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Psych 5500/6500

Measures of Variability

Fall, 2008

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Measures of Variability

We will look at three ways of measuring how much the scores differ from each other.1. Mean Absolute Deviation

2. Variance

3. Standard Deviation

These approaches are based upon the concept that you can tell how much the scores differ from each other by looking at how much they differ from the mean.

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Spatial Analogy

The two figures below represent the location of students within two rooms. In room A the variability of location is large, in room B it is small.

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Spatial Analogy (cont.)How might we want to measure the variability of

location? One solution would be to find the average distance each student is from every other student. These distances are reflected below.

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Spatial Analogy (cont.)

A simpler procedure would be to find the geographic center of the students, and measure the average distance each student is from that.

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Variability

In a similar fashion, if we want to know how close scores are to each other we could measure how close each score is to every other score. A simpler approach would be to measure how close each score is to the mean (which is located in the center of the scores). If all the scores are similar in value then they will all be close to the mean, if the scores differ a great deal some will be far from the mean.

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Variability: Example 1

10.Y

12. 11, 9, 10, 7, 11, Y :Sample

Y - = deviation from mean

11 - 10 = 1

7 - 10 = -3

10 - 10 = 0

9 - 10 = -1

11 - 10 = 1

12 - 10 = 2

Step 1: determine how far each score is from the mean

Y

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Mean (Interesting Property #2)

Given our spatial analogy, it would make sense to add up the distances (deviations) from the mean, and find the average deviation as our measure of variability. This does not work however, as the sum of the deviations from the mean (e.g. 1 + -3 + 0 + -1 + 1 + 2) always equals zero. The mean is the only value for which this is true (i.e. plug any number other than 10 into the table on the previous slide and you will not get a sum of deviations equal to zero). The problem here--the reason we end up with an average distance of zero--is that we allow there to be both negative and positive distances from the mean, it would make more sense not to allow that.

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Mean Absolute DeviationOne solution would be to use the absolute value of the

deviations, and find the average (mean) of those deviations as our measure of variability.

Y - = deviations |deviations|

11 - 10 = 1 1

7 - 10 = -3 3

10 - 10 = 0 0

9 - 10 = -1 1

11 - 10 = 1 1

12 - 10 = 2 2

Y

1.332)/61103(1 deviation mean

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VarianceAnother approach, that used by the ‘variance’, is to square

each deviation, and find the average (mean) of those.

Y - = deviations deviations²

11 - 10 = 1 1

7 - 10 = -3 9

10 - 10 = 0 0

9 - 10 = -1 1

11 - 10 = 1 1

12 - 10 = 2 4

Y

Mean squared deviation = (1+9+0+1+1+4)/6=2.67

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Variance (defined)

The variance, then, is the average squared distance each score is from the mean. This is technically stated as ‘the mean squared deviation from the mean’. The spatial analogy still applies, but we square each distance before finding the average.

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Variance (continued)The variance will be the mean of the squared deviations

(1+9+0+1+1+4)/6=16/6=2.67. If the scores are similar to each other the mean squared deviation will be small. If the scores differ a lot the mean squared deviation will be larger.

Y - = deviations deviations²

11 - 10 = 1 1

7 - 10 = -3 9

10 - 10 = 0 0

9 - 10 = -1 1

11 - 10 = 1 1

12 - 10 = 2 4

Y

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Sum of SquaresTo find the variance we need to sum the squared deviations

and divide by N, that sum of squared deviations has a name.

Y - = deviations deviations²

11 - 10 = 1 1

7 - 10 = -3 9

10 - 10 = 0 0

9 - 10 = -1 1

11 - 10 = 1 1

12 - 10 = 2 4

“Sum of the squared deviations” = 1+9+0+1+1+4=16This is usually abbreviated to “Sum of Squares”, or simply “SS”.

Y

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Variance (computation)

2.67 6

16

n

)Y-(Y

n

SS S

.S is sample theof variancefor the symbol The

)Y-(YSS :is and SS,for formula' aldefinition'

thecalled is SS compute toused weprocess The

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2

2

Note that as we are summing squared values there is no way forSS (or the variance) to be a negative number.

