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 $233$ to $279$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $210$ to $302$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 187$ to $325$.
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. $$ 256 - 23 = 233 $$ $$ 256 + 23 = 279 $$ The range of numbers is 233 to 279
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. $$ 256 - 2 \cdot 23 = 210 $$ $$ 256 + 2 \cdot 23 = 302 $$ The range of numbers is 210 to 302
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. $$ 256 - 3 \cdot 23 = 187 $$ $$ 256 + 3 \cdot 23 = 325 $$ The range of numbers is 187 to 325
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 187 and 210
- 13.5% of the data values will lie between 210 and 233
- 34% of the data values will lie between 233 and 256
- 34% of the data values will lie between 256 and 279
- 13.5% of the data values will lie between 279 and 302
- 2.35% of the data values will lie between 302 and 325