Variance and covariance from a geometrical point of view

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Variance and covariance from a

geometrical point of view

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Mean, median and mode are central tendency

measures for a given distribution

The average tells us what is the general

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Variance is a measure of the dispersion of a

distribution around the average

The variance tells us how different are the

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The variance is the area of a square

So, if the mean is expressed in meters, the

variance is expressed in square meters

The standard deviation is the square root of

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-8,17

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-8,17 -4,17

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-8,17 -4,17

1,33

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-8,17 -4,17

1,33

3,33

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-8,17 -4,17

1,33

3,33

4,33

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-8,17 -4,17

1,33

3,33

4,33

19,83

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-8,17 -4,17

1,33

3,33

4,33

19,83

21,83

24,83

26,83

X -8,17 -4,17 1,33 3,33

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-8,17 -4,17

1,33

3,33

4,33

19,83

21,83

24,83

26,83

X -8,17 -4,17 1,33 3,33

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-8,17 -4,17

1,33

3,33

4,33

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X

10,00

19,83

-8,17 -4,17

1,33

3,33

4,33

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X

10,00

19,83

-8,17 -4,17

1,33

3,33

4,33

10,00

19,83

21,83

24,83

26,83

X-

= 9,83

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X

10,00

19,83

-8,17 -4,17

1,33

3,33

4,33

10,00

19,83

21,83

24,83

26,83

X-

= 9,83

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X

10,00

19,83

-8,17 -4,17

1,33

3,33

4,33

10,00

19,83

21,83

24,83

26,83

(X-

)

2

X (X-m) (X-m)2 -8,17

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X (X-m) (X-m)2 -8,17

-4,17 1,33 3,33

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X (X-m) (X-m)2 -8,17

-4,17 1,33 3,33

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X (X-m) (X-m)2 -8,17

-4,17 1,33

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X (X-m) (X-m)2 -8,17

-4,17 1,33

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X (X-m) (X-m)2 -8,17

-4,17

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X (X-m) (X-m)2 -8,17

-4,17

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X (X-m) (X-m)2 -8,17

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X (X-m) (X-m)2 -8,17 -18,17 330,0 -4,17 -14,17 200,7 1,33 -8,67 75,1 3,33 -6,67 44,4

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X (X-m) (X-m)2 -8,17 -18,17 330,0 -4,17 -14,17 200,7 1,33 -8,67 75,1 3,33 -6,67 44,4

4,33 -5,67 32,1

2

330

200

,

7

75

,

1

44

,

4

32

,

1

96

,

7

140

220

283

,

4

158

,

1

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X (X-m) (X-m)2 -8,17 -18,17 330,0 -4,17 -14,17 200,7 1,33 -8,67 75,1 3,33 -6,67 44,4

4,33 -5,67 32,1

2

330

200

,

7

75

,

1

44

,

4

32

,

1

96

,

7

140

220

283

,

4

158

,

1

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The covariance expresses the closeness of two

variables

For example, reading fluency and reading

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The covariance is the average signed area of

rectangles

So, if the means are expressed in meters and

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X Y (X-x) (Y-y) (X-x)(Y-y)

-8,17 4,56 -18,2 -5,4 98,9

-4,17 -3,44 -14,2 -13,4 190,5

1,33 -0,94 -8,7 -10,9 94,9

3,33 18,56 -6,7 8,6 -57,0

4,33 0,06 -5,7 -9,9 56,4

19,83 5,56 9,8 -4,4 -43,7

21,83 27,06 11,8 17,1 201,8

24,83 17,06 14,8 7,1 104,7

26,83 21,56 16,8 11,6 194,5

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X

Y

(21.83, 27.06)

X Y (X-x) (Y-y) (X-x)(Y-y)

-8,17 4,56 -18,2 -5,4 98,9

-4,17 -3,44 -14,2 -13,4 190,5

1,33 -0,94 -8,7 -10,9 94,9

3,33 18,56 -6,7 8,6 -57,0

4,33 0,06 -5,7 -9,9 56,4

19,83 5,56 9,8 -4,4 -43,7

21,83 27,06 11,8 17,1 201,8

24,83 17,06 14,8 7,1 104,7

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X

Y

(x

1

, y

1

)

X Y (X-x) (Y-y) (X-x)(Y-y)

-8,17 4,56 -18,2 -5,4 98,9

-4,17 -3,44 -14,2 -13,4 190,5

1,33 -0,94 -8,7 -10,9 94,9

3,33 18,56 -6,7 8,6 -57,0

4,33 0,06 -5,7 -9,9 56,4

19,83 5,56 9,8 -4,4 -43,7

21,83 27,06 11,8 17,1 201,8

24,83 17,06 14,8 7,1 104,7

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X

Y

(x

1

, y

1

)

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X

Y

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X

Y

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X

Y

X Y (X-x) (Y-y) (X-x)(Y-y)

-8,17 4,56 -18,2 -5,4 98,9

-4,17 -3,44 -14,2 -13,4 190,5

1,33 -0,94 -8,7 -10,9 94,9

3,33 18,56 -6,7 8,6 -57,0

4,33 0,06 -5,7 -9,9 56,4

19,83 5,56 9,8 -4,4 -43,7

21,83 27,06 11,8 17,1 201,8

24,83 17,06 14,8 7,1 104,7

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X

Y

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X

Y

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The problem with the interpretation of

covariance is that it is dependent on the

variances of the variables

Maximum and minimum values for the

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For example,

variance for Reading Fluency (Wpm) was

259.45

variance for Reading Comprehension was

669.27

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An alternative is the correlation

The correlation coefficient

r

measures the

strength and direction of a linear relationship

between two variables

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To interpret its value, see which of the following values

your correlation

r

is closest to:

Exactly

1.

A perfect downhill (negative) linear relationship

0.70.

A strong downhill (negative) linear relationship

0.50.

A moderate downhill (negative) relationship

0.30.

A weak downhill (negative) linear relationship

0.

No linear relationship

+0.30.

A weak uphill (positive) linear relationship

+0.50.

A moderate uphill (positive) relationship

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Figure

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Referencias

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