What a histogram is
A histogram is a bar chart that shows how often measurements fall into each range (class). It reveals the center, the spread and the shape of a process distribution, which a list of numbers or an average cannot show.
It is one of the 7 basic quality tools. Unlike a Pareto chart, which ranks categories, a histogram works with continuous data (diameter, weight, time, torque) and keeps the bars in the natural order of the values.
Histogram visual map
The map summarizes the step-by-step method, the frequency table of the example and the typical distribution shapes.
How to build one: 6 steps
- Collect at least 50 measurements from the same process, period and gauge. Below 30 points the shape is misleading.
- Compute the range: R = largest value − smallest value.
- Choose the number of classes (k). Sturges' rule gives k = 1 + 3.322 × log10(n). A practical alternative is k ≈ √n.
- Compute the class width: h = R ÷ k, rounded to a convenient value at the gauge resolution.
- Set the class boundaries without overlap (use half a unit more precision than the measurement so no value lands on a boundary) and count each class.
- Draw adjacent bars with frequency on the vertical axis, and mark the mean and the specification limits.
k = 1 + 3.322 × log10(n) | h = R ÷ k
Worked example: shaft diameter
Illustrative example: 50 machined shafts measured in mm, specification 10.00 ± 0.10 mm (LSL = 9.90; USL = 10.10). The smallest value is 9.95 and the largest is 10.06.
- Range: R = 10.06 − 9.95 = 0.11 mm
- Sturges: k = 1 + 3.322 × log10(50) = 1 + 3.322 × 1.699 ≈ 6.64, so 6 to 7 classes
- Width: h = 0.11 ÷ 7 = 0.0157, rounded to 0.02 mm, which gives 6 classes
- First boundary: 9.945, half a unit below the smallest value to avoid ties
| Class (mm) | Frequency | % of total | Cumulative |
|---|---|---|---|
| 9.945 to 9.965 | 3 | 6.0% | 3 |
| 9.965 to 9.985 | 6 | 12.0% | 9 |
| 9.985 to 10.005 | 14 | 28.0% | 23 |
| 10.005 to 10.025 | 17 | 34.0% | 40 |
| 10.025 to 10.045 | 8 | 16.0% | 48 |
| 10.045 to 10.065 | 2 | 4.0% | 50 |
The frequencies add up to 50. The peak is in the 10.005 to 10.025 class (17 parts, 34%) and the distribution is symmetric and bell-shaped. The sample mean is 10.006 mm and the sample standard deviation is 0.0235 mm.
How to read the shape
| Shape | What it indicates | Likely action |
|---|---|---|
| Symmetric bell | Stable process, common causes only | Compare with the specification |
| Skewed | Natural limit on one side (e.g. times, burrs) | Check for a physical or process limit |
| Two peaks (bimodal) | Mix of two sources: two machines, shifts, lots | Stratify the data by source |
| Plateau (flat) | Several means mixed or drift | Investigate wear and setup |
| Truncated (sharp cut) | Nonconforming parts removed before measuring | Measure before sorting |
| Isolated island | Measurement error, wrong entry or rare event | Check the data and the cause |
After the shape, compare with the specification. In the example the smallest value (9.95) is 0.05 mm from the LSL and the largest (10.06) is 0.04 mm from the USL: the tails still fit, but with little margin. If a tail crossed 9.90 or 10.10, there would be nonconforming parts.
From histogram to capability
With a mean of 10.006 and s = 0.0235, potential capability is Cp = (10.10 − 9.90) ÷ (6 × 0.0235) = 0.20 ÷ 0.141 ≈ 1.42. The index that accounts for off-centering is Cpk = min[(10.10 − 10.006); (10.006 − 9.90)] ÷ (3 × 0.0235) = 0.094 ÷ 0.0705 ≈ 1.33. Details and acceptance criteria are in Cp and Cpk.
Cp = (USL − LSL) ÷ 6s | Cpk = min(USL − x̄; x̄ − LSL) ÷ 3s
These indices are only valid if the process is in control. A histogram does not show time order: use a control chart, which detects drift that the histogram hides.
When to use a histogram
- Capability studies: check whether the distribution fits inside the specification limits.
- Variation diagnosis: separate sources when two peaks appear.
- Measurement system checks: data with very few distinct values point to poor gauge resolution, a topic of MSA and Gauge R&R.
- Root cause work with continuous data: look at the shape first, then use an Ishikawa diagram to list causes.
Common mistakes
- Too little data: with 15 or 20 points almost any shape looks possible.
- Too many or too few classes: many classes make the chart ragged; few hide the shape.
- Unequal class widths without adjusting bar height distort the reading.
- Mixing machines, shifts or lots in one sample and concluding the process is bimodal or "wide".
- Confusing it with a bar chart: in a histogram the bars touch and the order is fixed.
- Ignoring time: a nice histogram can come from an unstable process.
Frequently asked questions
How many classes should a histogram have?
Use Sturges, k = 1 + 3.322 × log10(n), or the square root of n. For 50 data points, 6 or 7 classes.
How much data do I need?
The practical minimum is 50 measurements; with 100 the shape is more reliable. Below 30, avoid conclusions.
What is the difference between a histogram and a bar chart?
A histogram shows the distribution of a continuous variable in ordered classes with touching bars. A bar chart compares categories.
How do I calculate the class width?
Divide the range by the number of classes and round. In the example, 0.11 ÷ 7 = 0.0157, rounded to 0.02 mm.
What does a histogram with two peaks mean?
Two populations are mixed, such as two machines or two shifts. Split the data by source and redraw the chart.
Does a histogram replace a control chart?
No. It shows the distribution but not the time order. Use both: the chart for stability, the histogram for shape and capability.
Sources
- MONTGOMERY, D. C. Introduction to Statistical Quality Control. Wiley.
- ISHIKAWA, K. Guide to Quality Control. Asian Productivity Organization.
- AIAG. Statistical Process Control (SPC) Reference Manual. 2nd ed.
- WHEELER, D. J. Understanding Variation: The Key to Managing Chaos. SPC Press.
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About the author
Lean Manufacturing & Behavioral Management Specialist
Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.
