Z-Score

Description:

The z-score calculation is used in the MEASURE phase and CONTROL phase of a DMAIC Six Sigma project. The score converts a point of interest in terms of standard deviations from the population mean. It allows the comparison of observations from different normal distributions.

Other names for the z-score are z-values, normal scores, and standardized variables.

Objective:

Determine a baseline z-score in the MEASURE phase after the MSA has passed. This preliminary value provided in the project contract may need refinement, this exercise is done in the MEASURE phase to get an accurate starting point.

Another z-score is calculated in the CONTROL phase. Sometimes it is done at the end of the IMPROVE phase but either way it is the final score indicating the change (better or worse) relative to the baseline measurement.

The z-score is most often used in the z-test in the Student's t-test for a population whose parameters are actually known (not estimates). But since it is rare to know the entire population, the t-test is more commonly used.

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Short Term versus Long Term

It is possible to estimate the long or short term performance based on that value. A GB/BB can apply the 1.5 sigma shift if converting to or from long term sigma or short term sigma (if you are a believer in this 1.5 sigma shift).

If the z-score was calculated from a sample then the long-term performance can be estimated by adding 1.5 to your calulated value.

If the value you came up with was from a population it is a long term representation of the process performance. Therefore, if interested in the projected short-term performance then you would subtract 1.5 from your calculated value.

LONG TERM to SHORT TERM: SUBTRACT 1.5 SIGMA

SHORT TERM to LONG TERM: ADD 1.5 SIGMA

If the point of interest (x) is located on the mean then the z-score is zero. This seems logical since the point lies directly on the mean and zero standard deviations from it.

If the point of interest (original measurement, x) is below the mean then the z-score will be negative. The z-score is positive when the x is a value higher or to the right of the mean.

Converting Z-scores

When a population is normally distributed, the percentile rank may be determined from the z-score and statistical tables. It is important to understand the table being used as websites or textbooks display the values in different ways. This is most often the most challenging part to a newer GB/BB and common mistake made on exams and projects. Get comfortable with converting z-scores to percentile ranks, PPM's, and using tables.