Histogram

Goal

The histogram is a bar chart that shows the shape of a measurement's variation, which an average hides. The NIST statistics handbook lists five questions that the histogram answers. What distribution do the data come from? Where is their centre? How widely do they spread? Are they symmetric or skewed? Do they contain outliers? An average answers only the second question, and a table of figures answers none of them at a glance. The histogram is one of the charts of exploratory data analysis.

The deliverable is the chart, with its data and a written reading: the observed shape, the position against the customer's limits and the decision that follows. That decision may be to re-centre a setting, to separate two mixed populations, to check an outlier or to move on to calculating process capability.

Kaoru Ishikawa brought together seven basic quality tools for the Japanese quality circles: the check sheet, the Pareto chart, the fishbone diagram, the histogram, the control chart, the scatter diagram and stratification. His Guide to Quality Control teaches them to the people who do the work.

Usage

When to use it

• A measurement under a tolerance: see whether parts, weights or lead times stay between the customer's limits and with what margin.
• An average that reassures: an average lead time of 5 days can hide a tail of cases at 30 days.
• Two suspected populations: two peaks point to two machines, two teams or two request types to separate.
• Before a capability calculation or a statistical test: check the shape the calculation assumes, most often the bell curve.
• Before and after an improvement: two histograms on the same scale show whether the centre has moved and whether the spread has narrowed.
• The Measure phase of a DMAIC project: establish the baseline variation that the project must reduce.

When not to use it

• A question about change over time: the histogram loses the order of the measurements. Use the control chart.
• Categorical data: complaint reasons or failure types are counted and sorted in a Pareto chart.
• A few dozen values: the shape depends on where the bins are cut. Place each value on the axis in a dot plot instead.

Description

The elements

The measurement is a continuous variable, recorded on each unit: the diameter of each part, the lead time of each case. The horizontal axis carries it. A bin is an interval of that measurement. All bins have the same width and cover the range of the data without overlap. They follow one another without gaps, so the bars touch. The frequency of a bin is the number of values it holds. It gives the height of the bar, on the vertical axis.

The specification limits, lower and upper, written LSL and USL, bound the tolerance that the customer, the drawing or the contract sets. The analyst draws them as vertical lines, together with the target, most often midway between them.

Specification limits come from outside the process. The control limits of a control chart are calculated from the process's own data. An analyst who confuses the two declares compliant a stable process that makes parts outside the tolerance.

Building the histogram

1. Define the population. The analyst decides which measurement, on which units, over which period and with which instrument. One machine, one team or one request type forms a population. Several populations mixed together produce a shape that none of them has.
2. Record the values. The analyst aims for about a hundred values. With a few dozen, shifting the bins is enough to change the shape. The analyst also notes the order and date of each measurement, to draw the control chart and to trace an outlier.
3. Divide the range into bins. The range runs from the smallest to the largest value. No single rule sets the number of bins. Software proposes a width, which the analyst rounds to a readable value: 5 µm, 1 day or CHF 100. The analyst also decides where a value that falls on a boundary goes, for example into the bin that starts at that boundary, and applies the same rule throughout.
4. Count. The total of the frequency table equals the number of measurements. A difference signals a value that was missed or counted twice.
5. Draw. One bar per bin, no space between the bars, the vertical axis starting at zero. The analyst adds the specification limits and the target, then states below the chart the number of values, the period and the source.
6. Read and decide. The analyst names the shape, places the centre and the range against the limits and writes the decision below the chart.

Choosing the bin width

The bin width decides what the chart shows. With three or four bars, the skew and the two peaks of a mixture disappear. With bins that are too narrow, the bars alternate between high and low by chance, and the reader sees dips that the process does not produce. So the analyst draws two or three widths on the same data. A shape that holds from one width to the next belongs to the process. A shape that disappears belongs to the binning.

The resolution of the instrument sets a minimum width. A dial indicator that reads to the micrometre leaves every other 0.5 µm bin empty. The chosen width is a whole multiple of the resolution. Otherwise neighbouring bins cover unequal numbers of possible readings: the bars alternate with no cause in the process.

Reading the shape

NIST describes the typical shapes of a histogram. The causes given for each are common leads from quality practice: stratification or the control chart confirms or rules them out.

• Bell-shaped, symmetric around one peak: the variation comes from many small causes. The mean and the standard deviation summarise it.
• Skewed right, with a long tail towards large values: common for lead times, durations and amounts, which cannot fall below zero. The median describes the typical case better than the mean, which the tail pulls upwards. For the choice between mean and median, see descriptive and inferential statistics.
• Skewed left: the values run up against a ceiling, a yield close to 100% or a maximum score. The analyst looks for that ceiling.
• Bimodal, with two peaks: NIST gives two readings. The first is a mixture of two distributions centred in different places, commonly caused by two machines, two teams or two suppliers. Stratification then separates the data by source, and each source gets its own histogram. The second is a process that oscillates over time, which the control chart shows.
• Flat-topped, with short tails: the bars have similar heights across almost the whole range. Several populations with nearby centres overlap, or the setting drifted during the period. The control chart decides between the two.
• Outlier, a bar separated from the rest by empty bins: an entry error, a measurement error or a real event. The analyst traces the value back to its source before discarding or explaining it.

