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 $249$ to $345$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $201$ to $393$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 153$ to $441$.
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. $$ 297 - 48 = 249 $$ $$ 297 + 48 = 345 $$ The range of numbers is 249 to 345
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. $$ 297 - 2 \cdot 48 = 201 $$ $$ 297 + 2 \cdot 48 = 393 $$ The range of numbers is 201 to 393
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. $$ 297 - 3 \cdot 48 = 153 $$ $$ 297 + 3 \cdot 48 = 441 $$ The range of numbers is 153 to 441
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 153 and 201
- 13.5% of the data values will lie between 201 and 249
- 34% of the data values will lie between 249 and 297
- 34% of the data values will lie between 297 and 345
- 13.5% of the data values will lie between 345 and 393
- 2.35% of the data values will lie between 393 and 441