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 $225$ to $301$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $187$ to $339$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 149$ to $377$.
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. $$ 263 - 38 = 225 $$ $$ 263 + 38 = 301 $$ The range of numbers is 225 to 301
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. $$ 263 - 2 \cdot 38 = 187 $$ $$ 263 + 2 \cdot 38 = 339 $$ The range of numbers is 187 to 339
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. $$ 263 - 3 \cdot 38 = 149 $$ $$ 263 + 3 \cdot 38 = 377 $$ The range of numbers is 149 to 377
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 149 and 187
- 13.5% of the data values will lie between 187 and 225
- 34% of the data values will lie between 225 and 263
- 34% of the data values will lie between 263 and 301
- 13.5% of the data values will lie between 301 and 339
- 2.35% of the data values will lie between 339 and 377