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 $127$ to $189$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $96$ to $220$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 65$ to $251$.
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. $$ 158 - 31 = 127 $$ $$ 158 + 31 = 189 $$ The range of numbers is 127 to 189
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. $$ 158 - 2 \cdot 31 = 96 $$ $$ 158 + 2 \cdot 31 = 220 $$ The range of numbers is 96 to 220
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. $$ 158 - 3 \cdot 31 = 65 $$ $$ 158 + 3 \cdot 31 = 251 $$ The range of numbers is 65 to 251
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 65 and 96
- 13.5% of the data values will lie between 96 and 127
- 34% of the data values will lie between 127 and 158
- 34% of the data values will lie between 158 and 189
- 13.5% of the data values will lie between 189 and 220
- 2.35% of the data values will lie between 220 and 251