Empirical Rule Calculator
Answer:
For a bell-shaped (normal) distribution:
Approximately 68% of the data values will fall within 1 standard deviation of the mean, from $153$ to $231$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $114$ to $270$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 75$ to $309$.
Solution:
The first part of the empirical rule states that 68% of the data values will fall within 1 standard deviation of the mean. To calculate "within 1 standard deviation," you need to subtract 1 standard deviation from the mean, then add 1 standard deviation to the mean. That will give you the range for 68% of the data values. $$ 192 - 39 = 153 $$ $$ 192 + 39 = 231 $$ The range of numbers is 153 to 231
The second part of the empirical rule states that 95% of the data values will fall within 2 standard deviations of the mean. To calculate "within 2 standard deviations," you need to subtract 2 standard deviations from the mean, then add 2 standard deviations to the mean. That will give you the range for 95% of the data values. $$ 192 - 2 \cdot 39 = 114 $$ $$ 192 + 2 \cdot 39 = 270 $$ The range of numbers is 114 to 270
Finally, the last part of the empirical rule states that 99.7% of the data values will fall within 3 standard deviations of the mean. To calculate "within 3 standard deviations," you need to subtract 3 standard deviations from the mean, then add 3 standard deviations to the mean. That will give you the range for 99.7% of the data values. $$ 192 - 3 \cdot 39 = 75 $$ $$ 192 + 3 \cdot 39 = 309 $$ The range of numbers is 75 to 309
Finally, we can use the symmetry of the bell curve to further divide up the percentages.
- 2.35% of the data values will lie between 75 and 114
- 13.5% of the data values will lie between 114 and 153
- 34% of the data values will lie between 153 and 192
- 34% of the data values will lie between 192 and 231
- 13.5% of the data values will lie between 231 and 270
- 2.35% of the data values will lie between 270 and 309