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 $168$ to $226$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $139$ to $255$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 110$ to $284$.
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. $$ 197 - 29 = 168 $$ $$ 197 + 29 = 226 $$ The range of numbers is 168 to 226
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. $$ 197 - 2 \cdot 29 = 139 $$ $$ 197 + 2 \cdot 29 = 255 $$ The range of numbers is 139 to 255
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. $$ 197 - 3 \cdot 29 = 110 $$ $$ 197 + 3 \cdot 29 = 284 $$ The range of numbers is 110 to 284
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 110 and 139
- 13.5% of the data values will lie between 139 and 168
- 34% of the data values will lie between 168 and 197
- 34% of the data values will lie between 197 and 226
- 13.5% of the data values will lie between 226 and 255
- 2.35% of the data values will lie between 255 and 284