Answer:
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.
197
−
46
=
151
197
+
46
=
243
The range of numbers is 151 to 243
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.
197
−
2
⋅
46
=
105
197
+
2
⋅
46
=
289
The range of numbers is 105 to 289
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.
197
−
3⋅
46
=
59
197
+
3
⋅
46
=
335
The range of numbers is 59 to 335
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 59 and 105
13.5% of the data values will lie between 105 and 151
34% of the data values will lie between 151 and 197
34% of the data values will lie between 197 and 243
13.5% of the data values will lie between 243 and 289
2.35% of the data values will lie between 289 and 335
