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 $129$ to $229$.
Approximately 95% of the data values will fall within 2 standard deviations of the mean, from $79$ to $279$.
Approximately 99.7% of the data values will fall within 3 standard deviations of the mean, from $ 29$ to $329$.
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. $$ 179 - 50 = 129 $$ $$ 179 + 50 = 229 $$ The range of numbers is 129 to 229
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. $$ 179 - 2 \cdot 50 = 79 $$ $$ 179 + 2 \cdot 50 = 279 $$ The range of numbers is 79 to 279
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. $$ 179 - 3 \cdot 50 = 29 $$ $$ 179 + 3 \cdot 50 = 329 $$ The range of numbers is 29 to 329
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 29 and 79
- 13.5% of the data values will lie between 79 and 129
- 34% of the data values will lie between 129 and 179
- 34% of the data values will lie between 179 and 229
- 13.5% of the data values will lie between 229 and 279
- 2.35% of the data values will lie between 279 and 329