Statistics

Histogram

In French: Histogramme

The histogram shows the distribution of a measured quantity by grouping the values into classes. Its shape tells you about the process well before the calculation of a mean does.

Building it correctly

The range of values is cut into classes of equal width, then the number of observations falling into each is counted. The height of the bars represents those counts, and the bars touch — that is what conveys the continuity of the measured quantity.

The number of classes is the only setting, and it changes everything. Too few classes flatten the shape and make a double peak disappear; too many chop up the distribution and bring out a relief that is only sampling noise.

There is no ideal value, only a method: try several cuts and keep the one that shows a stable shape. A structure that disappears as soon as the number of classes is changed slightly is not a structure.

Reading the shape

Two peaks almost always betray a mixture: two machines, two shifts, two batches of material, two settings. It is the most profitable information a histogram gives, and it is invisible in any mean.

A distribution stretched on one side only often indicates a physical limit on one side and not on the other — durations do not go below a minimum but can lengthen without bound. That asymmetry rules out certain readings that assume a balanced distribution.

A sharp cut on one edge signals sorting: the parts beyond a limit were removed before measurement. The process therefore produces wider than what the chart shows, and reasoning on what remains leads to overestimating its capability.

Isolated values, far from the group, deserve to be identified one by one rather than discarded. Each corresponds to a dated event that can often be traced, and the event is more instructive than the value.

What it does not show

It ignores the order of the measurements. A gradual drift and a rapid alternation between two levels produce exactly the same histogram, whereas the causes and remedies have nothing in common.

That is why it is always accompanied by a chronological representation. The two charts answer two different questions: how the values are distributed, and how they evolve.

Nor does it say anything about conformity until the expected limits have been drawn on it. Once they are, the comparison between the observed spread and the admitted interval is read at a glance, and prepares the capability calculation.

The common mistakes

Choosing the number of classes that gives the nicest shape. It is a presentation decision disguised as analysis; the cut is chosen before knowing what one hopes to see.

Building a histogram on data from mixed sources without saying so. The shape obtained describes an object that does not exist, and the mean drawn from it corresponds to none of the real populations.

Discarding the extreme values to obtain a regular shape. Those values are frequently the most interesting subject of the chart.

On the shop floor — Precision machining

A diameter passed a conformity check without difficulty. Its histogram, plotted on three hundred parts, showed a clean cut at the upper limit: the parts beyond it were reworked in the workshop before being recorded. The process was producing far wider than announced, and the cost of the rework was counted nowhere.

The other tools illustrated by this trade: machining, mechanical engineering and finishing.

Frequently asked questions

How many measurements are needed to plot a useful histogram?

About fifty values make it possible to make out a shape, and about a hundred make it reliable. Below about thirty, the observed distribution depends mostly on the chance of the sample, and finding a shape in it is a matter of interpretation: two successive samples from the same process will then give two different pictures.

What is the difference between a histogram and a Pareto chart?

The histogram distributes a measurable quantity into value classes, and the order of the bars is imposed by the scale. The Pareto counts occurrences per category and ranks the bars from highest to lowest to designate the few causes that weigh the most. One describes a spread, the other ranks subjects.

What should be concluded from a histogram with two peaks?

That two mixed populations are probably being observed, and that the first action is to separate them rather than to analyse the whole. The sorting is done on the information available — machine, shift, batch of material, period — and it is generally enough to replot a histogram per subgroup to see which one explains the separation.

Should the tolerance limits be drawn on the histogram?

Yes as soon as they exist, because it is the comparison between the real spread and the admitted interval that gives the chart its meaning. Take care, however, not to confuse those limits, which come from outside the process, with the values the process produces spontaneously: they are two different kinds of information, and mixing them leads to believing that a process is under control because it is conforming.