Scaling Test ScoresDate: 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/ |
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