 
Overview: This histogram tutorial will provide information on how to construct and
interpret histograms for use in quality process control (Q.C.). The main areas that will
be covered in this tutorial are the following:
 Tutorial Instructions
 Histogram Background
 Creating a Histogram (interactively by example)
 Interpreting Histograms
 Recommended Additional Q.C. Topics and Software
Purpose: The purpose of this tutorial is to let you become familiar with graphical
histograms which are used widely in quality control (Q.C.). Histograms are effective Q.C.
tools which are used in the analysis of data. They are used as a check on specific process
parameters to determine where the greatest amount of variation occurs in the process, or
to determine if process specifications are exceeded. This statistical method does not
prove that a process is in a state of control. Nonetheless, histograms alone have been
used to solve many problems in quality control.
 Key Terms:
 Histogram 
 a vertical bar chart of a frequency distribution of data
 Q.C. Methodology 
 a statistical tool used in the analysis and determination of possible solutions to
quality control problems in industry
 Frequency Distribution 
 a variation in a numeric sample of data
 HISTORY  CONSTRUCTION
 INTERPRETATIONS  EXAMPLE  SOFTWARE  RELATED TOPICS 
The histogram evolved to meet the need for evaluating data that occurs at a certain
frequency. This is possible because the histogram allows for a concise portrayal of
information in a bar graph format.
The histogram is a powerful engineering tool when routinely and intelligently used. The
histogram clearly portrays information on location, spread, and shape that enables the
user to perceive subtleties regarding the functioning of the physical process that is
generating the data. It can also help suggest both the nature of, and possible
improvements for, the physical mechanisms at work in the process.
 OVERVIEW  CONSTRUCTION
 INTERPRETATIONS  EXAMPLE  SOFTWARE  RELATED TOPICS 
 Determine the range of the data by subtracting the smallest observed measurement from
the largest and designate it as R.
Example: Largest observed measurement = 1.1185 inches Smallest observed measurement = 1.1030 inches R = 1.1185 inches  1.1030 inches =.0155 inch
 Record the measurement unit (MU) used. This is usually controlled by the measuring
instrument least count.
Example: MU = .0001 inch
 Determine the number of classes and the class width. The number of classes, k, should be
no lower than six and no higher than fifteen for practical purposes. Trial and error may
be done to achieve the best distribution for analysis.
Example: k=8
 Determine the class width (H) by dividing the range, R, by the preferred number of
classes, k.
Example: R/k = .0155/8 = .0019375 inch The class width selected should be an oddnumbered multiple of the measurement unit, MU. This value should be close to the H value: MU = .0001 inch Class width = .0019 inch or .0021 inch
 Establish the class midpoints and class limits. The first class midpoint should be
located near the largest observed measurement. If possible, it should also be a convenient
increment. Always make the class widths equal in size, and express the class limits in
terms which are onehalf unit beyond the accuracy of the original measurement unit. This
avoids plotting an observed measurement on a class limit.
Example: First class midpoint = 1.1185 inches, and the class width is .0019 inch. Therefore, limits would be 1.1185 + or  .0019/2.
 Determine the axes for the graph. The frequency scale on the vertical axis should
slightly exceed the largest class frequency, and the measurement scale along the
horizontal axis should be at regular intervals which are independent of the class width.
(See example below steps.)
 Draw the graph. Mark off the classes, and draw rectangles with heights corresponding to
the measurement frequencies in that class.
 Title the histogram. Give an overall title and identify each axis.
Now you have a histogram!!
 OVERVIEW  HISTORY  INTERPRETATIONS  EXAMPLE  SOFTWARE  RELATED TOPICS 
When combined with the concept of the normal curve and the knowledge of a particular
process, the histogram becomes an effective, practical working tool in the early stages of
data analysis. A histogram may be interpreted by asking three questions:
 Is the process performing within specification limits?
 Does the process seem to exhibit wide variation?
 If action needs to be taken on the process, what action is appropriate?
The answer to these three questions lies in analyzing three characteristics of the
histogram.
 How well is the histogram centered? The centering of the data provides information on
the process aim about some mean or nominal value.
