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 $142$ to $234$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $96$ to $280$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 50$ to $326$.
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. $$ 188 - 46 = 142 $$ $$ 188 + 46 = 234 $$ The range of numbers is 142 to 234
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. $$ 188 - 2 \cdot 46 = 96 $$ $$ 188 + 2 \cdot 46 = 280 $$ The range of numbers is 96 to 280
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. $$ 188 - 3 \cdot 46 = 50 $$ $$ 188 + 3 \cdot 46 = 326 $$ The range of numbers is 50 to 326
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 50 and 96
- 13.5% of the data values will lie between 96 and 142
- 34% of the data values will lie between 142 and 188
- 34% of the data values will lie between 188 and 234
- 13.5% of the data values will lie between 234 and 280
- 2.35% of the data values will lie between 280 and 326