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Normal Distributions
and the Empirical Rule
OBJECTIVE: Use the empirical rule (68-95-97 rule) to
analyze data
Standard Deviation
Standard Deviation shows the
variation in data. If the data is close
together, the standard deviation will
be small. If the data is spread out, the
standard deviation will be large.
Standard Deviation is often denoted
by the lowercase Greek letter sigma,
.
Using your Calculator
 Find mean, standard deviation
 Given standard deviation find variance
 Given variance find standard deviation
 Example
The bell curve which represents a
normal distribution of data shows
what standard deviation represents.
One standard deviation away from the mean (  ) in
either direction on the horizontal axis accounts for
around 68 percent of the data. Two standard
deviations away from the mean accounts for roughly
95 percent of the data with three standard deviations
representing about 99 percent of the data.
Normal distributions: N (μ, σ)




Symmetric, single peaked and bell shaped.
Center of the curve are μ and M.
Mean, median, & mode are the same
Standard deviation σ controls the spread of the
curve.
Normal curves are a good description
of some real data:
 test scores (SAT, ACT, IQ)
 biological measurements
 also approximate chance outcomes like
tossing coins
The Empirical rule
(68-95-99.7 rule)
In the normal dist. with mean μ and
standard deviation σ.
 50% of the observations fall below the
mean.
 50% of the observations fall above the
mean.
The Empirical rule
(68-95-99.7 rule)
In the normal dist. with mean μ and
standard deviation σ.
 68% of the observations fall within 1σ of
the mean.
 95% of the observations fall within 2σ of
the mean.
 99.7% of the observations fall within 3σ of
the mean.
Behold the normal curve
68%
34%
0.15%
2.35%
13.5%
95%
99.7%
34%
13.5%
2.35% 0.15%
Exercise 1: Men’s Heights
The distribution of adult American men is
approximately normal with mean 69 inches
and standard deviation 2.5 inches. Draw
the curve and mark points if inflection.
Recall: mean 69 in. and standard deviation
2.5 in.
16%tile
2.5%tile
.15%tile
84% tile
97.5th %
2.35%
.15
99.85th %
61.5 64 66.5 69 71.5 74 76.5
a) What percent of men are taller than 74 inches?
74 is two standard dev. above the mean.
2.35 + .15 =2.5
b) Between what heights do the middle 95% of men fall?
69 5
Between 64 and 74 inches
mean 69 in. and standard deviation 2.5 in.
13.5
84%
.15 2.35
61.5 64
66.5 69
71.5 74 76.5
c) What percent of men are shorter than 66.5 inches?
16.0%
Exercise 2: SAT Verbal Scores
SAT verbal scores are normally distributed
with a mean of 489 and a standard
deviation of 93.
Recall: mean 489 in. and standard deviation
93 in.
16%tile
2.5%tile
.15%tile
84% tile
13.5%
97.5th %
99.85th %
210 303 396 489 582 675 768
a) What percentage lie between 303 and 582?
303 is two standard dev. below the mean. & 582 is one
std. dev above
68 + 13.5 = 81.5%
Recall: mean 489 in. and standard deviation
93 in.
16%
2.5%
.15%
84%
2.35%
..15%
97.5 %
99.85 %
210 303 396 489 582 675 768
b) What percentage is above 675?
675 is two standard dev. above the mean.
2.35 +.15 = 2.5%
What percentage is below 675
100 - 2.5 = 97.5
Recall: mean 489 in. and standard deviation
93 in.
16%
2.5%
.15%
84%
13.5%
97.5 %
99.85 %
210 303 396 489 582 675 768
b) If 3,500 students took the SAT verbal test, about how
many received between 396 and 675 points?
68% + 13.5% = 81.5% fall within this range.
3,500 * 81.5% = 2853