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 $223$ to $319$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $175$ to $367$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 127$ to $415$.
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. $$ 271 - 48 = 223 $$ $$ 271 + 48 = 319 $$ The range of numbers is 223 to 319
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. $$ 271 - 2 \cdot 48 = 175 $$ $$ 271 + 2 \cdot 48 = 367 $$ The range of numbers is 175 to 367
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. $$ 271 - 3 \cdot 48 = 127 $$ $$ 271 + 3 \cdot 48 = 415 $$ The range of numbers is 127 to 415
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 127 and 175
- 13.5% of the data values will lie between 175 and 223
- 34% of the data values will lie between 223 and 271
- 34% of the data values will lie between 271 and 319
- 13.5% of the data values will lie between 319 and 367
- 2.35% of the data values will lie between 367 and 415