Transcript Standard Deviation

Unit 3
Section 3-3 – Day 1
3-3: Measures of Variation

Range– the highest value minus the lowest
value.
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Variance– the average of the squares of the
distance each value is from the mean.
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The symbol R is used for range.
The symbol is σ2
Standard Deviation– the square root for the
variance.

The symbol is σ
Section 3-3
The Empirical Rule
(Normal Rule)
 Applies
when the distribution is bellshaped (or what is called normal).
Approximately 68% of the data falls within 1
standard deviation of the mean.
Approximately 95% of the data falls within 2
standard deviation of the mean.
Approximately 99.7% of the data falls within
3 standard deviation of the mean.
Section 3-3
Empirical Rule
Section 3-3
Finding the Variance
(Population)
 The
formula for finding the variance of a
population is:
X = individual value
μ = population mean
N = population size
Section 3-3
Finding the Variance (Sample)
 The
formula for finding the variance of a
population is:
X = individual value
n = sample size
Section 3-3
Steps for Finding Variance
 Find
the mean.
 Find the difference between each data
value and the mean.
 Square each difference.
 Find the sum of their squares.
 Divide the sum by the number of data
entries.
Section 3-3
Finding the Variance
 Find
the variance for the population
below:
Value
10
20
30
40
50
60
X-μ
(X-μ)2
Section 3-3
Finding the Variance
 Find
the variance for the amount of
European auto sales for a sample of 6
years as shown.
Value
11.2
11.9
12.0
12.8
13.4
14.3
X-μ
(X-μ)2
Section 3-3
Steps for Finding Standard
Deviation
 First,
set.
determine the variance of the data
 Then,
take the square root of the
variance.
Section 3-2
Finding the Standard
Deviation
 Find
the standard deviation for both of
our previous examples.
Section 3-3
Finding the Standard Deviation for
Grouped Data
 Formula:
( f iX )
f iX - [
]
å
n
2
s =
n -1
2
m
m
2
Section 3-3
Finding the Variance and Standard
Deviation for Grouped Data
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Make a table as shown
A
B
C
D
E
Class
Frequency
f
Midpoint
Xm
f*Xm
f*Xm2
Find the midpoints of each class and place them
in column C.
Multiply the frequency by the midpoint for each
class, and place the product in column D.
Multiply the frequency by the square of the
midpoint for each class, and place the product in
column E.
Find the sums of column B, D, and E.
Section 3-3
 To
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
find the Variance:
Take the Sum of E, subtract away the
quantity of the Sum of D squared divided
by the Sum of B.
Then, divide your value by the Sum of B
minus one.
 To
find the Standard Deviation, take the
square root of the variance.
Section 3-3
Finding the Variance and Standard
Deviation: Grouped Data
 Find
the variance and standard deviation
for the grouped data below:
Class
Frequency
f
5.5 – 10.5
1
10.5 – 15.5
2
15.5 – 20.5
3
20.5 – 25.5
5
25.5 – 30.5
4
30.5 – 35.5
3
35.5 – 40.5
2
Midpoint
Xm
f*Xm
f*Xm2
Section 3-3
Uses of the Variance and
Standard Deviation
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Variances and standard deviations can be used to
determine the spread of data.
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Variances and standard deviations are used to
determine the consistency of a variable.
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If they are large, the data is more dispersed
Useful in comparing two or more data sets to determine
which is more variable.
Example: manufacturing nuts and bolts, the variation in
diameters must be low so the parts fir together.
Variances and standard deviations are used to
determine the number of data values that fall within
a specific interval in a distribution.
Variance and standard deviations are used quite
often in inferential statistics.
Section 3-3
 Coefficient
of Variation – the standard
deviation divided by the mean.
 Notation:
CVar
 The result is expressed as a percentage.
Example:
The mean of the number of sales of cars
over a 3-month period is 87, and the
standard deviation is 5. The mean of the
commissions is $5225, and the standard
deviation is $773. Compare the variations
of the two.

Range Rule of Thumb
A rough estimate of the standard deviation.
 Standard deviation is approximately the range
divided by four.
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Chebyshev’s theorem – the proportion of
values from a data set that will fall within k
standard deviations of the mean will be at
least 1-1/k2, where k is a number greater than
1.
This can be applied to any data set regardless
of its distribution or shape
 Also states that three-fourths (or 75%) of the
data values will fall within 2 standard deviations
of the mean.
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Section 3-3
Chebyshev’s theorem
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A survey of local companies found that the mean
amount of travel allowance for executives was $0.25
per mile. The standard deviation was $0.02. Using
Chebyshev’s theorem, find the minimum
percentage of the data values that will fall between
$0.20 and $0.30.
Step 1: Subtract the mean from the larger
value
Step 2: Divide the difference by the standard
deviation (find k)
Step 3: Use Chebyshev’s theorem to find the %
Section 3-3
Homework
 Pg
130-131: 18, 21, 31, 32, 35