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Histogram: Calculate descriptive statistics

There are several statistics which are useful to describe and analyze a histogram. They are calculated to describe the area under the curve formed by its shape. These descriptive statistics can be calculated using software such as SQCpack.

Central location

The central location of a set of data points is where (on what value) the middle of the data set is located. Central location is commonly described by the mean, the median, and/or the mode. The mean is the average value of the data points. The median is the middle number in the data set when the data points are arranged from low to high. The mode is the value in the data set that occurs most often.

Spread

Both range and the standard deviation illustrate data spread. Range is calculated by subtracting the minimum data value from the maximum data value. The standard deviation is a measure that indicates how different the values are from each other and from the mean. There are two methods of calculating standard deviation using individual data points or using a samples average range. Both formulas are available here.

Skewness

Skewness is the measure of the asymmetry of a histogram (frequency distribution). A histogram with normal distribution is symmetrical. In other words, the same amount of data falls on both sides of the mean. A normal distribution will have a skewness of 0. The direction of skewness is “to the tail.” The larger the number, the longer the tail. If skewness is positive, the tail on the right side of the distribution will be longer. If skewness is negative, the tail on the left side will be longer. The formula for skewness is available here.

Kurtosis

Kurtosis is a measure of the combined weight of the tails in relation to the rest of the distribution. As the tails of a distribution become heavier, the kurtosis value will increase. As the tails become lighter the kurtosis value will decrease. A histogram with a normal distribution has a kurtosis of 0. If the distribution is peaked (tall and skinny), it will have a kurtosis greater than 0 and is said to be leptokurtic. If the distribution is flat, it will have a kurtosis value less than zero and is said to be platykurtic. The formula for kurtosis is available here.

Coefficient of variance

The coefficient of variance is a measure of how much variation exists in relation to the mean. It may also be described as a measure of the significance of the sigma in relation to the mean. The larger the coefficient of variance, the more significant the sigma, relative to the mean. For example, if the standard deviation is 10, what does it mean? If the process average (mean) is 1000, a sigma value of 10 is not very significant. However, if the average is 15, a standard deviation of 10 is VERY significant. The formula for coefficient of variance is available here.

Chi-square

In SPC, the chi-square statistic is used to determine how well the actual distribution fits the expected distribution. Chi-square compares the number of observations found in each cell in a histogram (actual) to the number of observations that would be found in an expected distribution. If the differences are small, the distribution fits the theoretical distribution. If the difference are large, the distribution probably does not fit the expected distribution.

Using Chi-square with the assumption of a normal distribution

The calculated chi-square is compared to the value in the table of constants for chi-square based on the number of “degrees of freedom.”
If the calculated chi-square is less than the value in the table, the chi-square test passes, affirming that the process has a normal distribution.
If the chi-square is larger than the value in the table, the chi-square test fails. At this confidence level, you either do not have enough data to judge the process, or you should reject the assumption that the process has a normal distribution.

Note: Theoretical percent outside of specifications may be misleading.

The formula for chi-square is available here along with the degrees of freedom table.

Follow these steps to interpret histograms.

Study the shape.
Calculate descriptive statistics.
Compare the histogram to the normal distribution.

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Histogram: Compare to normal distribution

Is the shape of the histogram normal? The following characteristics of normal distributions will help in studying your histogram, which you can create using software like SQCpack.

The first characteristic of the normal distribution is that the mean (average), median, and mode are equal.
A second characteristic of the normal distribution is that it is symmetrical. This means that if the distribution is cut in half, each side would be the mirror of the other. It also must form a bell-shaped curve to be normal. A bimodal or uniform distribution may be symmetrical; however, these do not represent normal distributions.
A third characteristic of the normal distribution is that the total area under the curve is equal to one. The total area, however, is not shown. This is because the tails extend to infinity. Standard practice is to show 99.73% of the area, which is plus and minus 3 standard deviations from the average.
The fourth characteristic of the normal distribution is that the area under the curve can be determined. If the spread of the data (described by its standard deviation) is known, one can determine the percentage of data under sections of the curve. To illustrate, refer to the sketches right. For Figure A, 1 times the standard deviation to the right and 1 times the standard deviation to the left of the mean (the center of the curve) captures 68.26% of the area under the curve. For Figure B, 2 times the standard deviation on either side of the mean captures 95.44% of the area under the curve. Consequently, for Figure C, 3 times the standard deviation on either side of the mean captures 99.73% of the area under the curve. These percentages are true for all data that falls into a normally distributed pattern. These percentages are found in the standard normal distribution table.
Once the mean and the standard deviation of the data are known, the area under the curve can be described. For instance 3 times the standard deviation on either side of the mean captures 99.73% of the data.