 How wide is the histogram? Looking at histogram width defines the variability of the
process about the aim.
 What is the shape of the histogram? Remember that the data is expected to form a normal
or bellshaped curve. Any significant change or anomaly usually indicates that there is
something going on in the process which is causing the quality problem.
Examples of Typical Distributions
NORMAL
 Depicted by a bellshaped curve
 most frequent measurement appears as center of distribution
 less frequent measurements taper gradually at both ends of distribution
 Indicates that a process is running normally (only common causes are present).
BIMODAL
 Distribution appears to have two peaks
 May indicate that data from more than process are mixed together
 materials may come from two separate vendors
 samples may have come from two separate machines.
CLIFFLIKE
 Appears to end sharply or abruptly at one end
 Indicates possible sorting or inspection of nonconforming parts.
SAWTOOTHED
 Also commonly referred to as a comb distribution, appears as an alternating jagged
pattern
 Often indicates a measuring problem
 improper gage readings
 gage not sensitive enough for readings.
SKEWED
 Appears as an uneven curve; values seem to taper to one side.
It is worth mentioning again that this or any other phase of histogram analysis must be
married to knowledge of the process being studied to have any real value. Knowledge of the
data analysis itself does not provide sufficient insight into the quality problem.
 OTHER CONSIDERATIONS
 Number of samples.
 For the histogram to be representative of the true process behavior, as a general rule,
at least fifty (50) samples should be measured.
 Limitations of technique.
 Histograms are limited in their use due to the random order in which samples are taken
and lack of information about the state of control of the process. Because samples are
gathered without regard to order, the timedependent or timerelated trends in the process
are not captured. So, what may appear to be the central tendency of the data may be
deceiving. With respect to process statistical control, the histogram gives no indication
whether the process was operating at its best when the data was collected. This lack of
information on process control may lead to incorrect conclusions being drawn and, hence,
inappropriate decisions being made. Still, with these considerations in mind, the
histogram's simplicity of construction and ease of use make it an invaluable tool in the
elementary stages of data analysis.
 OVERVIEW  HISTORY  CONSTRUCTION  EXAMPLE  SOFTWARE  RELATED TOPICS 
The following example shows data collected from an experiment measuring pellet penetration
depth from a pellet gun in inches and the corresponding histogram:
Penetration depth (inches) __________________________ 2 3 3 3 3 4 4 4 5 5 6 6
Some important things to remember when constructing a histogram:
 Use intervals of equal length.
 Show the entire vertical axes beginning with zero.
 Do not break either axis.
 Keep a uniform scale across the axis.
 Center the histogram bars at the midpoint of the intervals (in this case, the
penetration depth intervals).
 OVERVIEW  HISTORY  CONSTRUCTION  INTERPRETATIONS  SOFTWARE  RELATED TOPICS 
 Evaluation of
Histogramming Packages
 P. Ganev, S. Hellman, R. Jones . Table of Contents . 1.0 Introduction . 2.0 Evaluation
of the histogramming packages . 2.1 PAW . 2.1.1 General description of the package. .
2.1.2 Description of data formats required...
 Computer Software Development
 Software Development Projects . The Common Histogram Object . Show: An Implementation of
the Common Histogram Object . 3D Data Visualization . mailfeedback . Last Modified: Dec
24 1994
 pnmhisteq(1) USER COMMANDS
pnmhisteq(1)
 NAME pnmhisteq  histogram equalise a portable anymap SYNOPSIS pnmhisteq [gray] [rmap
pgmfile] [wmap pgmfile] [verbose] [pnmfile] DESCRIPTION pnmhisteq increases the contrast
of a portable graymap or pixmap through the technique of histogram ...
 E852 Data
Processing
 Production analyzer
 OVERVIEW  HISTORY  CONSTRUCTION  INTERPRETATIONS  EXAMPLE  RELATED TOPICS 
 Frequency histogram
 Bell shape
 Skewed distribution
 Uniform distribution
 Bimodal distribution
 Truncated distribution
 OVERVIEW  HISTORY  CONSTRUCTION  INTERPRETATIONS  EXAMPLE  SOFTWARE 
Tutorial Index
 
