Tunny Icon
TunnyDocs

The next-gen Grasshopper optimization tool.

Histogram

Overview

A histogram summarizes the distribution of a single variable by dividing its range into consecutive bins and counting how many observations fall into each one. It is the most direct way to see the shape of a distribution — whether it is skewed, has multiple peaks (multimodal), or contains outliers — that a single summary statistic such as the mean cannot reveal.

Binning Rules

The number (or width) of bins strongly affects how the distribution appears, so several rules exist to choose a reasonable default automatically.

Sturges' rule picks the bin count kk from the sample size alone:

k=⌈log⁡2n⌉+1k = \lceil \log_2 n \rceil + 1

It is derived assuming the data are approximately normally distributed, and tends to under-bin (produce too few, too-wide bins) as nn grows large, since kk only grows logarithmically.

Scott's rule instead derives the bin width hh from the data's spread, minimizing the integrated mean squared error under a normality assumption:

h=3.49 σ n−1/3h = 3.49 , \sigma , n^{-1/3}

where σ\sigma is the sample standard deviation, computed with the unbiased ( n−1n-1-corrected) estimator. Because it uses σ\sigma, it is sensitive to outliers, which inflate σ\sigma and widen the bins.

Freedman–Diaconis' rule replaces σ\sigma with the interquartile range (IQR, see Box Plot):

h=2⋅IQR⋅n−1/3h = 2 \cdot \mathrm{IQR} \cdot n^{-1/3}

Since the IQR is a robust spread measure unaffected by extreme values, this rule stays reliable even when the data contain outliers or are far from normal — making it a safer general-purpose default than Scott's rule.

If the computed width hh is non-positive or non-finite — e.g., the IQR collapses to 00 under Freedman–Diaconis when many values repeat, or σ=0\sigma = 0 under Scott's rule for near-constant data — Tunny Dashboard falls back to Sturges' rule instead of producing a degenerate bin count.

Characteristics

  • The apparent shape of a histogram (number of peaks, skewness) can change noticeably depending on the chosen bin count/width — a feature that looks like a real peak with one binning may vanish with another. Always be aware that a histogram is one particular view of the data, not the data itself.
  • A histogram is a hard-binned, discontinuous estimate of the underlying distribution. Kernel density estimation (KDE) smooths this into a continuous curve, trading a bin-boundary artifact for a bandwidth-choice artifact — neither is strictly "more correct," but a smooth KDE overlay is often easier to read alongside the histogram bars.
  • Regardless of the rule used, the resulting bin count is always clamped to [1,200][1, 200], so pathological inputs (or very large samples) never produce an unbounded or unusably fine set of bins.
  • Bins are half-open, [edgei,edgei+1)[\text{edge}i, \text{edge}{i+1}), except the last one, which is closed on both ends, [edgek−1,edgek][\text{edge}_{k-1}, \text{edge}_k] — the same convention as numpy.histogram — so the maximum value in the data is always counted in the last bin rather than dropped.

Where It Is Used in the App

  • Histogram widget: shows the distribution of a selected parameter or objective's values, helping identify skewness, multimodality, and where the optimizer's search has concentrated its exploration. The Fit selector overlays a maximum-likelihood parametric density — see Distribution fitting.

References

← PreviousStatisticsNext →Statistics: Box Plot