Follow these steps to interpret histograms.

Study the shape.
Calculate descriptive statistics.
Compare the histogram to the normal distribution.

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Affinity diagram

What is it?

An affinity diagram is the organized output from a brainstorming session. It is one of the seven management tools for planning. The diagram was created in the 1960s by Kawakita Jiro and is also known as the KJ method.

The purpose of an affinity diagram is to generate, organize, and consolidate information concerning a product, process, complex issue, or problem. Constructing an affinity diagram is a creative process that expresses ideas without quantifying them.

The affinity diagram helps a group to develop its own system of thought about a complex issue or problem. A group can use an affinity diagram at any stage where it needs to generate and organize a large amount of information. For example, members of a leadership team may use the diagram during strategic planning to organize their thoughts and ideas. Alternatively an improvement team can use the diagram to analyze the common causes of variation in its project. The diagram is flexible in its application and is easy to use.

What does it look like?

A completed affinity diagram is shown below. In the example, a bakery has recently expanded its business and opened a chain of retail outlets. A number of problems have arisen and the management team, involved with the retail outlets, has met to discuss the problems. The issues are complex so they have decided to complete an affinity diagram.

When is it used?

Use an affinity diagram when you can answer “yes” to all of the following questions:

Is the problem (or issue) complex and hard to understand?
If the problem or issue is relatively simple or easy to understand, a cause-and-effect diagram may be more appropriate.
Is the problem uncertain, disorganized, or overwhelming?
Complex issues often feel overwhelming due to their size.
Does the problem require the involvement and support of a group?
The process a group goes through to make an affinity diagram helps the group develop its own system of thought concerning the problem and builds consensus among the members.

Getting the most

Choose a facilitator.
The facilitator is responsible for leading the group through the steps to make the affinity diagram. It is beneficial to have a facilitator experienced in making affinity diagrams.
State the issue or problem.
Before beginning, the group should state the issue or problem to be addressed. It is often useful to state the problem in the form of a question. In the example, the question is, “What are the problems associated with our expansion into retail outlets?” It is essential that the group understands the aim of the session.
Brainstorm and record ideas.
Next, brainstorm ideas concerning the issue statement. Brainstorming for ideas to make an affinity diagram uses a mixture of traditional brainstorming and the Crawford slip method. In traditional brainstorming, individuals generate ideas, which they voice in turn. Ideas are given by each person in the group until no one has anything else to add. In the Crawford slip method, ideas are recorded on index cards, slips of paper, or sticky notes, in silence. There is no verbal exchange. Brainstorming for the affinity diagram uses a mixture of these two approaches.

The above article is an excerpt from the “Operational Definition” chapter of Practical Tools for Continuous Improvement Volume 2 Statistical Tools.

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Attributes data (counts)

What is it?

Attributes data is data that can be classified and counted. There are two types of attributes data: counts of defects per item or group of items (nonconformities ) and counts of defective items (nonconforming).

How is it used?

Attributes data is analyzed in control charts that show how a system changes over time. There are two chart options for each type of attributes data. These attributes control charts, and more, can be created easily using software packages such as SQCpack.

What type of attributes data do I have?

Counts of defective items (noncomforming)

What is it?

Nonconforming data is a count of defective units. It is often described as go/no go, pass/fail, or yes/no, since there are only two possible outcomes to any given check. It is also referred to as a count of defective or rejected units. For example, a light bulb either works or it does not. Track either the number failing or the number passing.

How is it used?

Nonconforming data is analyzed in p-charts and np-charts. Chart selection is based on the consistency of the subgroup size:

If the number inspected is always or usually the same, use an np-chart.
If the number inspected varies with each subgroup use a p-chart.

Count of defects per item (noncomformities)

What is it?

Nonconformities data is a count of defects per unit or group of units. Nonconformities can refer to defects or occurrences that should not be present but are. It also refers to any characteristic that should be present but is not. Examples of nonconformities are dents, scratches, bubbles, cracks, and missing buttons.

