How to Construct Histograms

[Steven Bonacorsi]

An important aspect of total quality is the identification and control
of all the sources of variation so that processes produce essentially
the same result again and again. A histogram is a tool that allows you
to understand at a glance the variation that exists in a process.
Although the histogram is essentially a bar chart, it creates a "lumpy
distribution curve" that can be used to help identify and eliminate
the causes of process variation. Histograms are especially useful in
the measure, analyze and control phases of the Lean Six Sigma
methodology.

What can it do for you?

A histogram will show you the central value of a characteristic
produced by your process, and the shape and size of the dispersion on
either side of this central value. The shape and size of the
dispersion will help identify otherwise hidden sources of variation.
The data used to produce a histogram can ultimately be used to
determine the capability of a process to produce output that
consistently falls within specification limits.

How do you do it?

1. Decide which Critical-To-Quality characteristic you wish to
examine. This CTQ must be measurable on a linear scale. That is, the
incremental value between units of measurement must be the same. For
example, a micrometer or a thermometer or a stopwatch can produce
linear data. Asking your customers to rate your performance from
"poor" to "excellent" on a five-point scale probably will not.

2. Measure the characteristic and record the results. If the
characteristic is continually being produced-such as voltage in a line
or temperature in an oven, or if there are too many items being
produced to measure all of them, you will have to sample. Take care to
ensure that your sampling is random.

3. Count the number of individual data points. Add the values for each
of the data points and divide by the number of points. This is the
mean (or average) value.

4. Determine the highest data value and the lowest data value.
Subtract the lower number from the higher. This is the range.

5. The next step is determining how many "classes" or bars your
histogram should have.

To make an initial determination, you can use this table:

Number of data points Number of classes

under 50 5 to 7

50 to 100 6 to 10

100 to 250 7 to 12

over 250 10 to 20

6. Divide the range by the trial number of classes you selected. The
resulting number will be your trial class interval (the horizontal
graduation or width) for each bar on your chart. You may round or
simplify this number to make it easier to work with, but the total
number of classes should be within those shown above. In determining
the number of classes and the class interval, consider how you are
measuring data. Increase or decrease the number of classes or modify
the class interval until there is essentially the same number of
measurement possibilities in each class.

7. Determine the class boundaries. You can do this by starting at the
center of the range. If you have an odd number of classes, center the
middle class approximately at the mid-point of the range, then
alternately add or subtract the class interval to define the other
class boundaries. If you have an even number of classes, begin the
process of adding or subtracting the class interval at approximately
the center of the range.

8. Tally the number of data points that fall in each of the classes.
Add the frequency totals for each class. This number should equal the
total number of data points. Divide the number of data points in each
class by the total number of data points. This will give you the
percentage of points falling in each class. Add the percentages of all
the classes. The result should be approximately 100.

9. Graph the results by beginning with the lowest measurement-value
class. Make the bar height correspond to the percentage of data points
that fall in that class. Draw the bar for the second class to the
right and touching the first bar. Again, make the height correspond to
the percentage of data points in that class. Continue in this way
until you have drawn in all the classes.

10. Draw a vertical dotted line through your histogram to represent
the mean value of all your data points.

11. If there are specification limits for the characteristic you are
studying, indicate them as vertical lines as well.

12. Title and label your histogram.

Now what?

The shape that your histogram takes tells a lot about your process.
Often, it will tell you to dig deeper for otherwise unseen causes of
variation.

The symmetrical or bell-shaped type of histogram: The mean value is in
the middle of the range of data. The frequency is high in the middle
of the range and falls off fairly evenly to the right and left. This
shape occurs most often.

The "comb" or multi-modal type of histogram: Adjacent classes
alternate higher and lower in frequency. This usually indicates a data
collection problem. The problem may lie in how a characteristic was
measured or how values were rounded. It could also indicate an error
in the calculation of class boundaries.

If the distribution of frequencies is shifted noticeably to either
side of the center of the range, the distribution is said to be
skewed. When the histogram is positively skewed. The mean value is to
the left of the center of the range, and the frequency decreases
abruptly to the left but gently to the right. This shape normally
occurs when the lower limit, the one on the Left, is controlled either
by specification or because values lower than a certain value do not
occur for some other reason.

If the skewness of the distribution is even more extreme, a clearly
asymmetrical, precipice-type histogram is the result. This shape
frequently occurs when a 100% screening is being done for one
specification limit.

If the classes in the center of the distribution have more or less the
same frequency, the resulting histogram looks like a plateau. This
shape occurs when there is a mixture of two distributions with
different mean values blended together. Look for ways to stratify the
data to separate the two distributions. You can then produce two
separate histograms to more accurately depict what is going on in the
process.

If two distributions with widely different means are combined in one
data set, the plateau splits to become twin peaks. The two separate
distributions become much more evident than with the plateau.
Examining the data to identify the two different distributions will
help you understand how variation is entering the process.

If there is a small, essentially disconnected peak along with a
normal, symmetrical peak, this is called an isolated-peak histogram.
It occurs when there is a small amount of data from a different
distribution included in the data set. This could also represent a
short-term process abnormality, a measurement error or a data
collection problem.

If specification limits are involved in your process, the histogram is
an especially valuable indicator for corrective action. The histogram
shows that the process is centered between the limits with a good
margin on either side. Maintaining the process is all that is needed.

When the process is centered but with no margin, It is a good idea to
work at reducing the variation in the process since even a slight
shift in the process center will produce defective material.

A process that would have produced material within specification
limits if it were centered is shifted to the left. Action must be
taken to bring the mean closer to the center of the specification
limits. A histogram that shows a process that has too much variation
to meet specifications no matter how it is centered. Action must be
taken to reduce variation in this process.

A process that is both shifted, in this case to the right, and has too
much variation. Action is necessary to both center the process and
reduce variation. This is a picture of the statistical variation in
your process. Not only can histograms help you know which processes
need improvement, they can also help you track that improvement.


Steven Bonacorsi is a Certified Lean Six Sigma Master Black Belt
instructor and coach. Steven Bonacorsi has trained hundreds of Master
Black Belts, Black Belts, Green Belts, and Project Sponsors and
Executive Leaders in Lean Six Sigma DMAIC and Design for Lean Six
Sigma process improvement methodologies.

Author for the Process Excellence Network (PEX Network / IQPC)

Process Excellence Network
Steven Bonacorsi, President of International Standard for Lean Six
Sigma(ISLSS)
Certified Lean Six Sigma Master Black Belt
47 Seasons Lane
Londonderry, NH 03053
Phone: +(1) (603) 401-7047
E-mail: sbona...@islss.com
Process Excellence Network: http://bit.ly/n4hBwu

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