Z Scores

Quick reference

• 00:04 Hi, I'm Ray Sheen.
• 00:05 We've discussed some basics associated with statistical process performance.
• 00:10 I now want to introduce the z value, and discuss the z transformation.
• 00:17 So let's talk about z-score or as often referred to, the standard score.
• 00:22 Z-score indicates where particular instance of a process performance is
• 00:26 with respect to the normal process performance distribution.
• 00:30 Is it above the average or below the average and by how much?
• 00:34 But what makes the z-score special is that it does not use units of process
• 00:39 performance.
• 00:40 Rather, it uses the units of standard deviation of the process performance.
• 00:44 So a z-score of 1 is 1 standard deviation above the mean and
• 00:49 a z-score of -2 is 2 standard deviations below the mean.
• 00:54 Now to do this, the value of process performance must be converted into
• 00:58 an equivalent value in the units of standard deviation.
• 01:02 You can see the formula here on this slide.
• 01:05 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 The z value will open up some great possibilities for analysis.
• 01:16 Let's take a look at a few of them here.
• 01:19 The z value can allow us to compare information about performance
• 01:23 between data sets.
• 01:25 We can take data from two otherwise dissimilar data sets.
• 01:28 And by using the z value, we can normalize them and possibly do some comparisons.
• 01:34 We really can compare apples to oranges.
• 01:37 Some things to watch for.
• 01:39 The absolute value of a parameter.
• 01:40 For illustration, let's say it's the weight of an apple.
• 01:43 May be exactly the same as the absolute value of the same parameter for
• 01:47 an item in a different set.
• 01:49 Let's say, it's the weight of an orange, but when we consider the mean and
• 01:52 standard deviation of apples and oranges, then they have very different z values.
• 01:57 By the same token, we could take two items with the same z value such as an apple
• 02:02 with a z-score of 1, meaning it is one standard deviation heavier than average,
• 02:07 and an orange with a z-score of 1,
• 02:09 meaning it is 1 standard deviation heavier than the average orange.
• 02:13 Although, they are the same z-scores, they are likely different actual weights,
• 02:17 because the average in standard deviation of the two distributions is different.
• 02:21 This illustrates that z-scores can be used to normalize data parameters
• 02:25 between distributions.
• 02:27 Now, let's discuss the z transformation.
• 02:31 With the z-score, I can always get a percentage value for
• 02:34 the parameter through a z transformation using the table on this page.
• 02:39 And don't worry if you can't read it.
• 02:41 This table can be looked up or Googled with no difficulty.
• 02:44 In fact, if you take one of the IASSC exams
• 02:47 this table will be made available to you for use during the exam.
• 02:51 The way this table works,
• 02:52 is that a value of zero is at the 50% point of the distribution.
• 02:57 50% of the data values are above the mean and 50% are below.
• 03:02 So the z-score is zero.
• 03:05 On this table,
• 03:06 take the absolute value of your z-score to the nearest tenth of a standard deviation.
• 03:11 Into the vertical axis, and then go across the columns until you reach the column for
• 03:16 the 100th value of your z-score.
• 03:18 That will lead you to the percentage of the distribution that's either above or
• 03:23 below the value represented by your z-score.
• 03:27 If it's a positive z-score, you're above the mean.
• 03:29 So add 50% to the table value and
• 03:32 that is a percentage of the distribution below that point.
• 03:37 So let's say my z-score is 1.25,
• 03:40 I start with the 1.2 row, I go over to the 0.05 column.
• 03:45 The value there is .3944.
• 03:48 Add 50% to that and I know that 89.44% of the process values are below this point.
• 03:57 And, of course, that means that 10.56% are above.
• 04:01 If the z value is negative, you're below the mean.
• 04:04 So, in that case,
• 04:05 you add 50% to determine how much of the distribution is above that point.
• 04:11 Let's do another example.
• 04:12 If my z-score is negative 2.12, I start with the 2.1 row,
• 04:18 I then go over to the 0.02 column, and the value is 0.4826,
• 04:24 add 50% to that and I get 98.26% of the distribution
• 04:29 is above that point which means that only 1.74% is below that point.
• 04:36 In addition to comparison between data sets.
• 04:39 The z-scores a tremendous aid in predicting physical process performance.
• 04:43 To do this we will use the z transformation table again.
• 04:47 As we Illustrated on the last slide, the z Transformation table gives us a percentage
• 04:51 of the distribution that are either above or below the value of the process.
• 04:56 So let's say that we've been told that we wanna have a process yield
• 04:59 that is above some value.
• 05:01 From my illustration, I will say that we want a yield above 98%.
• 05:06 We can go into the z transformation table and
• 05:09 see that a z-score of 2.06 will get us there.
• 05:13 I can now use the distribution mean and standard deviation to tell you what
• 05:18 the process value limit I should use in order to achieve a 98% yield rate.
• 05:22 Of course, we can come about this from the other direction.
• 05:27 I can start with a parameter value such as a tolerance limit.
• 05:30 On a drawing, convert that to a z-score, and then tell you what kind of yield or
• 05:35 scrap rate you can expected based upon that value.
• 05:38 Now, you may be thinking that hauling this table is a pain.
• 05:43 Well, don't worry, almost all the statistical software packages already have
• 05:47 a function built into them to do the lookup for you.
• 05:51 In fact, even Excel has a function called z-test, which has two arguments.
• 05:56 The data for the data set, so that you can calculate a mean and
• 06:00 standard deviation, and the actual data point in question.
• 06:06 Z-scores and z transformations are practical ways to normalize a parameter
• 06:11 value for comparison with other data sets for prediction of
• 06:15 real world process performance, based upon a specific data point.

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