How Do You Calculate a Z-Score/ Sigma Level?by Jeff Sauro | June 14, 2004 :: 2 Related Articles:: 44 Related Questions
The benefit of using a z-score in usability metrics was explained in "
What's
a Z-Score and why use it in Usability Testing?" this article
discusses different ways of calculating a z-score.
The short answer is: It depends on your data
and what you're looking for. If you've encountered the z-score in a
statistics book you usually get some formula like:
The above formula is for obtaining a z-score for an entire
population. Usability testing obviously samples a very small subset
of the population and thus the following formula is used:
Where x-bar and
s are used as estimators for
the population's true mean and standard deviation. Both formulas essentially
calculate the same thing:
Calculating a Z-Score Example
For example, lets say you took the GRE a
few weeks ago and got scores of 630 Verbal and 700 Quantitative. How
good are these scores? Which is better, the Verbal or Quantitative score?
Using a z-score can tell you how far you are from the mean and thus
how well you performed. If you know the mean and standard deviations
for a set of GRE test takers you can compare your scores.
ETS
publishes the means and standard deviations of a set of test takers
on the GRE website.
| |
Verbal |
Quantitative |
| Mean |
469 |
591 |
| StDev |
119 |
148 |
By plugging in your scores you get the following:
Verbal z = (630 -
469) ÷ 119 = 1.35σ
Quantitative z =
(700 - 591) ÷ 148 =
.736σ
To convert these sigma values into a percentage
you can look them up in a standard z-table, use the Excel formula
=NORMSDIST(1.35) or use the
Z-Score to Percentile Calculator
(choose 1-sided) and get the percentages : 91% Verbal and 77% Quantitative.
You can see where your score falls within the sample of other test takers
and also see that the verbal score was better than the quantitative
score. Assuming the sample data was normally distributed, here's how
the scores would look graphically:
Figure 1: Verbal Score
Figure 2: Quantitative Score
An
interactive Graph of the Standard Normal Curve similar to Figures 1 & 2 is available for you to visualize how the z-scores and the area under the normal curve correspond. The graphs also allow you to see the difference between one and two-sided (also called two-tailed) areas.
Z-Scores and Process Sigma
In Six Sigma the process sigma metric is derived using the same method as a z-score. However, in Six Sigma you are measuring the distance a sample mean is above a specification limit--there can be an upper and lower spec limit that a sample must fall between as well. As in the z-score, you still use the same normal-deviates from the z-table to approximate the area under the curve.
The process sigma metric is essentially a Z equivalent.
When testing software with users, task times
are usually a good metric that will reveal the individual differences
in performance. For task times there typically is only an upper spec
limit. That is, it usually doesn't matter how fast a user completes
a task, but it does matter if a user takes too long. For example, say
you and your product team determined that a task should be completed
in 120 seconds. 120 seconds becomes your Upper Spec Limit (USL). You
sampled 10 users and got these task times:
| Sample |
100
99
101
125
100
123
96
90
98
116 |
USL: 120
Mean: 104
StDev: 12
|
To calculate the process sigma you subtract the mean (104)
of the sample from the target (120) and divide by the sample standard
deviation (12). For Sample 1 the process sigma is -1.32σ. The
visual representation of the data can be seen below:
In the case of task times, a negative process
sigma is ideal--as you want more people completing the task below the
task time, not above it. You can simply drop the negative when communicating
the results in the event it causes confusion.
If you were to make radical improvements
to the UI and then sampled another set of ten users, here are more results:
| Sample 2 |
60
75
99
88
65
72
75
72
87
65 |
USL: 120
Mean: 75.8
StDev: 12.14 |
In the redesign, the average of the new sample
is well below the spec limit and the process sigma is now very high.
The corresponding defect area is now only .01% and the quality area
is 99.98%
Of course having users perform that much below the spec limit is not
very common due to the inherent variability in user performance.
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Related ArticlesWhat's a Z-Score and Why Use it in Usability Testing? Calculating a Sigma Level from Task Success
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Ask a Question
| June 19, 2008 | cochise180 wrote: |
| how do I calculate z score for a nominal of zero with usl 0.008" and lsl -0.008". i get good Cpk and Cp but bad Z score. |
|
| May 9, 2008 | Tim wrote: |
| Am confused as to how to calculate a negative Z score into a percentage using a standard Z score table |
|
| December 19, 2007 | anonomous wrote: |
| Why can't all math be explained this simple?
Thank you for this explanation! |
|
