Histogram, Pareto Diagram, Ishikawa Diagram, and Control Chart

1. Histogram
Pareto Diagram
Ishikawa Diagram
Control Chart
Nicola Ergo
AGH University of Science and Technology
Operational research in engineering

2. Histogram
A histogram is a graphical representation of the distribution of numerical data. It
is an estimate of the probability distribution of a continuous variable
(quantitative variable).
Purpose: to roughly assess the probability distribution of a given variable by
depicting the frequencies of observations occurring in certain ranges
of values.
Slide 1

3. Histogram
The vertical axis represents the frequency, that is the number of cases per unit of
the variable on the horizontal axis.
The horizontal axis represents the different variables that characterize the entire
range of values.
0
1
2
3
4
5
x1 x2 x3 x4 x5 x6
Frequency
Variables
Slide 2

4. Histogram
To construct a histogram:
• the first step is to divide the entire range of values into a series of intervals
and then count how many values fall into each interval; the intervals are
usually specified as consecutive, non-overlapping, they must be adjacent and
are usually equal size;
• the second step is to erect a rectangle over the interval with height
proportional to the frequency, so as to show the number of cases in each
interval.
Slide 3

5. Histogram
Example:
You measure the height of every tree in the orchard in centimeters (cm).
The heights vary from 100 cm to 340 cm.
You decide to put the results into groups of 50 cm:
• the 100 to just below 150 cm range;
• the 150 to just below 200 cm range;
• etc...
So a tree that is 260 cm tall is added to the "250-300" range.
And here is the result:
Slide 4

6. ParetoDiagram
A Pareto Diagram is a type of chart that contains both bars and a line graph,
where individual values are represented in descending order by bars, and the
cumulative total is represented by the line.
Purpose: highlight the most important among a set of factors.
Slide 5

7. The left vertical axis is the frequency of occurrence, but it can alternatively
represent cost or another important unit of measure.
The right vertical axis is the cumulative percentage of the total number of
occurrences, total cost, or total of the particular unit of measure.
The horizontal axis represent the
variables.
The line shows the cumulative
relative frequency. Due to the
values of the statistical variables
are placed in order of relative
frequency, the graph clearly
reveals which factors have the
greatest impact and where
attention is likely to yield the
greatest benefit.
Pareto Diagram
Slide 6

8. Example:
XYZ Clothing Store was seeing a steady decline in business. Before the manager
did a customer survey.
By collecting data and displaying it in a Pareto chart, the manager could see
which variables were having the most influence.
Pareto Diagram
Slide 7

9. 0%
20%
40%
60%
80%
100%
120%
Percent of total
Commulative percentage
Horizontal 80% line value
Following the Pareto Principle, those
are the areas where he should focus his
attention to build his business back up.
In this example,
parking difficulties,
rude sales people and
poor lighting were
hurting his business
most.
Pareto Diagram
Slide 8

10. Ishikawa Diagram
Ishikawa diagram (also called fishbone diagram, herringbone diagram, cause-
and-effect diagram, or Fishikawa) is causal diagram that show the causes of a
specific event or problem.
Purpose: to break down (in successive layers of detail) root causes that
potentially contribute to a particular effect.
Slide 9

11. Common uses of the Ishikawa diagram are product design and quality defect
prevention to identify potential factors causing an overall effect. Each cause or
reason for imperfection is a source of variation. Causes are usually grouped into
major categories to identify these sources of variation.
The categories typically include:
• people: anyone involved with the process;
• methods: how the process is performed and the specific requirements for
doing it, such as policies, procedures, rules, regulations and laws;
• machines: any equipment, computers, tools, etc. required to accomplish the
job;
• materials: raw materials, parts, pens, paper, etc. used to produce the final
product;
• measurements: data generated from the process that are used to evaluate its
quality;
• environment: the conditions, such as location, time, temperature, and culture
in which the process operates.
Ishikawa Diagram
Slide 10

