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 $148$ to $212$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $116$ to $244$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 84$ to $276$.
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. $$ 180 - 32 = 148 $$ $$ 180 + 32 = 212 $$ The range of numbers is 148 to 212
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. $$ 180 - 2 \cdot 32 = 116 $$ $$ 180 + 2 \cdot 32 = 244 $$ The range of numbers is 116 to 244
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. $$ 180 - 3 \cdot 32 = 84 $$ $$ 180 + 3 \cdot 32 = 276 $$ The range of numbers is 84 to 276
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 84 and 116
- 13.5% of the data values will lie between 116 and 148
- 34% of the data values will lie between 148 and 180
- 34% of the data values will lie between 180 and 212
- 13.5% of the data values will lie between 212 and 244
- 2.35% of the data values will lie between 244 and 276