Transcript Measures of Variability

Measures of Variability
Measures of Variability

Why are measures of variability important?
Why not just stick with the mean?
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Ratings of attractiveness (out of 10) – Mean = 5
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Everyone rated you a 5 (low variability)
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What could we conclude about attractiveness from this?
People’s ratings fell into a range from 1 – 10, that
averaged a 5 (high variability)
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What could we conclude about attractiveness from this?
Measures of Variability
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Measures of Variability
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Range
Interquartile Range
Average Deviation
Variance
Standard Deviation
Measures of Variability
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Range
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The difference between the highest and lowest
values in a dataset
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Heavily biased by outliers
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Dataset #1: 5 7 11
Range = 6
Dataset #2: 5 7 11 million
Range = 10,999,995
Measures of Variability
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Interquartile Range
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The difference between the highest and lowest
values in the middle 50% of a dataset
Less biased by outliers than the Range
Based on sample with upper and lower 25% of the
data “trimmed”
However this kind of trimming essentially ignores
half of your data – better to trim top and bottom 1
or 5%
Measures of Variability
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Average Deviation
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For each score, calculate deviation from the
mean, then sum all of these scores
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However, this score will always equal zero
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Dataset: 19, 16, 20, 17, 20, 19, 7, 11, 10, 19, 14, 11, 6,
11, 14, 19, 20, 17, 4, 11
X = 285 285/20 = 14.25
Data
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Diff. from Mean = 0
0/N = 0
www.randomizer.org
Mean
Diff. from Mean
19
14.25
4.75
16
14.25
1.75
20
14.25
5.75
17
14.25
2.75
20
14.25
5.75
19
14.25
4.75
7
14.25
-7.25
11
14.25
-3.25
10
14.25
-4.25
19
14.25
4.75
14
14.25
-.25
11
14.25
-3.25
6
14.25
-8.25
11
14.25
-3.25
14
14.25
-.25
19
14.25
4.75
20
14.25
5.75
17
14.25
2.75
4
14.25
-10.25
11
14.25
-3.25
Measures of Variability
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Variance
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Sample Variance (s2) = (X - X )2/(n -1)
Population Variance (σ2) = (X - X)2/N
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Note the use of squared units!
Gets rid of the positive and negative values in our “Diff.
from Mean” column before that added up to 0
However, because we’re squaring our values they will
not be in the metric of our original scale

If we calculate the variance for a test out of 100, a variance
of 100 is actually average variability of 10 pts. (100 = 10)
about the mean of the test
Data
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 (Diff. from Mean)2 =
493.75
Variance =
493.75/(20-1) = 25.99
Mean
Diff. from
Mean
(Diff. from
Mean)2
19
14.25
4.75
22.56
16
14.25
1.75
3.06
20
14.25
5.75
33.06
17
14.25
2.75
7.56
20
14.25
5.75
33.06
19
14.25
4.75
22.56
7
14.25
-7.25
52.56
11
14.25
-3.25
10.56
10
14.25
-4.25
18.06
19
14.25
4.75
22.56
14
14.25
-.25
.06
11
14.25
-3.25
10.56
6
14.25
-8.25
68.06
11
14.25
-3.25
10.56
14
14.25
-.25
.06
19
14.25
4.75
22.56
20
14.25
5.75
33.06
17
14.25
2.75
7.56
4
14.25
-10.25
105.06
11
14.25
-3.25
10.56
Measures of Variability
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Standard Deviation
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Sample Standard Deviation (s) =
√ [(X - X )2/(n -1)]
Population Standard Deviation (σ) =
√ [(X - X )2/N]
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Note that the formula is identical to the Variance except
that after everything else you take the square-root!
You can interpret the standard deviation without doing
any mental math, like you did with the variance
Variance = 25.99
Standard Deviation = √(25.99) = 5.10
Measures of Variability
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Standard Deviation
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Example: Bush/Cheaney – 55%
Kerry/Edwards – 40%
Margin of Error = 30%
Bush/Cheaney – 25% – 85%
Kerry/Edwards – 10% - 70%
Computational Formula for Variability
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Definitional Formula
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designed more to illustrate how the formula
relates to the concept it underlies
Computational Formula
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identical to the definitional formula, but different in
form
allows you to compute your variable with less
effort
particularly useful with large datasets
Computational Formula for Variability
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Definitional Formula for Variance:
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s2 = (X – X )2
N–1
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Computational Formula for Variance:
2

X 
2
X 
 s2 =
N
N 1
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All you need to plug in here is X2 and X
Standard deviation still = √ s2, no matter how it is
calculated
Computational Formula for Variability
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Definitional Formula for Standard Deviation:
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s = √ [(X – X )2]
[ N–1 ]
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Computational Formula for Standard
Deviation
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s=√
2



