Transcript 5.2 Properties of Normal Distribution

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 113
5.2 Properties of Normal Distribution
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Is it Normal?
Circumference of 10,000 adult wrists…
 YES--NORMAL
 High jump results for all High School track meets in America…
 YES--NORMAL
 Calories consumed by Sumo wrestlers compared to calories
consumed by the general public…
 SOMEWHAT NORMAL—SKEWED RIGHT COMPARED TO
GENERAL PUBLIC
 Prices of a particular Video game a different retail stores…
 YES--NORMAL
 Weights of 50 randomly selected baseballs…
 YES--NORMAL
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5.2 Properties of Normal Distribution

Remember what makes a variable
normally distributed?
1.
2.
3.
4.
The variable is determined by many different
factors (i.e. physiological variables or performance
variables)
The frequencies of the variable cluster around a
single peak that is near the mean of the
distribution.
The frequencies of the variable are symmetric
about the peak.
Large deviations from the mean are increasingly
rare.
5.2 Properties of Normal Distribution
THE EMPIRICAL RULE

Data sets having a normal, bell-shaped distribution, the following
properties apply:
http://www.stat.tamu.edu/~west/applets/empiricalrule.html
5.2 Properties of Normal Distribution
 68-95-99.7
Rule of Normal Distribution
68 % of the normal distribution occurs within 1
standard of deviation from the mean
 95 % of the normal distribution occurs within 2
standards of deviation from the mean
 99.7 % of the normal distribution occurs within
3 standards of deviation from the mean
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5.2 Properties of Normal Distribution
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Using 68-95-99.7 Rule for Normal Distribution:

If there are 600 total values. Approximately, how many
values occur within 1 standard of deviation from the
mean? 2 standards of deviation? 3 standards of
deviation?
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408, 570, and 598.2
5.2 Properties of Normal Distribution
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Using 68-95-99.7 Rule for Normal Distribution:
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If there are 325 values within 1 standard of deviation
from the mean, approximately, how many values exist in
the data set? Approximately, how many values occur
within 3 standards of deviation from the mean?
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478, and 477
5.2 Properties of Normal Distribution
 Unusual Values…
Values
that are more than 2 standards
of deviations away from the mean.
5.2 Properties of Normal Distribution
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EXAMPLE: Suppose you measure your heart rate
at noon every day for a year and record the data.
You discover that the data has a normal
distribution with a mean of 66 and a standard of
deviation of 4. On how many days was your heart
rate below 58 beats per minute?
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A heart rate of 58 is 8 below the mean (2 standards of deviation
below). From the 68-95-99.7 rule, 95% of the data occurs 2
standards of deviation from the mean. If we want to know how
many days our heart rate was below 58 this would imply 2.5% of
our data, because ((100%-95%)÷2=2.5%). Therefore, on 2.5% of
365 days in a year your heart rate is below 58 or roughly 9 days.
5.2 Properties of Normal Distribution
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STANDARD SCORES (Z-scores) –

The number of standard deviations between a particular
data value and the mean is called its standard score,
usually denoted z.
data value - mean
z  standard score =
standard deviation

Data values below the mean have a negative standard
score, and data values above the mean have a positive
standard score.
Z
xx

5.2 Properties of Normal Distribution
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Z-SCORE:
 The
Stanford-Binet IQ test is scaled so that scores
have a mean of 100 and a standard deviation of 16.
Find the z-scores for IQ’s of 85, 100, 125.
85  100
 85 z-score =
 0.94
16
 100
z-score =
100  100
 0.0
16
125  100
 1.56
 125 z-score =
16
5.2 Properties of Normal Distribution
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Z-SCORES and Percentiles:
TABLE 5.1 page 211.
 Standard
 i.e.
score ≈ percentile
Z-Score
Percentile
≈ -1.2 and -1.3
12th
≈ -0.7 and -0.65
25th
≈ -0.35 and -0.30
37th
≈ 0.0 and 0.0
50th
≈ 0.30 and 0.35
62nd
≈ 0.65 and 0.70
75th
≈ 1.1 and 1.2
87th
≈ 3.5 and above
99.98th
AreaOfANormalDistribution
.nbp
5.2 Properties of Normal Distribution
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MORE Z-SCORES…
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The heights of American women aged 18 to 24 are normally
distributed with a mean of 65 inches and a standard deviation of
2.5 inches. In order to serve in the U.S. Army, women must be
between 58 inches and 80 inches tall. What percentage of
women are ineligible to serve based on their height?
The z-score of 58 inches is -2.8 which corresponds to the 0.26
percentile from Table 5.1 on page 232. The z-score of 80 is 6.0
and corresponds to above the 99.98 percentile. Thus, 0.26% of
all women are too short, and 0.02% of all women are too tall.
Altogether, this means that 0.28% of all women are ineligible for
the army based on their height. Hence, roughly one out of 358
women are ineligible.
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5.2 Properties of Normal Distribution
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QUANITIFING THE POWER (Probability) OF NORMAL
DISTRIBUTION AND STANDARD DEVIATION:
μ ± 1σ → 0.682689492137 → 1 in 3 → weekly
μ ± 2σ → 0.954499736104 → 1 in 22 → monthly
μ ± 3σ → 0.997300203937 → 1 in 370 → yearly
μ ± 4σ → 0.999936657516 → 1 in 15,787 → every 60 years
(once in a lifetime)
μ ± 5σ → 0.999999426697 → 1 in 1,744,278 → every 5,000
years (once in history??)
μ ± 6σ → 0.999999998027 → 1 in 506,842,372 → every 1.5
million years (essentially never??)
Thus for a daily process, a 6σ event is expected
to happen less than once in a million years.
5.2 Properties of Normal Distribution
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HOMEWORK:
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# 23-26
#1-11
#13-24
#37, 38, 42