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 $156$ to $232$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $118$ to $270$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 80$ to $308$.
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. $$ 194 - 38 = 156 $$ $$ 194 + 38 = 232 $$ The range of numbers is 156 to 232
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. $$ 194 - 2 \cdot 38 = 118 $$ $$ 194 + 2 \cdot 38 = 270 $$ The range of numbers is 118 to 270
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. $$ 194 - 3 \cdot 38 = 80 $$ $$ 194 + 3 \cdot 38 = 308 $$ The range of numbers is 80 to 308
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 80 and 118
- 13.5% of the data values will lie between 118 and 156
- 34% of the data values will lie between 156 and 194
- 34% of the data values will lie between 194 and 232
- 13.5% of the data values will lie between 232 and 270
- 2.35% of the data values will lie between 270 and 308