The Standard Deviation Rule(p. 31 of OLI online text)
Also Known As: the 68 – 95 – 99.7 Rule
Also Known As: the Empirical Rule
- Approximately 68% of the observations fall within 1 standard deviation of the mean.
- Approximately 95% of the observations fall within 2 standard deviations of the mean.
- Approximately 99.7% (or virtually all) of the observations fall within 3 standard deviations of the mean.
Note about the diagram above, its horizontal axis:
• Under the horizontal axis, where it says:
mean
-3(SD)
It indicates that the point on the axis is at the location of
the value of “the mean minus 3 times the standard deviation”
• In a similar way, under the horizontal axis, where it says:
mean
+SD
It indicates that the point on the axis is at the location of
the value of “the mean plus the standard deviation”
• So, the red dots on the horizontal axis are marking the location of:
- The mean in the center,
- To the right of the mean there is a mark at a distance of one standard deviation over, then another standard deviation distance over, then a third standard deviation distance away from the mean.
- To the left of the mean there is a mark at a distance of one standard deviation over, then another standard deviation distance over, then a third standard deviation distance away from the mean.
Math& 146 – Introduction to StatisticsName ______
Module 1: Worksheet on the Standard Deviation Rule
Suppose that the distribution ofweekly incomes of a certain group of people is approximately normal with a mean of $700 and a population standard deviation of $100.
Here is a diagram of the normal distribution.
● Below the horizontal axis write somedata values (note: the data values are the amounts of income). Label the mean and places for the mean plus or minus 1, 2, and 3 standard deviations.
Later when you write percentages, write them above or “inside” the curve.
● Does the Standard Deviation Rule (aka the 68-95-99.7 rule) apply in this situation? ______
Why or why not?
a) What percentage of people have incomes between $500 and $900 per week? ______
b) 68% of all the people have weekly incomes between what two numbers? ______
c) What percentage of these people have weekly incomes above $600? ______
d) 95% of these people have weekly incomes between what two numbers? ______
e) What percentage of the people have weekly incomes below $600? ______
f) What percentage of the people have incomes between $400 and $800 per week? ______
g) 99.7% of the people have weekly incomes between what two numbers? ______
h) What percentage of the people have weekly incomes below $800? ______
i) What percentage of the people have weekly incomes above $1000? ______
j) What percentage of the people have weekly incomes below $500? ______
k) What percentage of the people have incomes between $500 and $1000 per week? ______
ANSWERS
• The 68-95-99.7 Rule applies because we were told the distribution of data is approximately normal.
a) 95%b) $600 and $800c) 84%d) $500 and $900
e) 16%f) 83.85% (or 83.9%)g) $400 and $1000h) 84%
i) 0.15% j) 2.5%k) 97.35% (or 97.4%)
Module 1 – Standard Deviation Rulep. 1