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 $136$ to $232$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $88$ to $280$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 40$ to $328$.
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. $$ 184 - 48 = 136 $$ $$ 184 + 48 = 232 $$ The range of numbers is 136 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. $$ 184 - 2 \cdot 48 = 88 $$ $$ 184 + 2 \cdot 48 = 280 $$ The range of numbers is 88 to 280
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. $$ 184 - 3 \cdot 48 = 40 $$ $$ 184 + 3 \cdot 48 = 328 $$ The range of numbers is 40 to 328
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 40 and 88
- 13.5% of the data values will lie between 88 and 136
- 34% of the data values will lie between 136 and 184
- 34% of the data values will lie between 184 and 232
- 13.5% of the data values will lie between 232 and 280
- 2.35% of the data values will lie between 280 and 328