12. How to create a “fish diagram”:
• create a head, which lists the problem or issue to be studied;
• create a backbone for the fish (straight line which leads to the head);
• identify at least four “causes” that contribute to the problem; connect these
four causes with arrows to the spine; these will create the first bones of the
fish;
• brainstorm around each “cause” to document those things that contributed
to the cause; use the 5 Whys or another questioning process such as the 4P’s
(Policies, Procedures, People and Plant) to keep the conversation focused;
• continue breaking down each cause until the root causes have been
identified.
Ishikawa Diagram
Slide 11

13. Example:
this example illustrates how a group might begin a “fish diagram” to identify all
the possible reasons a web site went down in order to discover the root cause.
Ishikawa Diagram
Slide 12

14. Control Chart
The control chart is a graph used to study how a process changes over time.
Data are plotted in time order. A control chart always has:
• a central line for the average;
• an upper line for the upper control limit UCL;
• a lower line for the lower control limit LCL;
• points representing a statistic (mean, range, proportion) of measurements of
a quality characteristic in samples taken from the process at different times.
By comparing current data to
these lines, you can draw
conclusions about whether the
process variation is consistent (in
control) or is unpredictable (out
of control, affected by special
causes of variation).
These lines are determined
from historical data.
Slide 13

15. When to Use a Control Chart:
• When controlling ongoing processes by finding and correcting problems as
they occur;
• When determining whether a process is stable (in statistical control);
• When analyzing patterns of process variation from special causes (non-
routine events) or common causes (built into the process);
• When determining whether your quality improvement project should aim to
prevent specific problems or to make fundamental changes to the process.
Control Chart
Slide 14

16. Control Chart Basic Procedure:
1. determine the appropriate time period for collecting and plotting data;
2. collect data, construct your chart and analyze the data;
3. look for “out-of-control signals” on the control chart.
When one is identified, mark it on
the chart and investigate the
cause. Document how you
investigated, what you learned,
the cause and how it was
corrected.
Control Chart
Slide 15

17. Controlled Variation
Controlled variation is characterized by a
stable and consistent pattern of
variation over time, and is associated
with common causes. A process
operating with controlled variation has
an outcome that is predictable within
the bounds of the control limits.
Uncontrolled Variation
Uncontrolled variation is characterized
by variation that changes over time and
is associated with special causes. The
outcomes of this process are
unpredictable.
Control Chart
Slide 16

18. Example:
A team in an accounting group has been working on improving the processing
of invoices. The team is trying to reduce the cost of processing invoices by
decreasing the fraction of invoices with errors. The team developed the
following operational definition for a defective invoice: an invoice is defective if
it has incorrect price, incorrect quantity, incorrect coding, incorrect address, or
incorrect name. The team decided to pull a random sample of 100 invoices per
day. If the invoice had one or more errors it was defective. The data from the
last 10 days are given in the table.
Control Chart
Slide 17

19. The next step is to calculate the average fraction defective p̄. To determine the
average, we add up all the np values and divide by the sum of all the n values.
The sum of the np values is 239; the sum of the n values is 1000. The average is
then calculated as shown below.
The next step is to determine the average subgroup size n̄. Since the subgroup
size is constant, the average subgroup size is 100. This average calculation is
shown in the second equation where k is the number of subgroups. The next
step is to calculate the control limits. The control limits calculations are shown
below.
Control Chart
Slide 18

20. Control Chart
Slide 19

21. Conclusion:
We have examined the variation in the % of invoices with errors from day to
day.
• On average, each day will have about 24% of the invoices with errors. Some
days it may be as high as 30% or as low as 20%. Only common causes of
variation are present.
• The process is in statistical control because all the values are within the UCL
and the LCL. This means that the process is consistent and predictable.
Note that this does not mean that the process is acceptable. Having 24% of
invoices with errors is not acceptable. The next step is to apply a problem-
solving model to reduce the number of errors. You should be using a Pareto
diagram with this control chart. The Pareto diagram is used to determine the
reason for errors and the frequency with which they occur.
Control Chart
Slide 20