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 $194$ to $268$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $157$ to $305$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 120$ to $342$.
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. $$ 231 - 37 = 194 $$ $$ 231 + 37 = 268 $$ The range of numbers is 194 to 268
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. $$ 231 - 2 \cdot 37 = 157 $$ $$ 231 + 2 \cdot 37 = 305 $$ The range of numbers is 157 to 305
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. $$ 231 - 3 \cdot 37 = 120 $$ $$ 231 + 3 \cdot 37 = 342 $$ The range of numbers is 120 to 342
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 120 and 157
- 13.5% of the data values will lie between 157 and 194
- 34% of the data values will lie between 194 and 231
- 34% of the data values will lie between 231 and 268
- 13.5% of the data values will lie between 268 and 305
- 2.35% of the data values will lie between 305 and 342