### 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 $232$ to $298$.

Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $199$ to $331$.

Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 166$ to $364$.

### 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. $$ 265 - 33 = 232 $$ $$ 265 + 33 = 298 $$ The range of numbers is 232 to 298

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. $$ 265 - 2 \cdot 33 = 199 $$ $$ 265 + 2 \cdot 33 = 331 $$ The range of numbers is 199 to 331

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. $$ 265 - 3 \cdot 33 = 166 $$ $$ 265 + 3 \cdot 33 = 364 $$ The range of numbers is 166 to 364

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 166 and 199
- 13.5% of the data values will lie between 199 and 232
- 34% of the data values will lie between 232 and 265
- 34% of the data values will lie between 265 and 298
- 13.5% of the data values will lie between 298 and 331
- 2.35% of the data values will lie between 331 and 364