Standard Deviation

Question

Suppose a normally distributed set of data has a mean of 104 and a standard deviation of 19. Use the 68-95-99.7 Rule to determine the percent of scores in the data set expected to be below a score of 161. Give your answer as a percent and include as many decimal places as the 68-95-99.7 rule dictates.

Answer 1

The 68, 95, 99.7 rule (also known as the empirical rule) states that "given a normal distribution, 68, 95 and 99.7 percent of the scores fall in intervals between 1, 2 and 3 standard deviations on either side of the mean respectively".

So, with a mean of 104 and a standard deviation of 19, the percent of scores falling below the score of 161 would be calculated by the following steps:

1. Get the difference between the score and the mean:

161-104 = 57

1. Calculate how many standard deviations above the mean this is:

57 / 19 = 3

Thus the score is 3 standard deviations above the mean.

1. We then use the empirical rule knowledge that 99.7% of the scores fall in an interval of 3 standard deviations on either side of the mean.
2. We divide 99.7% / 2 = 49.85% to get the percent of scores in that interval that are above the mean and less than 161.
3. Finally we add that result (49.85%) to 50% which includes all scores below the mean.

50% + 49.85% = 99.85%

In conclusion, the total % of scores below 161 is 99.85%.