## About this lesson

Z scores are a method of normalizing data from different data sets for comparison or prediction. Z scores normalize the data using the process standard deviation. The Z transformation table will convert Z scores into percentages.

## Exercise files

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## Quick reference

### Z Scores

Z scores are a method of normalizing data from different data sets for comparison or prediction. Z scores normalize the data using the process standard deviation. The Z transformation table will convert Z scores into percentages.

### When to use

Z scores are typically used when assessing basic process performance. They are also helpful when doing comparisons for data sets to apply insight from one data set to another. Finally, they are often used to set expectations on process yield based upon different tolerance levels or performance thresholds.

### Instructions

The Z score normalizes a process performance parameter using the mean and standard deviation for the parameter distribution.The formula that is used is:

The process parameter mean is subtracted from the value and the result is divided by the standard deviation. If the process parameter value is below the mean, the Z score is a negative number. If it is greater than the mean value, the Z Score is a positive number. When the parameter value equals the mean, the Z score is zero.

Through Z score normalization, dissimilar processes can be compared by comparing the process behavior when the Z scores are similar even though actual process parameter values are different. By the same token, the same process parameter value could have very different Z scores depending upon the distribution in which that value occurred.

The Z score is often used to create a Z transformation which means to change that Z score into a percentage value for process performance. For any given Z score, there is a known percentage of the process values that will be above that score and a known percentage that will be below that score due to the characteristics of a normal distribution. That Z transformation is a look-up table that shows the conversion. The Z transformation table starts at the mean in the upper left corner – which is a Z score value of zero. To simplify the table, it does not show negative Z scores values. So if you have a negative Z score, you must use the absolute value of the score. To find the percentage, first find the row that represents the Z score value in tenths and then follow across that row until the column that represents the Z score value in hundredths is reached. That is the decimal equivalent of the Z score on one side of the mean. If it is a positive Z score, add 50% to the value and you will have the percentage of parameter values that are less than the value represented by the Z score. If the Z score was negative, add 50% and you will have the percentage of the parameter values that are greater than the value represented by the Z score.

The Z transformation can be used to predict process yield for a given parameter value. In the same way, if a desired process yield is determined, the Z transformation can determine that process parameter value that should be used for tolerance limits.

### Hints & tips

- The Z transformation is normally a function within statistical software, such as Minitab. There is even a Z transformation function in Excel, “ZTest.”
- The reason for adding 50% to the Z transformation percentage is that the table starts at the mean. So one half of the normal distribution (50%) must be added to the table value.
- The Z score is used in some of the hypothesis tests that will be covered in other modules. A thorough understanding of Z scores will make it easier to work with those tests.
- If you are planning on sitting for one of the IASSC exams, the Z transformation table will be made available – so you don’t need to memorize it.

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