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 $195$ to $267$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $159$ to $303$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 123$ to $339$.
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. $$ 231 - 36 = 195 $$ $$ 231 + 36 = 267 $$ The range of numbers is 195 to 267
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. $$ 231 - 2 \cdot 36 = 159 $$ $$ 231 + 2 \cdot 36 = 303 $$ The range of numbers is 159 to 303
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. $$ 231 - 3 \cdot 36 = 123 $$ $$ 231 + 3 \cdot 36 = 339 $$ The range of numbers is 123 to 339
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 123 and 159
- 13.5% of the data values will lie between 159 and 195
- 34% of the data values will lie between 195 and 231
- 34% of the data values will lie between 231 and 267
- 13.5% of the data values will lie between 267 and 303
- 2.35% of the data values will lie between 303 and 339