Download The 68-95-99.7 Rule Section 5.9

Section 5.9
The 68-95-99.7 Rule
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
Because any particular normal distribution is
completely determined by its mean and
standard deviation, it is not surprising that all
normal distributions are the same in terms of
what proportion of observations are any
given number of standard deviations from
the mean. Here is an important rule based
on this fact.
The 68-95-99.7 Rule
for Normal Distributions
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
According to the 68-95-99.7 rule, in any
normal distribution:
1. 68% of the observations fall within 1
standard deviation of the mean.
2. 95% of the observations fall within 2
standard deviations of the mean.
3. 99.7% of the observations fall within 3
standard deviations of the mean.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
Example 1
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
The heights of women between the ages of
18 and 24 are roughly normally distributed,
with mean 64.5 inches and standard
deviation 2.5 inches. Using the 68-95-99.7
Rule find the 2.5% of the tallest women and
the 2.5% of the shortest women.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
2 standard deviations = 2(2.5) = 5 inches
95% of the women are between
(64.5 – 5) inches and (64.5 + 5) inches or
between 59.5 inches and 69.5 inches.
So the other 5% of the women have outside
the range of 59.5 inches to 69.5 inches.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
Since normal distributions are symmetric we
know that 2.5% of these women will be
above 69.5 inches and 2.5% of these
women will be below 59.5 inches.
This is our answer.
Example 2
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
The distribution of scores on tests such as
the SAT college entrance exams is close to
normal. Scores on each of the three
sections (math, critical reading, writing) of
the SAT are adjusted so that the mean
score is about 500 and the standard
deviation is about 100. Using this
information answer the following questions.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
A.
How high must a student score to fall in
the top 25%?
This question is asking for the third
quartile.
Q3 = (.67)(standard deviation)
= (.67)(100)
= 67 points above the mean
= 500 + 67 = 567
Scores above 567 are in the top 25%.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
B.
What percent of scores fall between
200 and 800?
These scores are 3 standard deviations
from the mean.
200 = 500 – 3(100)
800 = 500 + 3(100)
The percent of scores are 99.7%.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
C.
What percent of the scores are above
700?
700 is 2 standard deviations above the
mean.
700 = 500 + 2(100)
95% of the scores are between
500 – 2(100) and 500 + 2(100) or 300
to 700.
2.5% of scores are above 700 points.
Chapter 5: Exploring Data: Distributions
The 68-95-99.7 Rule
SAT scores have Normal distribution
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