Step 1. Check to see that your process is stable
You need at least 30 points. Assemble your data in a spreadsheet. We will call your data series X. Have the X values run down the page. Number the points from 1 to N - the number of points. Calculate the process average (X-bar) = the sum of the Xs divided by N.
Plot your data (y-axis) against its number in the series (x-axis). Look at this plot. Are there big jumps in the series or does it move at random from one side of the process average to the other. If there are big jumps, this could indicate a change in process average. That is, big jumps in the series may indicate that the process is not in 'statistical control'. You need at least 30 points in 'control' to be able to go on.
Step 2. Check to see that your process has approximately a normal distribution
If it is in control, calculate the average and standard deviation. Calculate the Z-score for each point.
Sort the Z-scores - ascending order. Number them 1, 2, 3 ... These are called their Z-rank.
Plot using a scatter plot, your Z-scores (y-axis) against their Z-ranks (x-axis).
If your data has a Normal distribution, the plot should approximate a straight line. The more it is curved, the less your data has a Normal distribution.
Step 3. If it is not a normal distribution, transform it
The best transformation is to use an average. Averages always have a normal distribution. If you can, have the average mean something. However, you could just put the points in groups of (say) three and take the average of each group and use those averages as your new points. Warning: If you do that, don't put any one data point in more than one group.
Other transformations are more tricky to deal with. However, you can use
- powers (X squared, X cubed), square root, cube root, inverse (1 / X). There is a 'ladder' of transformation (X to the power -3, -2, -1, -1/2, 1/2, 1, 2, 3).
- logs (log(X) or ln(X)), exponential (e to the power X, 10 to the power X)
- sin, cos, tan.
All have their place. To see if you have the appropriate transformation, go back to Step 1 above and use a Z-score plot of the transformed data. Don't get too carried away with this - keep it simple. If you are going to use a transformation, make it a simple one.
Step 4. Calculate Cpk and Step 5. Calculate percentage of the process distribution within specifications
Complete the data fields in the table below.
Notes
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When you calculate the probability and use it, you assume that your data comes from a "Normal" distribution. This is a fair assumption when your data is a physical measurement (e.g. feet or meters). It is not a good assumption if your data is a count of something or it is a time measurement (e.g. time to arrive or time to do something).
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A very quick way to obtain a normal distribution is to use 'batches'. Average the results of each batch and then take the average of those averages. The batch approach brings the 'central limit theorem' into play. By the 'central limit theorem' the averages are normally distributed.
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Cp is the simplest estimate of Process Capability. It relates to the spread of the process relative to the specification width. It does not look at how well the process average is centered within the specifications.
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If Cp is less than 100%, defects are being made. If Cp is greater than 100%, the process is less than the specification, however, defectives might be made if the process is not centered on the target value. Many customers recommend values of 1.33 or greater. This allows a margin of error in case a process shift occurs that is not immediately detected.
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The equations for process capability/performance indices are basically very simple; however, they are very sensitive to the input value for process standard deviation. Unfortunately there can be differences of opinion on how to determine standard deviation for a given situation. Motorola considers Cp and Cpk to be measures of long-term capability. An internal or external process is considered to be Six Sigma if Cp is equal to or greater than 2.0 and Cpk is equal to or greater than 1.5. A process with these indices has a calculated defect rate of 3.4 parts per million.
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Cpm takes into account variation between the process average and a target value. If the process average and the target are the same value, Cpm will be the same as Cpk. If the average drifts from the target value, Cpm will be less than Cpk.
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Confidence intervals calculated above assume normal distribution and use an alpha of 5%. (Ie willing to be wrong 5% of the time.)