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 $229$ to $273$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $207$ to $295$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 185$ to $317$.
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. $$ 251 - 22 = 229 $$ $$ 251 + 22 = 273 $$ The range of numbers is 229 to 273
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. $$ 251 - 2 \cdot 22 = 207 $$ $$ 251 + 2 \cdot 22 = 295 $$ The range of numbers is 207 to 295
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. $$ 251 - 3 \cdot 22 = 185 $$ $$ 251 + 3 \cdot 22 = 317 $$ The range of numbers is 185 to 317
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 185 and 207
- 13.5% of the data values will lie between 207 and 229
- 34% of the data values will lie between 229 and 251
- 34% of the data values will lie between 251 and 273
- 13.5% of the data values will lie between 273 and 295
- 2.35% of the data values will lie between 295 and 317