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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.
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.
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 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.
The estimated standard deviation for the example is:
The estimated standard deviation is calculated to one more decimal place than the original data.
Sketch a picture of a normal distribution. Begin by drawing a horizontal line (axis). Next, draw a normal (bell-shaped) curve centered on the horizontal axis. Then draw a vertical line from the horizontal axis through the center of the curve, cutting it in half. This line represents the overall average of the data and is always located in the center of a normal distribution. Label the line with the value for the overall average and its symbol. The value of the overall average in the example is 10.00 and the symbol for the overall average from the chart is . The example completed through this step follows.
Variables control chart pairs, which can be creating using SQCpack, illustrate central location and variability.
To look for process variability, study the range, standard deviation (sigma), or moving range chart of the control chart pair. These will show how data points within the subgroup differ from each other. Interpret for variability first. If out-of-control conditions exist here, address them before continuing. Too much variability in the subgroups can be a difficult challenge, but until this variability is reduced, it does little good to work on the target or central location.
To understand this, consider a marksman. If the pattern of shot varies wildly, one time tight and another time loose, all the marksman can do is aim at the middle of the target and hope for the best (see Figure A). If he can tighten up the shot pattern, though, he can place shots to his choosing inside the target (see Figure B).
Use the average (X-bar), median, or individuals (X) chart to analyze the central location of the process. This indicates where the middle of the subgroup is.
Here the marksman’s shot pattern is tight, showing little variability, but where is it placed in relation to the bullseye? Figure C shows the variability is tight or precise, but there is no accuracy. Figure D shows a marksman whose shots are both precise and on target or accurate.
If you are considering an individuals and moving range chart, keep in mind that you are looking at actual readings from the system, not averages or medians. Individual readings may not be normally distributed for a stable system. They may be skewed if the system is naturally bounded on one side. Characteristics such as flatness and timeliness are bounded by zero.
There is the tendency to recalculate control limits whenever a change is made to the process. However, you should extend the existing control limits out over the new data until you see evidence that the change has had an impact on the data, such as shifting or out-of-control evidence. Then recalculate limits using only data subgroups collected after the change was made.
There are no hard and fast rules for recalculating control limits, but here are some thoughts to help you decide.
The purpose of any control chart is to help you understand your process well enough to take the right action. This degree of understanding is possible only when the control limits appropriately reflect the expected behavior of the process. When the control limits no longer represent the expected behavior, you have lost your ability to take the right action. Merely recalculating the control limits is no guarantee that the new limits will properly reflect the expected behavior of the process.
Ideally, you should be able to answer yes to all of these questions before recalculating control limits.
To create control charts and easily recalculate control limits, try software products like SQCpack.
See also: >> Analyze for special cause variation >> Declare the system stable or unstable >> What do the chart pairs mean (variables control charts only)
If the system fails any of the tests for control, it is out-of-control or unstable.
If your system is unstable, does the special cause of variation create a favorable output? Does it improve the process? If yes, find the special cause(s) of variation. Try to replicate it and incorporate it into the process.
If not, and it creates a negative output or hurts the process, find the special cause(s) of variation and take steps to eliminate it. After you remove special causes of variation, continue collecting, charting, and analyzing the data. Once the process has been stabilized, consider performing capability analysis to compare how the process is running in relation to its specifications.
If it does not fail any of the tests for control, the process is stable.
If your process is stable, is the data normal distributed? Is it capable of producing output that is within specifications? You may want to create a histogram and perform capability analysis to learn more about your process.
If the process is stable, but failing to meet specification requirements, look for the source(s) of common cause variation. Can these be eliminated or reduced? How can the system be improved?
Note: A process that is currently stable, may not remain stable indefinitely. A new batch of raw material, a new operator, or new equipment, can change the output. Therefore, you should continue collecting, charting, and analyzing data for stable processes.
Capability analysis can be done using software products such as SQCpack.
See also: >> Analyze for special cause variation >> When do you recalculate control limits >> What do the chart pairs mean (variables control charts only)
Note: Use this test for control sparingly. There is a tendency to overcontrol the system when using this test. Use it only when there is some doubt about the system’s stability.
When four out of five consecutive points lie beyond the 1-sigma limit on one side of the average, the system is declared unstable.
See also: >> Any nonrandom pattern >> Too close to the average >> Too far from the average >> Cycles >> Trends >> Sawtooth >> Clusters >> 2 of 3 points beyond 2 sigma