Histograms: A Tutorial

Histograms

Overview, Purpose &, Key Terms
History & Background
Creating a Histogram
Interpretation
Example
Software
Related Topics

Overview

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.

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 & Background

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.

Creating a Histogram

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 odd-numbered 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 one-half 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!!

Interpretations

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 bell-shaped 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 bell-shaped 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).

BI-MODAL

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.

CLIFF-LIKE

Appears to end sharply or abruptly at one end
Indicates possible sorting or inspection of non-conforming parts.

SAW-TOOTHED

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.

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 time-dependent or time-related 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.

Example

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).

Software

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 . mail-feedback . Last Modified: Dec 24 1994
Histogram: Where can we grab your image? . Example: http://www.vrl.com/Imaging/test2.gif . webmaster@vrl.com
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 ...
ien163: Press here to go to the top of the ien 'tree'. . Echo Delay Measurements With GGP Packets IEN 163 R.G.Jones November 1980 University College London GGP IEN Echo 163 Delays 1. Introduction This IEN is intended to facilitate discussion of internet ...
E852 Data Processing: a1.Oct14 relnotes: Production analyzer version 14-Oct-1994 release notes . Source code . The source code is located in /home/lemond/e852/analyzer_production_1994 . Please see the README file for further information. . Bugs . This section documents the ...
LAMPF E1179 Experiment: This is the home page of the LAMPF E1179 experiment summarizing the results of exclusive data analysis. The total cross sections and reduced amplitudes are reported in D. Pocanic et at., Phys. Rev. Lett.,72, 1156 (1994). . The one- and ...