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 $231$ to $321$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $186$ to $366$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 141$ to $411$.
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. $$ 276 - 45 = 231 $$ $$ 276 + 45 = 321 $$ The range of numbers is 231 to 321
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. $$ 276 - 2 \cdot 45 = 186 $$ $$ 276 + 2 \cdot 45 = 366 $$ The range of numbers is 186 to 366
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. $$ 276 - 3 \cdot 45 = 141 $$ $$ 276 + 3 \cdot 45 = 411 $$ The range of numbers is 141 to 411
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 141 and 186
- 13.5% of the data values will lie between 186 and 231
- 34% of the data values will lie between 231 and 276
- 34% of the data values will lie between 276 and 321
- 13.5% of the data values will lie between 321 and 366
- 2.35% of the data values will lie between 366 and 411