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N or (N-1) ?We have defined the variance of our sample as being:

N

SSS2

You may have encountered a similar, but different, formula forvariance that has (N-1) in the denominator. That is actuallysomething different, and we will be covering it in the nextlecture. Note that when SPSS gives you ‘variance’ it uses the(N-1) formula.

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Variance: Example 2Let’s look at another sample:

Example 2: Y = 5, 10, 18, 8, 7, 12

Compare that to our first sample where:Example 1: Y = 11, 7, 10, 9, 11, 12

Note that example 2 has greater variability among its scores, as variance measures variability the variance of example 2 should be greater than the variance of example 1.

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Example 2 (Cont.)

Y = 5, 10, 18, 8, 7, 12. Again the mean is 10.

Y - = deviations deviations²

5 - 10 = 5 25

10 - 10 = 0 0

18 - 10 = 8 64

8 - 10 = -2 4

7 - 10 = -3 9

12 - 10 = 2 4

SS= 106

17.67 6

106 S2

Y

Note how much the score of 18added to the SS when its deviationof 8 was squared.

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Effect of scores that are far from the mean

Because the variance is the average squared distance each score is from the mean, scores that are far from the mean have a disproportionate effect on the variance. A score that is 1 away from the mean adds 12 =1 to the SS, a score that is 10 away from the mean adds 102 = 100 to the SS.

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Mean (Interesting Property #3)

The mean of the scores will give you the smallest possible ‘sum of squared deviations’. In other words, if you used any number other than the mean (10) to compute SS in the previous examples then the resulting value of SS would have been larger.

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Variance: Example 3Y = 10, 10, 10, 10, 10, 10. Again the mean is 10. Note

the scores are identical, variance should be zero.

Y - = deviations deviations²

10 - 10 = 0 0

10 - 10 = 0 0

10 - 10 = 0 0

10 - 10 = 0 0

10 - 10 = 0 0

10 - 10 = 0 0

SS= 00

6

0 S2

Y

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Formulas for SS

2)Y-(YSS :formula alDefinition

n

YYSS :formula nalComputatio

2

2

The definitional formula for SS has the advantage of makingit clear just exactly what ‘SS’ is, the ‘sum of the squareddeviations’:

The definitional formula has the disadvantage of being slowand cumbersome for large data sets. A much faster way tocompute SS using a calculator is with the computational formula.

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Computational Formula

166006166

3600616SS

6n 360060Y

601211910711Y

6161211910711Y

n

YYSS :formula nalComputatio

12 11, 9, 10, 7, 11, Y

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2222222

2

2

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Interpreting VarianceSo what does knowing, for example, that a sample

has a variance of ’10’ tell us about the sample? Well, it tells us that the average squared distance each score is from the mean is 10.

The variance also has meaning when it comes to comparing two samples. If sample A had a variance of 6 and sample B had a variance of 8, then the scores in sample B varied more than the scores in sample A.

And finally, if the variance of sample equals zero that tells us that all the scores were identical.

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Standard Deviation

The last measure of variability we will consider is the standard deviation. It is simply the positive square root of the variance. Its symbol is ‘S’.

0 0 S :3 Example

4.20 17.67 S :2 Example

1.63 2.67 S :1 Example

S S 2

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Interpreting Standard Deviation

So what does knowing, for example, that a sample has a standard deviation of ‘4’ tells us about the sample? Well, it tells us that the square root of the average squared deviation from the mean is ‘4’. As we will see in a future lecture, knowing the standard deviation is both interesting and comprehendible. Until then...

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Interpreting Standard Deviation (cont.)

...at least you know that, as with the variance, if sample A has a standard deviation of 3 and sample B has a standard deviation of 5, then the scores in sample B differed more than the scores in sample A.

And that if a sample had a standard deviation of zero that means that all of the scores were identical.