Common mistake 1 with Histogram

Reading a histogram of a small sample as a distribution. With thirty observations the shape is mostly noise.

Common mistake 2 with Histogram

Choosing the bin count until a feature appears.

Common mistake 3 with Histogram

Comparing histograms with different bin widths or different sample sizes side by side.

Common mistake 4 with Histogram

Overlaying histograms of groups with very different sizes. The larger group dominates the picture, and densities rather than counts are the only way to compare shapes.

EconStatsCraft › Histogram

Histogram

The shape of a single variable, which decides what can be done with it: skew argues for a log, bimodality for a missing group, and a spike at zero for a censored outcome that OLS will get wrong.

What it needs

Variable

The intuition

A histogram shows the shape of one variable by chopping its range into bins and counting. The shape you see is not a property of the data alone - it is a property of the data AND the bin width, and changing the bins can turn one hump into two or hide a gap entirely. That dependence is the whole thing to understand about histograms.

When to use it

Before modelling any variable, and on the residuals afterwards. It takes seconds and it catches problems no test will name for you.

Where the estimator comes from

With bin width h and origin x_0, bin k counts the observations in [x_0+kh,\ x_0+(k+1)h). The histogram is a density estimator whose mean integrated squared error has the familiar bias-variance shape: wide bins smooth away real features, narrow bins show sampling noise as structure. Optimal-width rules follow from minimising it - Scott's h=3.49\,\hat{\sigma}n^{-1/3} assumes near-normality, and Freedman-Diaconis h=2\,\mathrm{IQR}\cdot n^{-1/3} uses the interquartile range instead and so survives outliers. Both shrink only as n^{-1/3}, so quadrupling the sample buys a bin width smaller by only about a third.

What it assumes

The bin width is a choice, and it changes the picture. Too few bins hides structure, too many turns the data into noise. Move it and see whether the feature survives.
The variable is continuous enough for bands to mean something. For a handful of distinct values, a bar chart of counts is the honest chart.
The bins start somewhere, and that choice matters as much as their width. The same data binned from 0 and from 0.5 can show one mode or two.
The sample is representative of the distribution you mean to describe. A histogram of the observations you happen to have is not a histogram of the population unless the sampling says so.

Statistical properties

The bin width is a modelling choice with no default answer. Always look at two or three widths before believing a shape, and say which you showed.
The ORIGIN matters as well as the width. Shifting all bins by half a width can move a mode or merge two, which is why kernel density estimates - which have no origin - are often steadier.
Freedman-Diaconis is the safer automatic rule because the interquartile range is robust; Scott's rule uses the standard deviation and is pulled wide by a single outlier.
Counts depend on the sample size, so two histograms of different-sized samples are not comparable unless converted to density or proportion.
Unequal bin widths must be plotted as DENSITY, not count, or the area no longer represents frequency and the tallest bar is simply the widest one.
A histogram of a discrete or heavily rounded variable produces artefacts - spikes at round numbers - that say more about how the data was recorded than about the distribution.

How to read the output

Right skew argues for a log. Most economic variables - income, wealth, firm size, city size - are right skewed, which is why so much of applied economics is done in logs.
Two humps mean two populations mixed together. Find the variable that separates them; it usually belongs in the model.
A spike at zero or at a boundary means censoring, and OLS will get it wrong. That is what Tobit is for.
A smooth density is easier to compare across groups; a histogram is more honest about how much data there actually is.
Read the axis: counts and densities look identical and mean different things, and only a density histogram is comparable across groups of different sizes.
The tails are where the shape is least determined. A single bar at the far right may be one observation, and whether it is an outlier or the start of a tail is not answerable from this chart.
Compare against a reference distribution rather than by eye. A normal curve overlaid, or a Q-Q plot beside it, settles questions about shape that a histogram only raises.

A worked reading

Freedman-Diaconis and Scott disagree most exactly when it matters. On a skewed sample with one large outlier, Scott's rule uses the inflated standard deviation and produces bins wide enough to merge two genuine modes into one hump; Freedman-Diaconis uses the interquartile range, which the outlier does not move, and keeps them separate. Neither picture is wrong - they are estimates at different smoothing levels - but only one of them shows the feature, and a reader given a single histogram has no way to know a choice was made.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Two modes at one bin width, one at another	The structure may be noise	Try several widths and a kernel density estimate
A tall spike at a round number	Rounding at collection	A recording artefact, not a feature of the population
Nearly all mass in one bar	A skewed variable or an outlier	Take logs, or plot the log scale
A very jagged histogram	Bins too narrow for the sample	Widen; noise is being shown as structure
Comparing two samples of different sizes	Counts are not comparable	Use density or proportion
Unequal bins drawn as counts	The area no longer means frequency	Plot density

Common mistakes

Reading a histogram of a small sample as a distribution. With thirty observations the shape is mostly noise.
Choosing the bin count until a feature appears.
Comparing histograms with different bin widths or different sample sizes side by side.
Overlaying histograms of groups with very different sizes. The larger group dominates the picture, and densities rather than counts are the only way to compare shapes.

Where to read this properly

Scott (1979), "On Optimal and Data-Based Histograms", Biometrika 66(3).
Freedman and Diaconis (1981), "On the Histogram as a Density Estimator", Zeitschrift fur Wahrscheinlichkeitstheorie 57(4).
Silverman, Density Estimation for Statistics and Data Analysis (1986).

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Histogram

What it needs

The intuition

When to use it

Where the estimator comes from

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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