Section June 5, 2013 Summer 2013 - Math 1040 (1040)
Section
Summer 2013 - Math 1040
June 5, 2013
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Roadmap
Today we will work via examples. The arithmetic will involve sums, differences, division, and square roots.
Range.
Population variation.
Sample variation.
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Spread
The spread of data includes the range of data, deviations, standard deviations, and variance.
Range of a data set is the maximum value minus the minimum value. We saw this when choosing class widths.
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Age in years of patients for a clinic:
34 27 50 45 41 37 24
57 40 38 62 44 39 440
The range of this data is 62 − 24 = 38 years.
Spread
The spread of data includes the range of data, deviations, standard deviations, and variance.
Range of a data set is the maximum value minus the minimum value. We saw this when choosing class widths.
Age in years of patients for a clinic:
34 27 50 45 41 37 24
57 40 38 62 44 39 440
The range of this data is 62 − 24 = 38 years.
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Example - Salaries
Compute the ranges for salaries in thousands of dollars for each company.
Company A:
41 38 39 45 47 41 44 41 37 42
Company B:
40 23 41 50 49 32 41 29 52 58
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Deviations
Deviations are the difference of data entries and the mean.
Deviation of x = ( x − µ ) (population)
Deviation of x = ( x − ¯ ) (sample)
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For the patient data, the mean is 41 .
3 years. The deviation of 24 years is
(24 − 41 .
3) = − 17 .
3 years. The deviation for 62 is (62 − 41 .
3) = 20 .
7 years.
Deviations
Deviations are the difference of data entries and the mean.
Deviation of x = ( x − µ ) (population)
Deviation of x = ( x − ¯ ) (sample)
For the patient data, the mean is 41 .
3 years. The deviation of 24 years is
(24 − 41 .
3) = − 17 .
3 years. The deviation for 62 is (62 − 41 .
3) = 20 .
7 years.
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Example - Company B - Deviations
The mean of the salaries, in thousands of dollars, is $41,500. (41.5).
Below is a table of the values and their devations.
x 23 29 32 40 41 41 49 50 52 58 x − µ -18.5
-12.5
-9.5
-1.5
-0.5
-0.5
7.5
8.5
10.5
16.5
The sum of the deviations is always zero, so the average of the deviations is not useful. We can, however, find the absolute deviations or the squares of deviations. We will compute a table for the latter values.
( x − µ )
2
( x − µ )
2
342.25
156.25
90.25
2.25
0.25
0.25
56.25
72.25
110.25
272.25
Note the sum of squares SS x is 1102.5 “thousands-of-dollars-squared”.
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Variance
Population variance is given by
Sample variance is given by
σ
2
=
P ( x − µ )
2
N s
2
=
P ( x − x )
2 n − 1
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Variances are averages of the square-deviations.
The square roots of variance are called standard deviations.
Variance
Population variance is given by
Sample variance is given by
σ
2
=
P ( x − µ )
2
N s
2
=
P ( x − x )
2 n − 1
Variances are averages of the square-deviations.
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The square roots of variance are called standard deviations.
Variance
Population variance is given by
Sample variance is given by
σ
2
=
P ( x − µ )
2
N s
2
=
P ( x − x )
2 n − 1
Variances are averages of the square-deviations.
The square roots of variance are called standard deviations.
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Corporation B - Variance
Recall that µ = 41 .
5 thousands of dollars.
( x − µ ) 2
( x − µ ) 2
342.25
156.25
90.25
2.25
0.25
0.25
56.25
72.25
110.25
272.25
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The sum of squares SS x is 1102.5 “thousands-of-dollars-squared”. The average of the square-deviations is
σ
2
=
P ( x − µ )
2
N
=
1102 .
5
10
= 110 .
25
The variance is σ
2
= 110 .
25 “thousands-of-dollars-squared”.. The standard deviation is σ = 110 .
25 = 10 .
5 thousands of dollars.
Corporation B - Variance
Recall that µ = 41 .
5 thousands of dollars.
( x − µ ) 2
( x − µ ) 2
342.25
156.25
90.25
2.25
0.25
0.25
56.25
72.25
110.25
272.25
The sum of squares SS x is 1102.5 “thousands-of-dollars-squared”. The average of the square-deviations is
σ
2
=
P ( x − µ )
2
N
=
1102 .
5
10
= 110 .
25
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The variance is σ
2
= 110 .
25 “thousands-of-dollars-squared”.. The standard deviation is σ = 110 .
25 = 10 .
5 thousands of dollars.
Corporation B - Variance
Recall that µ = 41 .
5 thousands of dollars.
( x − µ ) 2
( x − µ ) 2
342.25
156.25
90.25
2.25
0.25
0.25
56.25
72.25
110.25
272.25
The sum of squares SS x is 1102.5 “thousands-of-dollars-squared”. The average of the square-deviations is
σ
2
=
P ( x − µ )
2
N
=
1102 .
5
10
= 110 .
25
The variance is σ
2
= 110 .
25 “thousands-of-dollars-squared”.. The standard deviation is σ = 110 .
25 = 10 .
5 thousands of dollars.
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Interpreting Standard Deviation
The salary data.
µ = 41 .
5 thousands of dollars and σ = 10 .
5 thousands of dollars.
x 23 29 32 40 41 41 49 50 52 58
How many data entires are within one standard deviation ( ± σ ) of the mean? Two standard deviations ( ± 2 σ )? Three ( ± 3 σ )?
Remember: Standard deviation is a measure of the typical amount an observation deviates from the mean. High standard deviation means data is spread out. Low standard deviation means data is close together.
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Assignements
Assignment:
1
2
Read pages 80 - 86.
Try exercise 1-31 odd on page 91.
Vocabulary: Range, deviations, variance, sums of squares, standard deviation.
Understand: Range, deviations, and the different formulats for population variance and sample variance. Sums of square deviations is SS x
. Taking square roots of a variance gets a standard deviation.
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