Histogram

1. Collect the data points from the process under study, recording each measurement with consistent precision.
2. Determine the range of the dataset by subtracting the minimum observed value from the maximum observed value.
3. Select the number of bins. A common starting point for sample sizes of moderate scale is the square root of the total number of observations. For example, the square root of forty is approximately six. Practitioners often round the number of bins up or down to align the bin widths and boundary edges with standard, easily interpretable decimal increments.
4. Calculate the bin width by dividing the total range by the chosen number of bins, adjusting the interval size slightly upward if necessary to cover the entire range of values and provide convenient edge boundaries.
5. Define the boundary rules for data points that fall precisely on an edge. A standard convention places a boundary value into the upper bin rather than the lower bin.
6. Sort each observation into its corresponding bin, count the total number of observations per bin, and construct the chart by drawing adjacent vertical bars whose heights match those frequencies.

Selecting an appropriate bin width is essential to revealing the true structure of the data. Too few bins aggregate diverse measurements together and hide underlying features. Sorting forty items into only two bins, for example, indicates only that most items are near the center, obscuring skewness, secondary clusters, or isolated values. Conversely, choosing too many narrow bins disperses the counts, resulting in a fragmented display where random gaps appear between individual measurements. The same forty values look different at intervals of 0.01 mm, 0.02 mm, and 0.05 mm, and one of those widths will show features that the others hide.

Common distribution patterns include:

• Unimodal symmetric distributions: A single central peak with frequencies decreasing evenly on both sides indicates a stable, single source of random variation centered on a specific average.
• Skewed distributions: When measurements trail off more gradually in one direction, the distribution is skewed. A right-skewed distribution has a longer tail extending toward higher values, while a left-skewed distribution has a longer tail extending toward lower values. This asymmetry often indicates a physical or operational boundary, such as zero processing time or an absolute mechanical stop.
• Bimodal or multimodal distributions: The presence of two or more distinct peaks suggests that data from two or more separate operating conditions, machines, operators, shifts, or material batches have been mixed into a single sample.
• Isolated values: A bar separated from the main distribution by one or more empty bins, such as the single shaft at 10.08 mm, represents an outlier. Isolated values warrant direct investigation to determine whether they stem from measurement errors, raw material defects, or special causes in the production process.
• Truncated distributions: A histogram that ends abruptly on one side with a tall bar rather than tapering off gradually indicates that parts beyond that threshold have been sorted out, reworked, or discarded prior to measurement.

In a healthcare setting, patient wait times in an outpatient clinic averaged 15 minutes, which met the organizational target. When staff plotted individual patient arrival-to-consultation times, the histogram revealed a right-skewed distribution with a long tail reaching 60 minutes. While the mathematical average appeared acceptable, the chart showed that a notable subset of patients experienced excessive delays due to morning scheduling bottlenecks.

In an administrative environment, document processing times for loan applications averaged 3 days, matching the service level plan. Constructing a histogram of processing times showed two distinct clusters. Most documents completed within 1 to 2 days, but a second group required 7 to 8 days. The bimodal shape prompted an investigation that found applications containing missing signatures followed an entirely separate review workflow that lacked dedicated staffing.

Why do the bars in a histogram touch each other?

A histogram plots continuous numerical data across contiguous intervals called bins, so adjacent bars touch to reflect an unbroken measurement scale. Standard bar charts display separate categorical groups and leave spaces between bars to show the categories are distinct. On a histogram, a space appears only when an interval contains zero recorded observations.

How does bin width affect the appearance of a histogram?

Choosing too few wide bins aggregates diverse measurements together, concealing features like skewness or secondary clusters. Conversely, setting bins too narrow scatters counts across many intervals, creating fragmented displays with artificial gaps between individual values. Selecting an appropriate width balances these extremes so the true shape of the distribution becomes visible.

What does a bimodal distribution indicate on a histogram?

A bimodal distribution displays two distinct peaks, indicating that data from two different operating conditions have been combined into a single dataset. This shape typically occurs when measurements mix outputs from different machines, operators, work shifts, or material batches. Investigating each group separately reveals the distinct causes behind each peak.

Why is a histogram unable to show if a process is drifting over time?

A histogram groups measurements solely by magnitude and discards the sequence in which data was collected. Because all observations are pooled into frequency bins regardless of time, the chart cannot show whether values are drifting upward, shifting abruptly, or cycling. Tracking time-dependent behavior requires pairing the histogram with a time-ordered tool like a run chart or a control chart.

How does a histogram display outliers?

An outlier appears as an isolated bar separated from the main body of the distribution by one or more empty bins. This gap highlights measurements that fall far outside the normal spread of process variation. Such isolated values point investigators toward special causes, raw material defects, or measurement errors.