Objective Selection of the Bin Width for a Time Histogram#

Stage 2 · 🔗 Associations & Correlation · Lesson 09 of 56 · beginner

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Important

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The bin-width problem#

A histogram summarises a distribution by counting values into bins, but the picture depends entirely on the bin width you choose. Too wide and you smooth away real structure; too narrow and random noise looks like signal. For a time histogram — counting events (like taxi pickups) into time intervals — the width sets whether you see a genuine daily rhythm or a jagged mess. Choosing it should not be guesswork.

Rules of thumb#

Several rules pick a width from the data automatically. Sturges’ rule sets the number of bins from \(\lceil \log_2 n \rceil + 1\), assuming roughly normal data. Scott’s rule chooses width \(h = 3.49\,\hat{\sigma}\,n^{-1/3}\), and the Freedman–Diaconis rule \(h = 2\,\mathrm{IQR}\,n^{-1/3}\), which uses the interquartile range and so resists outliers. Each aims to balance detail against noise.

An objective criterion#

For a time histogram specifically, the Shimazaki–Shinomoto method turns the choice into an optimisation. It picks the width \(\Delta\) that minimises a cost estimating the error between the histogram and the true underlying rate:

\[C(\Delta) = \frac{2\bar{k} - v}{\Delta^2},\]

where \(\bar{k}\) is the mean and \(v\) the variance of the bin counts. Sweeping \(\Delta\) and taking the minimum gives a width derived from the data, not chosen by eye.

Why it matters#

The theme is objectivity. A histogram is one of the first plots an analyst makes, and an arbitrary bin width can quietly manufacture or hide patterns. A principled rule makes the picture reproducible — the same data yields the same histogram for everyone — which is exactly the standard the rest of this course holds its methods to.

Hint