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 $214$ to $314$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $164$ to $364$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 114$ to $414$.
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. $$ 264 - 50 = 214 $$ $$ 264 + 50 = 314 $$ The range of numbers is 214 to 314
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. $$ 264 - 2 \cdot 50 = 164 $$ $$ 264 + 2 \cdot 50 = 364 $$ The range of numbers is 164 to 364
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. $$ 264 - 3 \cdot 50 = 114 $$ $$ 264 + 3 \cdot 50 = 414 $$ The range of numbers is 114 to 414
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 114 and 164
- 13.5% of the data values will lie between 164 and 214
- 34% of the data values will lie between 214 and 264
- 34% of the data values will lie between 264 and 314
- 13.5% of the data values will lie between 314 and 364
- 2.35% of the data values will lie between 364 and 414