Transcript Normal Distributions

LSP 121
Normal Distributions
What is a Normal Distribution?
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Very common, very special type of distribution
Most data values are clustered near the mean (a
single peak)
Distribution is symmetric
Tapering tales as you move away from the mean
Looks like a bell curve
The 68-95-99.7 Rule
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About 68% (68.3%), or just over 2/3, of the data
points fall within 1 standard deviation (+ or -) of
the mean
About 95% (95.4%) of the data points fall within 2
standard deviations of the mean
About 99.7% of the data points fall within 3
standard deviations of the mean
Questions
How many percent lie between mean -1 standard deviation and
mean + 1 standard deviation?
68%
How many percent lie between mean + 1 stdev and mean +3
stdev? 15.85%
How many percent lie greater than mean + 3 stdev?
0.15%
Example
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SAT exams were designed to produce normal
distributions with a mean of 500 and a standard
deviation of 100.
Thus, 68% of the students scored between 400 and 600
95% of the students scored between 300 and 700
99.7% scored between 200 and 800
What if someone scored 720 on the SAT? What
percentage of students scored less than or equal to 720?
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Use Excel’s NORMDIST function (see next slide)
=NORMDIST(x, mean, stdev, true)
For our problem: =NORMDIST(720, 500, 100, TRUE)
Answer = 0.986097, or 98.6097%
What percentage scored greater than 720?
Normal Dstn and its Inverse
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In Excel, the normal
distribution function is
given by NORMDIST
and takes four
arguments:
NORMDIST(x,mean,SD,true),
where x is a value, mean is the
average of all values, SD is the
standard deviation of all values,
and true is a constant.
NORMDIST returns the cumulative
probability that a value is ≤ x.
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In Excel, the inverse of
the normal distribution
function is given by
NORMINV and it takes
three arguments:
NORMINV(p,mean,SD),
where p is a probability, mean
and SD are as before.
NORMINV returns a value x, with
the property that p% of the
values are ≤ x.
Another Example
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A survey finds that prices paid for two-year-old Ford
Explorers are normally distributed with a mean of
$16,500 and a standard deviation of $500. Consider
a sample of 10,000 people who bought two-year-old
Ford Explorers.
How many people paid between $16,000 and $17,000?
 =NORMDIST(16000,16500,500,true) yields 0.158655
 =NORMDIST(17000, 16500, 500, true) yields 0.841345
 Subtract: 0.841345 – 0.158655 yields 0.682689
 Therefore, 0.682689*10000 or 6827 people paid
between $16000 and $17000.
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Another Example
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How many paid less than $16,000?
 =NORMDIST(16000,
16500, 500, true) yields
0.158655, or 15.8655 %
 Or use the graph
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What is another way of saying “What percentage
of values are less than or equal to some value X?”
(see next slide)
Percentiles
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The pth percentile of a data set is the
smallest value, x, in the set with the
property that at least p% of the data
values are less than or equal x.
In a normal distribution, a z score of 0 is
the mean. At the mean, 50% (or 0.50) of
all the values are less than or equal to the
mean. The mean is the 50th percentile.
Example
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Cholesterol levels in men 18 to 24 years of age are
normally distributed with a mean of 178 and a
standard deviation of 41.
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what percentile is a man with a cholesterol level of
190?
Using Excel’s Normal Distribution function:
=NORMDIST(190,178,41,true) returns 0.61, or 61st
percentile
Standard Scores
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The number of standard deviations a data value
lies above or below the mean is called its Standard
Score, or z-score, or simply z.
The standard score of the mean is z=0
The standard score of a data value 1.5 standard
deviations above the mean is z=1.5
Standard Scores
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The standard score of a data value 2.4 standard
deviations below the mean is z = -2.4
In general:
z = (data value – mean) / standard deviation
Example
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The Stanford-Binet IQ test is designed so
that scores are normally distributed with
a mean of 100 and a standard deviation
of 16. What are the z-scores for IQ
scores of 95 and 125?
z = (95 - 100) / 16 = -0.31
z = (125 - 100) / 16 = 1.56 Thus, an IQ score of 125
lies 1.56 standard deviations above the mean.
Inverse Normal Distribution Function
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What if you know the mean and standard deviation
of a set of data and you want to know a particular
percentile, x?
We use Excel’s NORMINV
For example, if a set of scores has a mean of 76, a
standard deviation of 12, what is the 66th
percentile?
Answer: x = NORMINV(0.66,76,12) = 80.95