Distribution and correlation

Histograms and scatter plots answer questions that averages cannot: what shape is this data, and how do these two variables move together. Both reward looking at the raw observations, and both have display parameters that quietly change the conclusion.

Why plotting beats summarising

Anscombe quartet is the standard demonstration and still the most efficient one. Four datasets share the same mean, variance, correlation, and fitted regression line. Plotted, one is a clean linear relationship, one is a curve, one is linear with a single outlier dragging the fit, and one is a vertical stack of points with one distant observation creating the entire correlation. Every summary statistic agrees; every plot disagrees.

The same holds for a single variable. A mean and a standard deviation are consistent with a symmetric distribution, a strongly skewed one, or two separate populations mixed together, and those have very different implications. A histogram distinguishes them immediately.

This is why exploratory plotting comes before modelling rather than after. The choices a model makes — whether a linear fit is appropriate, whether outliers need handling, whether the data should be split — are visible in a plot and invisible in a summary table.

Binning is a smoothing decision

A histogram divides a continuous range into bins and counts observations in each. The bin width is a parameter, and the same data at different widths can look unimodal, bimodal, or uniform, with every version technically accurate.

Too few bins oversmooths. A genuinely bimodal distribution — two populations mixed, which is common in real data — collapses into a single broad hump if the bins are wide enough to span both peaks. That is a substantive finding destroyed by a display setting.

Too many bins turns noise into structure. With 50 observations across 30 bins, most bins hold zero, one, or two values, and the jagged profile that results is sampling randomness rendered as if it were shape.

Common rules give different answers. Sturges, the base-2 logarithm of the sample size plus one, assumes roughly normal data and tends to undercount bins for large or skewed samples. Freedman-Diaconis uses the interquartile range and is more robust to outliers, usually giving more bins. Scott uses the standard deviation. All are heuristics.

The practical defence is to look at more than one binning. A feature that survives across several bin counts is probably real; a second peak that appears at one setting and vanishes at the next is probably not. Bin boundaries matter as well as bin count — shifting all boundaries by half a width can change the shape, most visibly when the data clusters on round numbers because a human reported or rounded it.

Reading distribution shapes

A symmetric single peak suggests a process with a stable centre and random variation, which describes many measurement and manufacturing processes.

Right skew with a long upper tail is extremely common for anything bounded below by zero and unbounded above: incomes, response times, order values, file sizes. For skewed data the mean sits above the median and is a poor summary, which is why median income is the reported figure. A log transform often makes such data approximately symmetric and easier to work with.

Two peaks almost always mean two populations. Response times with a fast peak and a slow peak usually indicate two code paths — a cache hit and a cache miss. Splitting the data and analysing each population separately is nearly always more informative than describing the mixture, and the mixture average often describes no actual case.

A flat profile is either a genuinely uniform process, which is rare outside random number generation, or bins wide enough to flatten everything. Isolated bars far from the main body are outliers, and a histogram is frequently where you first spot a sentinel value like 999 or a unit error sitting in the data.

Scatter plots and overplotting

A scatter plot places each observation by position on two common scales, the most accurately read encoding available, and shows every point rather than a summary. Neither axis needs to include zero, because position rather than length carries the meaning and readers are comparing points to each other.

The failure mode is density. With enough points, markers overlap and dense regions saturate into a solid mass that shows the outline of the data and nothing about where within it observations concentrate. Rounded or discrete data makes it worse, since many observations share exactly the same coordinates and stack invisibly — a hundred points at one location look identical to one.

Transparency lets density show through as darkness and works to a few thousand points. Smaller markers delay saturation. Jittering separates coincident points in discrete data at the cost of showing slightly wrong positions. Beyond roughly ten thousand points, binning into a heatmap or hexbin is more honest than any of these, because it shows density directly.

Adding a third variable as marker size gives a bubble chart, with one perceptual caveat: people read the area of a circle, not its radius, and area grows with the square of the radius. Mapping value linearly to radius makes a doubled value look quadrupled. Correct implementations map to area. Even done right, size is a weak encoding good for roughly three distinguishable levels, so it belongs on a secondary variable while the important two stay on the axes.

What a pattern does not establish

A visible relationship establishes association, and association has several explanations of which causation is one.

Both variables may be driven by a third. Ice cream sales and drowning deaths correlate because both rise with temperature. The direction may be reversed from the assumed one, which is common in observational business data — a feature that engaged users adopt is not necessarily a feature that creates engagement. And with enough variables, relationships appear by chance: testing twenty pairs at a 5 percent threshold yields about one spurious result on average.

Two structural traps deserve specific attention. Selection effects can create or reverse a relationship, because if the data only includes cases that passed some filter, the pattern among survivors can be the opposite of the pattern in the population. And Simpson paradox means an aggregated relationship can reverse the relationship within every subgroup — if your points are group averages, the trend you are seeing may hold for no individual at all.

None of this is an argument against plotting. The scatter plot is the right tool for finding these questions; it is simply not the tool that answers them, and the gap between the two is where most over-interpretation lives.

Frequently asked questions

Chart makers referenced in this guide