar charts for continuous outcomes in their figure guidance.

The practical rule of thumb is short:

• Fewer than about twenty observations per group — plot every point (a dot plot or a raincloud); do not summarise.
• Tens to a few hundred per group — a box plot or a violin, better still with sampled points on top.
• Thousands per group — points turn into a smear; use a violin or a histogram, and put the sample size on the figure.
• Always — say in the caption what the error bar is and what the sample size was.

When a bar chart is the right choice

The bar chart is not a bad plot; it is a plot used in the wrong place. Its proper job is counts and proportions: how many people, or what share, fall in each level of a categorical variable. There the bar length starts at zero, genuinely means “how many”, and is both additive and comparable. That case is covered in the chi-squared test and comparing proportions.

For a continuous outcome, even if all you want is the summary, two options beat a bar chart:

• Plot the mean and its confidence interval as a point with an interval, with no bar. The bar itself carries no information while consuming the whole vertical axis.
• For a skewed variable, report the median and the IQR instead. Which of the two to report is decided in the “mean or median” section of Table 1 and the standardised mean difference.

Common misuse

Rerun every number on this page

/opt/homebrew/bin/Rscript figures/scripts/B1-05-distribution-plots.R

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