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 $137$ to $215$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $98$ to $254$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 59$ to $293$.
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. $$ 176 - 39 = 137 $$ $$ 176 + 39 = 215 $$ The range of numbers is 137 to 215
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. $$ 176 - 2 \cdot 39 = 98 $$ $$ 176 + 2 \cdot 39 = 254 $$ The range of numbers is 98 to 254
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. $$ 176 - 3 \cdot 39 = 59 $$ $$ 176 + 3 \cdot 39 = 293 $$ The range of numbers is 59 to 293
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 59 and 98
- 13.5% of the data values will lie between 98 and 137
- 34% of the data values will lie between 137 and 176
- 34% of the data values will lie between 176 and 215
- 13.5% of the data values will lie between 215 and 254
- 2.35% of the data values will lie between 254 and 293