HistogramPareto Diagram

Ishikawa DiagramControl Chart

Nicola Ergo

AGH University of Science and TechnologyOperational research in engineering

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

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.

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Slide 2

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

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

Pareto Diagram

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

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

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

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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

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

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

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

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

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

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

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

Controlled VariationControlled 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 VariationUncontrolled 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

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

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

Control Chart

Slide 19

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

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