### 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 $286$.

Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $205$ to $313$.

Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 178$ to $340$.

### 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. $$ 259 - 27 = 232 $$ $$ 259 + 27 = 286 $$ The range of numbers is 232 to 286

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. $$ 259 - 2 \cdot 27 = 205 $$ $$ 259 + 2 \cdot 27 = 313 $$ The range of numbers is 205 to 313

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. $$ 259 - 3 \cdot 27 = 178 $$ $$ 259 + 3 \cdot 27 = 340 $$ The range of numbers is 178 to 340

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 178 and 205
- 13.5% of the data values will lie between 205 and 232
- 34% of the data values will lie between 232 and 259
- 34% of the data values will lie between 259 and 286
- 13.5% of the data values will lie between 286 and 313
- 2.35% of the data values will lie between 313 and 340