Statistics Task

Question

The scores for all high school seniors taking the verbal section of the scholastic aptitude test (sat) in a particular year had a mean score of 490 and a standard deviation of 100. The distribution of sat scores is bell-shaped.

• what does the Empirical Rule suggest?

• about 68% of students scored between what 2 values?

• about 95% of the students scored between 390 and 590 on this Sat test?

• one student scored 795 on this test. How did this student do compared to the rest of the scores?

Answer 1

The empirical rule suggests that since your data is assumed to normally distributed (nice bell curve), than almost all of the data will fall within 3 standard deviations of the mean. This can be taken further to say that 99.7% of the data falls within 3 standard deviations, 95% within 2, and 68% within 1 standard deviation from the mean.

So to find the values between which 68% of scores fall, we take our mean (xbar=490) and subtract the standard deviation (σ=100) to find the lower bound. To find the upper bound we just add the standard deviation to the mean. So 68% of the scores fall between 490-100 = 390 and 490+100 =590

The student who scored 795 did INCREDIBLY well. Their score is more than 3 standard deviations above the mean. You would not expect many values this high because it is at the very tail end of the bell curve.