### 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 $101$ to $143$.

Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $80$ to $164$.

Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 59$ to $185$.

### 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. $$ 122 - 21 = 101 $$ $$ 122 + 21 = 143 $$ The range of numbers is 101 to 143

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. $$ 122 - 2 \cdot 21 = 80 $$ $$ 122 + 2 \cdot 21 = 164 $$ The range of numbers is 80 to 164

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. $$ 122 - 3 \cdot 21 = 59 $$ $$ 122 + 3 \cdot 21 = 185 $$ The range of numbers is 59 to 185

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 59 and 80
- 13.5% of the data values will lie between 80 and 101
- 34% of the data values will lie between 101 and 122
- 34% of the data values will lie between 122 and 143
- 13.5% of the data values will lie between 143 and 164
- 2.35% of the data values will lie between 164 and 185