Download 9.4 Standard Deviation

Chapter 1
9.3 – Measures of Dispersion
Objective: TSW calculate the range
and standard deviation from a set of
data.
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Measures of Dispersion
Sometimes we want to look at a measure of
dispersion, or spread, of data.
Two of the most common measures of
dispersion are the range and the standard
deviation.
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Range
For any set of data, the range of the set is given
by
Range = (greatest value in set) – (least value in set).
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Example: Find the median and mean
for each set of data below.
Now, let’s find the range.
Set A
1 2
7
12 13
Set B
5 6
7
8
9
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Standard Deviation
One of the most useful measures of
dispersion, the standard deviation, is based
on deviations from the mean of the data.
To find the deviations of each number:
1. Find the mean of the set of data
2. Subtract the number from the mean
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Example: Deviations from the Mean
Find the deviations from the mean for all data values
of the sample 1, 2, 8, 11, 13.
Solution
The mean is 7. Subtract to find deviation.
Data Value
Deviation
1
2
8
11 13
The sum of the deviations for a set is always 0.
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Standard Deviation
Variance, to find:
1. Square each deviation
2. add the squares up
3. divide by (n-1) where n = number of items in the set.
standard deviation – the square root of the variance.
Gives an average of the deviations from the mean. Which
is denoted by the letter s. (The standard deviation of a
,
population is denoted the lowercase Greek letter
sigma.)
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Calculation of Standard Deviation
Let a sample of n numbers x1, x2,…xn have mean x .
Then the sample standard deviation, s, of the
numbers is given by
s
2
(
x

x
)

n 1
.
The smaller the number for
standard deviation – the closer your
data is together.
The higher the number for standard
deviation, you data is further apart.
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Two standard deviations, or two
sigmas, away from the mean (the
red and green areas) account for
roughly 95 percent of the data
points. Three (3) standard
deviations (the red, green and blue
areas) account for about 99 percent
of the data points.
If this curve were flatter and more
spread out, the standard deviation
would have to be larger in order to
account for those 68 percent or so
of the points. That's why the
standard deviation can tell you
how spread out the examples in a
set are from the mean.
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Calculation of Standard Deviation
The individual steps involved in this calculation are
as follows
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Calculate the mean of the numbers.
Find the deviations from the mean.
Square each deviation.
Sum the squared deviations.
Divide the sum in Step 4 by n – 1.
Take the square root of the quotient in
Step 5.
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Example
Find the standard deviation of the sample
1, 2, 8, 11, 13.
The mean =_____.
Data Value
Deviation
1
2
8
11
13
(Deviation)2
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Example: Find the Standard Deviation
12, 13, 16, 18, 18, 20 mean = _______
Data Value
Deviation
12
13
16
18
18
20
(Deviation)2
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Example: Find the Standard
Deviation
125, 131, 144, 158, 168, 193
Data Value
Deviation
125 131
144
158
168
193
(Deviation)2
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Homework
Worksheet
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