How is it used?

Nonconformities data is analyzed in u-charts and c-charts. Chart selection is based on the consistency of the subgroup size:

If the number inspected is always or usually the same, use a c-chart.
If the number inspected varies with each subgroup use a u-chart.

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Capability analysis: Can a process produce output within spec?

A process that is in control is not necessarily producing an output that meets customer or engineering requirements. To find out if your process is capable of producing outputs that are in spec, you should perform capability analysis.

Capability analysis is a set of calculations used to assess whether a system is statistically able to meet a set of specifications or requirements. To complete the calculations, a set of data is required, usually generated by a control chart; however, data can be collected specifically for this purpose. Easily create control charts and perform capability analysis using software like SQCpack.

Specifications or requirements are the numerical values within which the system is expected to operate, that is, the minimum and maximum acceptable values. Occasionally there is only one limit, a maximum or minimum. Customers, engineers, or managers usually set specifications. Specifications are numerical requirements, goals, aims, or standards. It is important to remember that specifications are not the same as control limits. Control limits come from control charts and are based on the data. Specifications are the numerical requirements of the system.

All methods of capability analysis require that the data is statistically stable, with no special causes of variation present. To assess whether the data is statistically stable, a control chart should be completed. If special causes exist, data from the system will be changing. If capability analysis is performed, it will show approximately what happened in the past, but cannot be used to predict capability in the future. It will provide only a snapshot of the process at best. If, however, a system is stable, capability analysis shows not only the ability of the system in the past, but also, if the system remains stable, predicts the future performance of the system.

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Capability analysis for attributes

ttributes capability measures are taken directly from attributes control charts. No additional calculations are required.

The capability for a p-chart is the average proportion of nonconforming items (p-bar). The capability for an np-chart is the average number of nonconforming items generated by the system (np-bar). The capability for a c-chart is the average number of nonconformities per subgroup (c-bar). The capability for a u-chart is the average number of nonconformities per unit (u-bar).

A weakness in capability estimates for attributes data is that they do not suggest why a system is either capable or not. For instance, there is no way of knowing whether the system is incapable because it is not centered, it is too close to a specification limit, or it exhibits too much unit-to-unit variation. Further studies must be done to learn how to improve the system.

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Any point lying outside the control limits

This is the quickest and easiest test for system stability. Look above the upper control limit and below the lower control limit to see whether any points fall in those regions of the chart. If you are looking at a chart pair (X-bar and R, X-bar and s, or X and MR), look at both charts.

Points falling outside the control limits may be the result of a special cause that was corrected quickly, either intentionally or unintentionally. It may also point to an intermittent problem. The chart below shows two points outside the control limits.

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How do I compare the Cp/Pp and Cpk/Ppk?

Assume the original target is a Cpk or Ppk of 1.0

	If Cpk or Ppk is less than 1.0	If Cpk or Ppk is greater than 1.0

If Cp or Pp is less than 1.0	Variation in the process should be reduced.	Not mathematically possible. Check for an error in calculations.

If Cp or Pp is greater than 1.0	The process should be centered within its specifications.	Fine tune and improve the process continuously. Increase the Cpk target.

See also:
>> Cpk
>> Cp
>> Cr
>> Cpm
>> Ppk
>> Pp
>> Pr
>> Capability indices

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control chart interpretation, and other quality metrics.

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Capability indices: Cp

The Cp index is used to summarize a system’s ability to meet two-sided specification limits (upper and lower). Like Cpk, it uses estimated sigma and, therefore, shows the system’s potential to meet the specifications. However, it ignores the process average and focuses on the spread. If the system is not centered within the specifications, Cp alone may be misleading.

The higher the Cp value, the smaller the spread of the system’s output. Cp is a measure of spread only. A process with a narrow spread (a high Cp) may not meet customer needs if it is not centered within the specifications.

If the system is centered on its target value, Cp should be used in conjunction with Cpk to account for both spread and centering. Cp and Cpk will be equal when the process is centered on its target value. If they are not equal, the smaller the difference between these indices, the more centered the process is.

See also:
>> How do I compare the Cp/Pp and Cpk/Ppk?
>> Cpk
>> Cr
>> Cpm
>> Ppk
>> Pp
>> Pr
>> Capability indices