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### Scaling Test Scores

```
Date: 09/28/97 at 17:09:09
From: Hung Lo
Subject: Standard deviation

How do professors and teachers utilize the standard deviation to scale
scores? For example, a teacher intends that the class will have a mean
of 85 and standard deviation of 6. What would be the scaled score of a
student who got 96 with a class average of 90 (for that test only)?
What would the score be if the teacher had intended the score to be a
standard deviation of 16 at first hand. How does the intended standard
deviation affect the actual scaled score?
```

```
Date: 09/29/97 at 12:22:09
From: Doctor Statman
Subject: Re: Standard deviation

Hi!

The key to scaling scores is to convert to a standard score, usually
called a z-score. This is accomplished by taking a raw score, then
subtracting the mean, and then dividing that different by the standard
deviation.

For example, if the raw score is X and the mean is M and the standard
deviation is S, then the Z-score is:

Z = (X-M)/S

So let me try to work through your example:

A student got a 96 on a test that had a class average of 90. You
didn't give me a standard deviation, and I need one, so I will use 6.

That student's Z-score would be: Z = (96-90)/6 = 1.00
In other words, the student fell one standard deviation above the
mean.

Now, if the teacher wants to recenter the scores to a mean of 85 with
a standard deviation of 6, then the Z score can be converted back to a
raw score as follows.  The Z score is 1.00 which represents one
standard deviation above the mean, so 85 + 1.00 * 6 = 91.  The new
re-centered raw score would be 91.

If the new standard deviation was supposed to be 16, then the new raw
score would be 85 + 1.00 * 16 = 101, if that score is possible.

Z-scores are like a common currency. If you wanted to convert from
Japanese yen to German francs, you might convert to dollars first.
Z-scores are like the dollars of statistics.

Hope this helps.  Best wishes!

Sincerely,

-Doctor Statman,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Statistics

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