The test scores for a very large statistics class have a bell-shaped distribution with a mean of 70 points.?

Question

If 16% of all the students in the class scored above 85, what is the standard deviation of the scores?
If 95% of the scores are between 60 and 80, what is the standard deviation?

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Answer 1

It is known that with a normal (bell-shaped) distribution, 68% of the observations fall within +-1 standard deviation from the mean, 95% fall within +-2 standard deviations from the mean. (You can look this up from a normal table, but statistical geeks just memorize it.)

So, if 68% fall within one SD, that means that 32% fall outside this range, with 16% on each tail. If 16% of your scores are above 85, then 85 is one SD from your mean , and your standard deviation = 85 - mean = 85-70 = 15.

If 95% of the scores fall between 60 and 80, that means that 80 is +2 SD from the mean, and 60 is -2 SD from the mean. If the mean is 70, then the SD is 5.