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 $122$ to $170$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $98$ to $194$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 74$ to $218$.
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. $$ 146 - 24 = 122 $$ $$ 146 + 24 = 170 $$ The range of numbers is 122 to 170
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. $$ 146 - 2 \cdot 24 = 98 $$ $$ 146 + 2 \cdot 24 = 194 $$ The range of numbers is 98 to 194
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. $$ 146 - 3 \cdot 24 = 74 $$ $$ 146 + 3 \cdot 24 = 218 $$ The range of numbers is 74 to 218
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 74 and 98
- 13.5% of the data values will lie between 98 and 122
- 34% of the data values will lie between 122 and 146
- 34% of the data values will lie between 146 and 170
- 13.5% of the data values will lie between 170 and 194
- 2.35% of the data values will lie between 194 and 218