Measures of Variability. Why are measures of variability important? Why not just stick with the mean? Ratings of attractiveness (out of 10) – Mean =

Measures of Variability


Why are measures of variability important? Why not just stick with the mean? Ratings of attractiveness (out of 10) – Mean = 5

Everyone rated you a 5 (low variability) What could we conclude about attractiveness from this?

People’s ratings fell into a range from 1 – 10, that averaged a 5 (high variability) What could we conclude about attractiveness from this?


Measures of Variability Range Interquartile Range Average Deviation Variance Standard Deviation


Range The difference between the highest and lowest

values in a dataset

Heavily biased by outliers

Dataset #1: 5 7 11

Range = 6

Dataset #2: 5 7 11 million

Range = 10,999,995


Interquartile Range 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%


Average Deviation For each score, calculate deviation from the

mean, then sum all of these scores

However, this score will always equal zero 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

Diff. from Mean = 0

0/N = 0

www.randomizer.org

Data 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


Variance Sample Variance (s2) = (X - )2/(n -1) Population Variance (σ2) = (X - )2/N

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

X

X

(Diff. from Mean)2 = 493.75

Variance =

493.75/(20-1) = 25.99

Data 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


Standard Deviation Sample Standard Deviation (s) =

√ [(X - )2/(n -1)] Population Standard Deviation (σ) =

√ [(X - )2/N] 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

X

X


Standard Deviation Example: Bush/Cheaney – 55%

Kerry/Edwards – 40%

Margin of Error = 30%

Bush/Cheaney – 25% – 85%

Kerry/Edwards – 10% - 70%

Computational Formula for Variability Definitional Formula

designed more to illustrate how the formula relates to the concept it underlies

Computational Formula 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 Definitional Formula for Variance:

s2 = (X – )2

N – 1

Computational Formula for Variance:

s2 =

All you need to plug in here is X2 and X Standard deviation still = √ s2, no matter how it is

calculated

X

1

22

NN

XX

Computational Formula for Variability Definitional Formula for Standard Deviation:

s = √ [(X – )2] [ N – 1 ]

Computational Formula for Standard Deviation

s = √ ( )

X

1

22

NN

XX

Computational Formula for Variability Example:

For the following dataset, compute the variance and standard deviation.

1 2 2 3 3 3 4 5

X = 23 X2 = 77

s2 = 77 – (23)2

_____8__8 – 1

s2 = 77 – 66.125 = 1.55 7

Data (X) X2

1 1

2 4

2 4

3 9

3 9

3 9

4 16

5 25


What do you think will happen to the standard deviation if we add a constant (say 4) to all of our scores?

What if we multiply all the scores by a constant?


Characteristics of the Standard Deviation Adding a constant to each score will not alter the

standard deviation 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

Add 5 to all scores, the new data ranges from 6 – 15 In both cases the range is 9


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) 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 i.e. 2 x 5 = 10 – difference of 8 pts.

5 x 5 = 25 – difference of 20 pts.


Characteristics of the Standard Deviation Generally, the larger the dataset, the smaller the

range/standard deviation More scores = more clustering in the middle –

REMEMBER: more central scores are more likely to occur

Smaller Dataset

s = 3.96482

Larger Dataset

s = 2.75609

Graphically Depicting Variability Boxplot/Box-and-

Whisker Plot Median

Hinges/1st & 3rd Quartiles

H-Spread

Whisker

Outlier

103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

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Whisker Plot Median


H-Spread

Whisker

Outlier

103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

re

60

50

40

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Whisker Plot Median


H-Spread

Whisker

Outlier

103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

re

60

50

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{


Whisker Plot Median


H-Spread

Whisker

Outlier

103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

re

60

50

40

30

20

10

0

-10

199

211

1581669214

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Whisker Plot Median


H-Spread

Whisker

Outlier

103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

re

60

50

40

30

20

10

0

-10

199

211

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Graphically Depicting Variability Percentile – the point below which a certain

percent of scores fall i.e. If you are at the 75th%ile (percentile), then

75% of the scores are at or below your score

Graphically Depicting Variability Quartile – similar to %ile, but splits distribution into

fourths i.e. 1st quartile = 0-25% of distribution, 2nd = 26-50%, 3rd =

51-75%, 4th = 76-100%

Graphically Depicting Variability Interpreting a

Boxplot/Box-and-Whisker Plot Off-center median = Non-

symmetry Longer top whisker =

Positively-skewed distribution

Longer bottom whisker = Negatively-skewed distribution

103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

re

60

50

40

30

20

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0

-10

199

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Graphically Depicting Variability

BDI2 Total Score for Females

50.045.040.035.030.025.020.015.010.05.00.0

40

30

20

10

0

Std. Dev = 10.80

Mean = 12.4

N = 135.00103135N =

Gender

MaleFemale

BD

I2 T

ota

l Sco

re

60

50

40

30

20

10

0

-10

199

211

1581669214

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Graphically Depicting Variability Boxplot/Box-and-Whisker Plot

Hinge/Quartile Location = (Median Location+1)/2

Data: 1 3 3 5 8 8 9 12 13 16 17 17 18 20 21 40 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


H-Spread = Upper Hinge – Lower Hinge H-Spread = 18-5 = 13

Whisker = H-Spread x 1.5 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 = 19.5Lower whisker from 5 to 1Higher whisker from 18 to 21

Outliers Value of 40 extends beyond upper whisker


16N =

1.00

30

20

10

0

-10

Measures of Variability. Why are measures of variability important? Why not just stick with the mean? Ratings of attractiveness (out of 10) – Mean =

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