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The Z transformation is a technique that converts a data value into the number of standard deviations it is above or below the mean of the distribution. This provides an opportunity to compare between data sets and to estimate process yield rates.

Quick reference

Z Transformation

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 used 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 capability and 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 a given Z score, there is a known percentage of the process values in a normal curve 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 does that conversion.  Normally, the Z transformation table is a pair of two tables. The Right Tail Z Transformation Table is used when the Z score is positive and the Left Tail Z Transformation Table is used when the Z score is negative. 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 and indicates the percentage of the distribution that is below that Z score value.

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 the 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 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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  • 00:05 Hi, I'm Ray Sheen.
  • 00:06 We've discussed some basics associated with statistical process performance.
  • 00:11 I now want to introduce the Z value and discuss the Z-transformation.
  • 00:16 So let's talk about the Z-score, or is often referred to the standard score.
  • 00:21 The Z-score indicates where a particular instance of a process performance is with
  • 00:26 respect to the normal process performance distribution.
  • 00:29 Is it above or below the average, and by how much?
  • 00:33 But what makes the Z-score special is that it does not use the units of
  • 00:37 the process performance,
  • 00:39 rather it uses the units of standard deviation of the process performance.
  • 00:44 So a Z-score of 1 is one standard deviation above the mean, and
  • 00:49 a Z-score of -2 is two standard deviations below the mean.
  • 00:53 So to do this, the value of process performance must
  • 00:57 be converted into an equivalent value of units of standard deviation.
  • 01:01 You can see the formula here on the side.
  • 01:04 You just subtract the value of the mean from your current process value and
  • 01:08 divide the result by the value of the standard deviation.
  • 01:11 In this diagram, the Z=1.5 line is shown.
  • 01:15 It is 1.5 standard deviations above the mean value.
  • 01:19 The Z value will open up some great possibilities for an analysis.
  • 01:23 Let's take a look at a few of them.
  • 01:26 Z value can also be used to compare information about performance
  • 01:30 between datasets.
  • 01:32 We can take data from two, otherwise, dissimilar datasets.
  • 01:36 And by using the Z value, we can normalize them and possibly do some comparisons.
  • 01:42 We really can compare apples to oranges.
  • 01:45 Some things to watch for, the absolute value of a parameter,
  • 01:49 let's say it's the weight of an apple,
  • 01:51 may be exactly the same as the absolute value of that same parameter for
  • 01:55 an item that's in a different dataset, such as the weight of an orange.
  • 01:59 But after we consider the mean and standard deviation of apples and
  • 02:04 oranges, they may be very different Z values.
  • 02:07 By the same token, we could take two items with the same Z-score,
  • 02:11 such as an apple with a Z-score of 1,
  • 02:13 meaning it's one standard deviation heavier than the average apple.
  • 02:17 And an orange with a Z-score of 1,
  • 02:19 meaning it's one standard deviation heavier than the average orange.
  • 02:24 Although they may have the same Z-score, they're likely to be a different weight,
  • 02:29 because the average and standard deviation of the two distributions are different.
  • 02:34 This illustrates that the Z-score can be used to normalize data parameters
  • 02:39 between distributions.
  • 02:40 Now, let's discuss the Z- transformation.
  • 02:44 With a Z-score, I can also get a percentage value for
  • 02:47 the parameter through a Z-transformation using the table on this page.
  • 02:51 And don't worry if you can't read it.
  • 02:54 This cable can be looked up on Google with no difficulty.
  • 02:57 If you're going to take one of the Isaac exams,
  • 03:00 the table is available to you during the exam.
  • 03:02 The way to use this table is to start with the row that has your Z-score at
  • 03:07 the nearest tenth, then look across the table to find the column
  • 03:11 with the value in your hundredths place.
  • 03:14 This table is the left side of the normal distribution, so
  • 03:18 the z-score is always negative.
  • 03:20 The value in the table will be the percentage of the data points that are to
  • 03:25 the left of the Z-score.
  • 03:27 That means the percentage of the distribution that has a lower value
  • 03:31 notice.
  • 03:32 If we go in with the Z-score of 0,
  • 03:34 that's in the lower left corner of this table, the value is 50%.
  • 03:38 What that means is that you're right at the middle of the distribution, and
  • 03:43 half the data points are below that value, and of course that means half or above.
  • 03:47 In case you're wondering, there is a table for the positive Z-score also.
  • 03:52 This is the right side of the normal curve.
  • 03:54 And if you're looking for the 0 z-score, well,
  • 03:57 it's in the upper left corner on this table.
  • 03:59 So that's where the 50% value is sitting.
  • 04:02 As we have said, the Z value is directly related to the percentage of the data
  • 04:06 points in the distribution.
  • 04:08 Let's do a quick example.
  • 04:10 If I have a Z-score of 2.12, I start with the 2.1 row.
  • 04:15 I then go over to the third column to get the value for 2.12.
  • 04:21 The percentage value is 98.3%.
  • 04:25 That is the percentage of the points that are to the left of the Z-score and,
  • 04:30 of course, it means that 1.7% of the points are at the right of that Z-score.
  • 04:35 You can also go the other direction with this table.
  • 04:38 If you want to find the Z-score that would be associated with having 15%
  • 04:42 of the data points to the right of the value you are selecting.
  • 04:46 That would mean that the percentage value in the table is 85%.
  • 04:49 The closest value I can find to 85% is a Z-score of 1.04.
  • 04:54 The value in that table position is actually 85.083%.
  • 05:01 Z-scores and Z-transformations are practical ways to normalize
  • 05:06 a parameter value for comparison with other data sets or for prediction
  • 05:10 of real world process performance based upon a specific data point.

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