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