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 $246$ to $344$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $197$ to $393$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 148$ to $442$.
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. $$ 295 - 49 = 246 $$ $$ 295 + 49 = 344 $$ The range of numbers is 246 to 344
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. $$ 295 - 2 \cdot 49 = 197 $$ $$ 295 + 2 \cdot 49 = 393 $$ The range of numbers is 197 to 393
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. $$ 295 - 3 \cdot 49 = 148 $$ $$ 295 + 3 \cdot 49 = 442 $$ The range of numbers is 148 to 442
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 148 and 197
- 13.5% of the data values will lie between 197 and 246
- 34% of the data values will lie between 246 and 295
- 34% of the data values will lie between 295 and 344
- 13.5% of the data values will lie between 344 and 393
- 2.35% of the data values will lie between 393 and 442