Histogram

Definition

A histogram is a bar chart that shows the frequency distribution of measured data. The horizontal axis represents measurement intervals (bins), and the vertical axis represents the number of data points falling into each interval. The resulting shape reveals the process’s behavior: its central tendency, its spread, its symmetry, and any abnormalities.

Where a control chart shows process behavior over time (sequence matters), a histogram shows the overall shape of the data (sequence is discarded). Both are essential — the control chart tells you if the process is stable, the histogram tells you what the stable process looks like.

Japanese Origin

ヒストグラム (hisutoguramu) is written in katakana, indicating a borrowed foreign term. The word “histogram” comes from the Greek HistoS (mast, upright bar) and gramma (drawing, writing). The term was coined by the English mathematician Karl Pearson in 1891.

In Japanese QC literature, it may also be referred to as 度数分布図 (dosū bunpu zu, “frequency distribution diagram”) or 柱状図 (chūjō zu, “column-shaped diagram”).

History

The histogram predates the Japanese quality movement by decades — frequency distributions were a standard statistical tool in the 19th century. What Ishikawa did was make the histogram a practical shop-floor tool by teaching frontline workers how to construct, read, and interpret histograms without formal statistical training.

At Toyota and in Japanese industry — Histograms are used extensively in process capability analysis. When a new process is established or an existing process is investigated, the first question is: what does the distribution of output look like? Is it centered within the specification limits? How much of the output falls near the limits? Is the distribution normal (bell-shaped) or does it show signs of problems (skew, multiple peaks, truncation)?

Connection to check sheets — The frequency distribution check sheet is designed so that data collection and histogram construction happen simultaneously. As the operator measures each part and marks the corresponding interval on the check sheet, the histogram takes shape visually on the form itself.

How to Construct

1. Collect data — Measure at least 50-100 data points from the process. More data produces a more reliable histogram.
2. Determine the range — Find the maximum and minimum values.
3. Choose the number of intervals (bins) — A common rule of thumb: for n data points, use approximately √n intervals. For 100 data points, 10 intervals is typical.
4. Calculate interval width — Range ÷ number of intervals.
5. Tally the data — Count how many data points fall into each interval.
6. Draw the bars — Each bar’s height represents the frequency (count) for that interval.
7. Add specification limits — Overlay the upper specification limit (USL) and lower specification limit (LSL) as vertical lines to see how the distribution relates to the customer’s requirements.

How to Read a Histogram

Shape reveals process behavior:

• Bell-shaped (normal distribution) — A symmetric, single-peaked histogram centered within specifications indicates a stable, capable process.
• Skewed left or right — The process center is shifted toward one specification limit. This may indicate tool wear, machine drift, or a physical boundary.
• Bimodal (two peaks) — Two distinct peaks suggest two different conditions are mixed in the data — two machines, two material lots, two operators. Stratify the data to separate them.
• Truncated (cliff edge) — A sharp cutoff on one side suggests that the process produces out-of-spec parts that are being sorted out (100% inspection removing the tail). The process is actually wider than the histogram shows.
• Flat/uniform — No clear central peak suggests the process is not controlled — output is essentially random within the range.

Position relative to specifications:

• Centered and well within limits — Good capability. There is a safety margin.
• Centered but close to limits — The process is capable today but vulnerable to any drift.
• Shifted toward one limit — Parts are being produced near or beyond one specification. Adjust the process center.
• Wider than the specification range — The process is not capable of meeting the specification consistently. Reduce variation through process improvement, or widen the specification if the product can tolerate it.

Common Mistakes

Not collecting enough data. A histogram with 15-20 data points is unreliable — the shape is dominated by random sampling effects. Collect at least 50 points, preferably 100, for a meaningful distribution.

Mixing data from different conditions. If the data includes parts from two different machines, two different shifts, or two different material lots, the histogram may show a misleading shape (bimodal or flat). Always stratify data from known different conditions and create separate histograms.

Confusing the histogram with a control chart. The histogram shows the overall shape of the data but not when individual measurements occurred. A process can have a beautiful bell-shaped histogram and still be out of control (if the mean shifted during the data collection period). Use both tools: the control chart for time-based behavior, the histogram for overall distribution.

Not overlaying specification limits. A histogram without specification limits shows the process but not whether the process meets requirements. Always include USL and LSL to assess capability.

Using software defaults without thought. Statistical software automatically selects bin widths and counts. These defaults may not be appropriate for the data. Review the bin selection and adjust if the histogram shape is obscured by too few or too many bins.