X
( X 2 
N
N 1
)
Computational Formula for Variability
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Example:
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For the following dataset,
compute the variance and
standard deviation.
1
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X2
Data (X)
1
1
2
4
2
4
X = 23
X2 = 77
3
9
3
9
s2 = 77 – (23)2
_____8__
8–1
s2 = 77 – 66.125 = 1.55
7
3
9
4
16
5
25
2
2
3
3
3
4
5
Measures of Variability
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What do you think will happen to the standard
deviation if we add a constant (say 4) to all of
our scores?
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What if we multiply all the scores by a
constant?
Measures of Variability
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Characteristics of the Standard Deviation
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Adding a constant to each score will not alter the
standard deviation
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i.e. add 3 to all scores in a sample and your s will remain
unchanged
Let’s say our scores originally ranged from 1 – 10
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Add 5 to all scores, the new data ranges from 6 – 15
In both cases the range is 9
Measures of Variability
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However, multiplying or dividing each score by a
constant causes the s to be similarly multiplied or
divided by that constant (and s2 by the square of
the constant)
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i.e. divide each score by 2 and your s will decrease from
10 to 5
in multiplication, higher numbers increase more than
lower ones do, increasing the distance between the
highest and lowest score, which increases the variability
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i.e. 2 x 5 = 10 – difference of 8 pts.
5 x 5 = 25 – difference of 20 pts.
Measures of Variability
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Characteristics of the Standard Deviation
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Generally, the larger the dataset, the smaller the
range/standard deviation
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More scores = more clustering in the middle –
REMEMBER: more central scores are more likely to
occur
Smaller Dataset
Larger Dataset
s = 3.96482
s = 2.75609
Graphically Depicting Variability
Boxplot/Box-andWhisker Plot
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Median
60
50
1
52
40
14
92
166
158
211
199
30
 Hinges/1st
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H-Spread
&
3rd
Quartiles
BDI2 Total Score
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20
10
0
-10
N=
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Whisker
Gender
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Outlier
135
103
Female
Male
Graphically Depicting Variability
Boxplot/Box-andWhisker Plot
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Median
60
50
1
52
40
14
92
166
158
211
199
30
 Hinges/1st
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H-Spread
&
3rd
Quartiles
BDI2 Total Score
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20
10
0
-10
N=
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Whisker
Gender
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Outlier
135
103
Female
Male
Graphically Depicting Variability
Boxplot/Box-andWhisker Plot

Median
60
50
1
52
40
14
92
166
158
211
199
30
 Hinges/1st
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H-Spread
&
3rd
Quartiles
BDI2 Total Score
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20
{
10
0
-10
N=

Whisker
Gender
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Outlier
135
103
Female
Male
Graphically Depicting Variability
Boxplot/Box-andWhisker Plot

Median
60
50
1
52
40
14
92
166
158
211
199
30
 Hinges/1st

H-Spread
&
3rd
Quartiles
BDI2 Total Score

20
10
0
-10
N=

Whisker
Gender

Outlier
135
103
Female
Male
Graphically Depicting Variability
Boxplot/Box-andWhisker Plot

Median
60
50
1
52
40
14
92
166
158
211
199
30
 Hinges/1st

H-Spread
&
3rd
Quartiles
BDI2 Total Score
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20
10
0
-10
N=
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Whisker
Gender
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Outlier
135
103
Female
Male
Graphically Depicting Variability
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Percentile – the point below which a certain
percent of scores fall
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i.e. If you are at the 75th%ile (percentile), then
75% of the scores are at or below your score
Graphically Depicting Variability
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Quartile – similar to %ile, but splits distribution into
fourths
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i.e. 1st quartile = 0-25% of distribution, 2nd = 26-50%, 3rd =
51-75%, 4th = 76-100%
Graphically Depicting Variability
Interpreting a
Boxplot/Box-andWhisker Plot
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Off-center median = Nonsymmetry
Longer top whisker =
Positively-skewed
distribution
Longer bottom whisker =
Negatively-skewed
distribution
BDI2 Total Score
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60
50
1
52
40
14
92
166
158
211
199
30
20
10
0
-10
N=
Gender
135
103
Female
Male
Graphically Depicting Variability
60
40
50
1
52
40
14
92
166
158
211
30
199
30
20
20
10
10
0
Std. Dev = 10.80
Mean = 12.4
-10
N=
Gender
135
103
Female
Male
N = 135.00
0
0.0
5.0
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0
BDI2 Total Score for Females
Graphically Depicting Variability
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Boxplot/Box-and-Whisker Plot
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Hinge/Quartile Location = (Median Location+1)/2
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Data: 1 3 3 5 8 8 9 12 13 16 17 17 18 20
21 40
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Median Location = (16+1)/2 = 8.5
Hinge Location = (8.5+1)/2 = 4.75 (4 since we drop the
fraction)
Hinges = 5 and 18
Graphically Depicting Variability
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H-Spread = Upper Hinge – Lower Hinge
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H-Spread = 18-5 = 13
Whisker = H-Spread x 1.5
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Since the whisker always ends at an actual data point, if we,
say calculated the whisker to end at a value of 12, but the data
only has a 10 and a 15, we would end the whisker at the 10.
Whiskers = 12x1.5 = 19.5
Lower whisker from 5 to 1
Higher whisker from 18 to 21
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Outliers
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Value of 40 extends beyond upper whisker
Graphically Depicting Variability
30
20
10
0
-10
N=
16
1.00