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 $191$ to $269$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $152$ to $308$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 113$ to $347$.
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. $$ 230 - 39 = 191 $$ $$ 230 + 39 = 269 $$ The range of numbers is 191 to 269
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. $$ 230 - 2 \cdot 39 = 152 $$ $$ 230 + 2 \cdot 39 = 308 $$ The range of numbers is 152 to 308
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. $$ 230 - 3 \cdot 39 = 113 $$ $$ 230 + 3 \cdot 39 = 347 $$ The range of numbers is 113 to 347
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 113 and 152
- 13.5% of the data values will lie between 152 and 191
- 34% of the data values will lie between 191 and 230
- 34% of the data values will lie between 230 and 269
- 13.5% of the data values will lie between 269 and 308
- 2.35% of the data values will lie between 308 and 347