Variance and covariance from a geometrical point of view

(1)

Variance and covariance from a

geometrical point of view

(2)

(3)

• Mean, median and mode are central tendency

measures for a given distribution

• The average tells us what is the general

(4)

• Variance is a measure of the dispersion of a

distribution around the average

• The variance tells us how different are the

(5)

• 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

(6)

(7)

-8,17

(8)

-8,17 -4,17

(9)

-8,17 -4,17

1,33

(10)

-8,17 -4,17

1,33

3,33

(11)

-8,17 -4,17

1,33

3,33

4,33

(12)

-8,17 -4,17

1,33

3,33

4,33

19,83

(13)

(14)

(15)

(16)

-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

(17)



-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

(18)



-8,17 -4,17

1,33

3,33

4,33

(19)



X

10,00

19,83

-8,17 -4,17

1,33

3,33

4,33

(20)



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

(21)



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

(22)



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

(23)

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

-4,17 1,33 3,33

(24)

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

-4,17 1,33 3,33

(25)

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

-4,17 1,33

(26)

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

-4,17 1,33

(27)

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

-4,17

(28)

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

-4,17

(29)

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

(30)

(31)

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

(32)

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

(33)

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

(34)

(35)

• The covariance expresses the closeness of two

variables

• For example, reading fluency and reading

(36)

(37)

• The covariance is the average signed area of

rectangles

• So, if the means are expressed in meters and

(38)

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



(39)

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

(40)

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

(41)

X

Y

(x

₁

, y

₁

)

(42)

(43)

(44)

X

Y

(45)

(46)

X

Y

(47)

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

(48)

X

Y

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X

Y

(50)

(51)

• The problem with the interpretation of

covariance is that it is dependent on the

variances of the variables

• Maximum and minimum values for the

(52)

• For example,

–

variance for Reading Fluency (Wpm) was

259.45 –

variance for Reading Comprehension was

669.27

(53)

(54)

(55)

• An alternative is the correlation

• The correlation coefficient

r

measures the

strength and direction of a linear relationship

between two variables

(56)

(57)

(58)

• 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

(59)

Variance and covariance from a geometrical point of view

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