Placing the shape against the limits

Once the shape is named, the analyst compares the spread of the bars with the interval between the limits. NIST calls this the graphical assessment of capability. If the whole distribution fits between the two limits, the process holds the tolerance at the time of recording.

An adjustment corrects an offset. A distribution wider than the interval produces scrap, and no adjustment removes it: the team then reduces the variation or renegotiates the tolerance. When the two defects combine, re-centring comes first, because it costs the least.

Process capability turns this reading into indices, Cp and Cpk. NIST points out that these indices assume normally distributed data from a stable process. It also asks for about 50 independent values to calculate them. The histogram checks the shape first: a bimodal or strongly skewed histogram distorts the value of the indices.

What the histogram does not show

The histogram loses the order of the measurements. A process that drifts during recording produces a wide bell that looks the same as that of a stable process with a wide spread. The shape predicts tomorrow's output only if the process is stable. The control chart checks that stability. The analyst draws both on the same data.

The analyst compares two histograms on the same scale and with the same bins. If the two data sets do not hold the same number of values, the vertical axis shows percentages. If the software sets each chart's axes and bins on its own, it makes a change of shape appear that comes only from the binning.

The histogram and its neighbours

AI considerations

An assistant writes the code that draws the histogram, in Python or R, with the specification limits and several bin widths side by side. The analyst checks that the code reads the right column in the right unit over the right period. The analyst also checks that the limits drawn are those of the drawing or the contract.

A language model given values pasted into the conversation can return wrong frequencies without saying so. The analyst counts in the spreadsheet or in code. The analyst checks that the frequencies add up to the number of measurements.

The model names a shape and proposes causes: two peaks, so two machines. It does not know whether the workshop has two machines or whether a supplier changed during the period. Its causes remain hypotheses, which stratification confirms or rules out.

Lead times or amounts linked to individuals are personal data under the Federal Act on Data Protection (FADP). Before passing them to an external AI service, the analyst removes the identifiers: the histogram needs only the measurement.

Examples

A histogram places production against the tolerance. A bar-turning company in the Jura Arc machines shafts of 4.00 mm diameter, with a tolerance of ±0.03 mm. Inspection measures 100 shafts from one machine during one shift, with a dial indicator that reads to 0.5 µm. The table and the chart show the deviation from nominal size, in micrometres.

Every shaft is within the tolerance. The mean, calculated on the bin midpoints, is 8.1 µm, against a target of 0. The right tail comes within 5 µm of the upper limit, while at least 15 µm separate the lowest bin from the lower limit. Lowering the setting by about 8 µm gives margin on both sides without touching the spread.

Visualisations

The histogram of the example, the catalogue of shapes, the comparison of bin widths and the four positions against the limits are diagrams: the shape of the bars and their place on the scale carry the information. The frequency table, the comparison with the neighbouring tools and the cost are made of rows and columns, so they are tables.

Cost

Tooling

On the shop floor, a check sheet is enough. The operator ticks each measurement in its bin, and the columns of ticks form the histogram as the recording goes on.

A spreadsheet draws the histogram from a column of values. Excel's Histogram chart type calculates the bins and lets the analyst set their width. It is good enough for a first look. It cannot be combined with any other chart type, so it does not draw the limits. To add them, the analyst calculates the frequencies with COUNTIFS or FREQUENCY, plots columns with no gap between them and adds each limit as a scatter series.

Minitab and JMP draw the histogram with the limits and calculate capability in the same analysis. R and Python, with ggplot2 or matplotlib, do the same work in code, which the team reviews and reruns on the next month's data.

Sources

• Ishikawa, Kaoru, Guide to Quality Control, Asian Productivity Organization, original Japanese edition 1968: the histogram among the seven basic quality tools, how to build it and how to read it.
• NIST/SEMATECH, e-Handbook of Statistical Methods, §1.3.3.14 "Histogram": the definition, the five questions the histogram answers and the typical shapes, each described on its own page.
• NIST/SEMATECH, e-Handbook of Statistical Methods, §3.4.6 "Assessing Process Capability": reading the histogram against the specification limits and moving on to the Cp and Cpk indices.
• NIST/SEMATECH, e-Handbook of Statistical Methods, §6.1.6 "What is Process Capability?": the conditions for the Cp and Cpk indices, namely a normal distribution, a stable process and about 50 independent values.