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AP Statistics
Empirical Rule
What does a population that is
normally distributed look like?
 = 80 and  = 10
X
50
60
70
80
90
100
110
Normal Distribution Features
• Symmetric, single-peaked, and bell-shaped.
• Reported: N   ,  
• Changing µ changes the location of the curve on
the horizontal axis with no change in spread.
• Changing  changes the spread of the curve.
•  is located at the inflection point of the curve.
Empirical Rule
68%
95%
99.7%
68-95-99.7% RULE
Empirical Rule—restated
68% of the observations fall within 1 standard
deviation of the mean in either direction.
95% of the observations fall within 2 standard
deviations of the mean in either direction.
99.7% of the observations fall within 3
standard deviations of the mean in either
direction.
Remember values in a data set must appear
to be a normal bell-shaped histogram,
dotplot, or stemplot to use the Empirical
Rule!
Empirical Rule
34% 34%
68%
47.5%
47.5%
95%
49.85%
49.85%
99.7%
Average American adult male
height is 69 inches (5’ 9”) tall with a
standard deviation of 2.5 inches.
Empirical Rule-- Let H~N(69, 2.5)
What is the likelihood that a randomly selected
adult male would have a height less than 69 inches?
Answer: P(h < 69) = .50
P represents Probability
h represents one adult
male height
Using the Empirical Rule
Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult
male will have a height between 64 and 74 inches?
P(64 < h < 74) = .95
Using Empirical Rule-- Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult
male would have a height of less than 66.5 inches?
P(h < 66.5) = 1 – (.50 + .34) = .16
OR  .50 - .34 = .16
Using Empirical Rule--Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult
male would have a height of greater than 74 inches?
P(h > 74) = 1 – P(h  74)
= 1 – (.50 + .47.5)
= 1 - .975
= .025
Using Empirical Rule--Let H~N(69, 2.5)
What is the probability that a randomly selected
adult male would have a height between 64 and
76.5 inches?
P(64 < h < 76.5) = .475 + .4985
= .9735
Empirical Rule
More about Normal Curves
• Percentiles divide the area of the density
curve into fractions out of 100.
• Median: 50th percentile.
• Q1: 25th percentile.
• Q3: 75th percentile.
Why are Normal Curves Important?
• They provide good descriptions of some
distributions of real data (SAT, ACT,
characteristics of biological populations).
• They are a good approximation of the
results of chance outcomes (sum the two
die rolls).
• Many statistical procedures are based on
normal distributions.
However!
• Many distributions are not normal
(income). Don’t assume every situation is
approximated by a normal distribution.
• HW: Textbook 2.7, 2.8, 2.13, 2.15