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Check for Understanding
The Empirical Rule
1. What does the Empirical Rule tell us?
2. Graduate School Exam (GRE) Verbal Scores are Normally distributed with a mean of 544 and a
standard deviation of 103.
a. Sketch this Normal distribution and label the scores -3, -2, -1, 0, 1, 2, and 3 standard
deviations away from the mean.
b. Using your sketch, give an interval that contains the middle 95% of GRE scores.
c. What percent of scores fall above 647?
3. MCAT scores (the medical school entrance exam) are Normally distributed with a mean of 28.1.
The top 2.5% of students scored above 32.1. What is the standard deviation of this distribution?
Answers:
1. What does the Empirical Rule tell us?
In a Normal distribution, about 68% of observations fall within one standard deviation of the mean,
about 95% fall within two standard deviations of the mean, and about 99.7% of observations fall
within three standard deviations of the mean.
2. Graduate School Exam (GRE) Verbal Scores are Normally distributed with a mean of 544 and a
standard deviation of 103.
a. Sketch this Normal distribution and label the scores -3, -2, -1, 0, 1, 2, and 3 standard
deviations away from the mean.
235 338 441 544 647 750 853
GRE scores
b. Using your sketch, give an interval that contains the middle 95% of GRE scores.
The middle 95% of scores fell between 338 and 750.
c. What percent of scores fall above 647?
About 16% of scores fell above 647.
If 68% of scores are in the middle, that leaves out 32%, or 16% on either side of the first standard
deviation away.
3. MCAT scores (the medical school entrance exam) are Normally distributed with a mean of 28.1.
The top 2.5% of students scored above 32.1. What is the standard deviation of this distribution?
The standard deviation is about 2.0.
With 95% of the data within two standard deviations of the mean, only 2.5% of the data lies more
than two standard deviations higher than the mean and more than two standard deviations lower
than the mean, because 2.5% + 2.5% = 5%. Since the top 2.5% scored 4 points better than the mean,
two standard deviations must equal 4 points, and one standard deviation must equal 2 points.