### 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 $255$ to $297$.

Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $234$ to $318$.

Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 213$ 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. $$ 276 - 21 = 255 $$ $$ 276 + 21 = 297 $$ The range of numbers is 255 to 297

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. $$ 276 - 2 \cdot 21 = 234 $$ $$ 276 + 2 \cdot 21 = 318 $$ The range of numbers is 234 to 318

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. $$ 276 - 3 \cdot 21 = 213 $$ $$ 276 + 3 \cdot 21 = 339 $$ The range of numbers is 213 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 213 and 234
- 13.5% of the data values will lie between 234 and 255
- 34% of the data values will lie between 255 and 276
- 34% of the data values will lie between 276 and 297
- 13.5% of the data values will lie between 297 and 318
- 2.35% of the data values will lie between 318 and 339