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6.2. Calculate Cpk

Cpk takes into account the center of the data relative to the specifications, as well as the variation in the process. Cpk is simple to calculate. The smaller of the two Z values is selected. This is known as Zmin . When Zmin has been selected, it is divided by 3. The formula is:

The Z values for the example are Zupper of 2.00 and Zlower of 5.00, therefore Zmin is 2.00. Cpk for the example is:

If the Cpk formula is written in full, it becomes more apparent how Cpk works.

This is the smaller of:

Graphically this can be drawn for the example as follows:

The diagram clearly shows that the overall average is too close to the upper specification. By taking the smaller of the two Z values, Cpk is always looking at the worst side, where the specification is closest to the overall average. Since it is looking only at half the picture, instead of dividing by 6 as in Cp, it is divided by 3.

A Cpk value of one indicates that the tail of the distribution and the specification are an equal distance from the overall average, as shown below:

A Cpk of less than one, as in the example, means that some of the data is beyond the specification limit. A Cpk greater than one indicates that the data is within the specification. The larger the Cpk, the more central and within specification the data.

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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6.1. Calculate Cp

Cp is an index used to assess the width of the process spread in comparison to the width of the specification. It is calculated by dividing the allowable spread by the actual spread. The allowable spread is the difference between the upper and lower specification limits. The actual spread is 6 times the estimated standard deviation. Plus or minus 3 times the estimated standard deviation contains 99.73 percent of the data and is commonly used to describe actual spread.

Cp for the example is:

A Cp of one indicates that the width of the process and the width of the specification are the same. A Cp of less than one indicates that the process spread is greater than the specification. This means that some of the data lies outside the specification. A Cp of greater than one indicates that the process spread is less than the width of the specification. Potentially this means that the process can fit inside the specification limits. The following diagrams show this graphically.

In fact, the Cp states how many times the process can fit inside the specification. So a Cp of 1.5 means the process can fit inside the specification 1.5 times. A Cp greater than one is obviously desirable. However, the example has a Cp greater than one and yet it still has data outside the specification. This is due to the position of the overall average relative to the specification. When the overall average is away from the center of the specification, the system can still produce data outside the specification even though the Cp is greater than one, as in the example below:

To overcome this problem, Cpk was created.

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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4.5. Calculate how much data is outside the specifications

As indicated in the previous step, some of the distribution is outside the specification limit. The question is, how much? To determine the percentage that falls outside the specification limits, it is necessary to find how many estimated standard deviations exist between the overall average and each specification limit. The number of standard deviations is known as the Z value. Z values are used to determine the percentage of output that is outside the specification limits using the standard normal distribution table.

>> 5.1. Find the percentage above the upper specification
>> 5.2. Find the percentage below the lower specification
>> 5.3. Determing the total percentage outside the specifications

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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5.3. Determining the total percentage outside the specifications

The total percentage outside the specification limits or requirements is found by adding the percentage outside the upper and lower specification limits. The total percent of output located outside the specification limits for the example is:

2.28 + 0 = 2.28%

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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5.2. Find the percentage below the lower specification

The Z value for the lower specification is found by subtracting the lower specification from the overall average, and then dividing by the estimated standard deviation. The Z value for the lower specification is denoted as Zlower. The lower specification for the example is 0, the overall average is 10.00, and the estimated standard deviation is 2.00. Thus, the value for Zlower for the example is:

This means that the lower specification is located 5.00 estimated standard deviations away from the overall average. Look up the Z value in the standard normal distribution table as previously described. Since the table shows Z values up to only 4, the proportion and percentage outside of the specification is taken as 0. If any of the data is outside the specification, add the percentage to the diagram.

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5.1. Find the percentage above the upper specification

The first step in determining the percentage above the upper specification is to calculate the Z value for the upper specification. This is found by subtracting the overall average from the upper specification, and then dividing by the estimated standard deviation. The Z value for the upper specification is denoted as Zupper. The upper specification for the example is 14, the overall average is 10.00, and the estimated standard deviation is 2.00. Thus, the value of Zupper for the example is:

This means that the upper specification is located 2.00 estimated standard deviations away from the overall average. Look up the Z value in the standard normal distribution table to find the estimated proportion of output that is outside the upper specification.

Z values are listed along the left and top of the table. The whole number (number to the left of the decimal) and the tenths digit (first number to the right of the decimal) are listed on the left hand side of the table, and the hundredths digit (second number to the right of the decimal) is along the top. The table shows Z values only up to 4. If the Z value is greater than 4, the proportion outside the specification is virtually 0. In the example, the Z value is 2.00. To find the percentage outside the specification, go down the left hand side of the table to 2.0 and then across to the column marked x.x0. The number is 0.0228, which is the proportion outside the specification. To convert the proportion to a percentage, multiply it by 100. The percentage outside the upper specification is 2.28 percent. Place this percentage on the diagram.

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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4. Draw the specification limits on the distribution

Draw vertical lines on the distribution to represent the lower and upper specification limits. In the example, the lower specification limit (LSL) is 0 minutes (on time) and the upper specification limit (USL) is 14 minutes. Estimate where the two lines should be located in reference to the overall average and the tails of the curve. Label each specification with its abbreviation and value. The example completed through this step follows.

The diagram shows whether any portion of the curve is beyond the specifications. In the example, some of the distribution is beyond the upper specification. If the overall average of the distribution is outside the specification, refer to “Variation – Capability analysis where the overall average is outside the specification” later in this section.

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3. Determine the location of the tails for the distribution

he next step is to determine where (at what value) the tails or ends of the curve are located. These values can be estimated by adding and subtracting three times the estimated standard deviation from the overall average. Remember, from the histogram section, that for a normal distribution, plus or minus three times the standard deviation from the overall average includes 99.73 percent of the area under the curve.

The calculation for the location of the left tail is:

For the example the left tail is:

The right tail is calculated as follows:

For the example the right tail is:

Add the values to the distribution drawn earlier. The example completed through this step follows.

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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2. Calculate the estimated standard deviation

The next stage is to calculate the position of the tails of the distribution that has just been drawn. However, in order to calculate the position of the tails, the standard deviation is required. In this version of capability analysis where data has been collected over a period of time, an estimated standard deviation is used. The symbol for the estimated standard deviation is (read “sigma hat”). The formula for the estimated standard deviation is:

is calculated when constructing a control chart. Substitute M for if an X-MR chart has been completed. In the example, the value is 4.653. The denominator (d2) is a weighting factor whose value is based on the subgroup size, n, from the control chart. The value for d2 in the example, based on a subgroup size of 5, is 2.326. A short listing of the d2 values for other subgroup sizes follows. The full table of values is given in the appendix.

n
2
3
4
5
6
7
8
9
10
d2 
1.128
1.693
2.059
2.326
2.534
2.704
2.847
 2.970
 3.078

 

The estimated standard deviation for the example is:

The estimated standard deviation is calculated to one more decimal place than the original data.

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