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 $143$ to $225$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $102$ to $266$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 61$ to $307$.
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. $$ 184 - 41 = 143 $$ $$ 184 + 41 = 225 $$ The range of numbers is 143 to 225
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. $$ 184 - 2 \cdot 41 = 102 $$ $$ 184 + 2 \cdot 41 = 266 $$ The range of numbers is 102 to 266
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. $$ 184 - 3 \cdot 41 = 61 $$ $$ 184 + 3 \cdot 41 = 307 $$ The range of numbers is 61 to 307
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 61 and 102
- 13.5% of the data values will lie between 102 and 143
- 34% of the data values will lie between 143 and 184
- 34% of the data values will lie between 184 and 225
- 13.5% of the data values will lie between 225 and 266
- 2.35% of the data values will lie between 266 and 307