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 $202$ to $250$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $178$ to $274$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 154$ to $298$.
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. $$ 226 - 24 = 202 $$ $$ 226 + 24 = 250 $$ The range of numbers is 202 to 250
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. $$ 226 - 2 \cdot 24 = 178 $$ $$ 226 + 2 \cdot 24 = 274 $$ The range of numbers is 178 to 274
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. $$ 226 - 3 \cdot 24 = 154 $$ $$ 226 + 3 \cdot 24 = 298 $$ The range of numbers is 154 to 298
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 154 and 178
- 13.5% of the data values will lie between 178 and 202
- 34% of the data values will lie between 202 and 226
- 34% of the data values will lie between 226 and 250
- 13.5% of the data values will lie between 250 and 274
- 2.35% of the data values will lie between 274 and 298