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 $192$ to $262$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $157$ to $297$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 122$ to $332$.
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. $$ 227 - 35 = 192 $$ $$ 227 + 35 = 262 $$ The range of numbers is 192 to 262
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. $$ 227 - 2 \cdot 35 = 157 $$ $$ 227 + 2 \cdot 35 = 297 $$ The range of numbers is 157 to 297
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. $$ 227 - 3 \cdot 35 = 122 $$ $$ 227 + 3 \cdot 35 = 332 $$ The range of numbers is 122 to 332
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 122 and 157
- 13.5% of the data values will lie between 157 and 192
- 34% of the data values will lie between 192 and 227
- 34% of the data values will lie between 227 and 262
- 13.5% of the data values will lie between 262 and 297
- 2.35% of the data values will lie between 297 and 332