What is the approximate percentage score of less than 140 using the 68-95-99.7 rule if a set of test scores is normally distributed with a mean of 100 and a standard deviation of 20?

Answer 1

The 68-95-99.7 rule states that in a normally distributed set of data, approximately 68% of all observations lie within one standard deviation either side of the mean, 95% lie within two standard deviations and 99.7% lie within three standard deviations.

Or looking at it cumulatively:

0.15% of the data lie below the mean minus three standard deviations

2.5% of the data lie below the mean minus two standard deviations

16% of the data lie below the mean minus one standard deviation

50 % of the data lie below the mean

84 % of the data lie below the mean plus one standard deviation

97.5% of the data lie below the mean plus two standard deviations

99.85% of the data lie below the mean plus three standard deviations

A normally distributed set of data with mean 100 and standard deviation of 20 means that a score of 140 lies two standard deviations above the mean. Hence approximately 97.5% of all observations are less than 140.