Worksheets for Advanced Statistics

Worksheet - Variance-Covariance Matrix and Correlation Matrix

Section 1 - Raw Data

Variance

Co-Variance

Standardized Covariance (Correlation)

Secrtion 2 - Standardized Data (z-values via standard normal deviate)

Secrtion 3 - Optional - Ranked Data

Raw data (i.e., Raw score matrix)

Calculate the mean vector and standard deviation

	Σ	Mean	σ²	σ
x1	13	2.6	3.3	1.8166
x2	15	3.0
x3	19	3.8

Obtain a listing of deviations from the mean vector (i.e., deviation matrix)

Square the deviations (i.e., Squared Deviation matrix)

		Individuals	Σ
Squared Deviations from Variable Means	x1	2.56
	x2
	x3

Average the summed squared deviations - you lose a degree of freedom for every parameter that you estimate from your sample. You need an estimate of the parametric mean to calculate the deviations, you therefore get your estimated average squared dispersion (i.e., variance) when dividing by N-1 and not N.

		Σ	N-1	Mean
Mean Squared Deviations from Variable Means	x1	13.2	4	3.3
	x2
	x3

Multiply the deviations from two different variables and average them

		Individuals	Σ	N-1	Mean
Deviations from Variable Means	x12	1.6	7	4	1.75
	x13
	x23

Divide the Covariance by the two standard deviations of the respective variables

		Σ	σ₁*σ₂	Σ/ (σ₁*σ₂)
Mean Squared Deviations from Variable Means	x12	1.75	1.8166*1	0.9633
	x13
	x23

Arrange results into a SS-CrossProducts Matrix

Term				Result
	x1	x2	x3	x1	x2	x3
x1	Σ(X₁²)	Σ(X₁X₂)	Σ(X₁X₃)	13.2	7	.
x2	Σ(X₁X₂)	Σ(X₂²)	Σ(X₂X₃)	7	.	.
x3	Σ(X₁X₃)	Σ(X₂X₃)	Σ(X₃²)	.	.	.

Arrange results into a Variance-Covariance Matrix

Term				Result
	x1	x2	x3	x1	x2	x3
x1	Var₁₁=Σ(X₁²)/(N-1)			3.3	1.75	.
x2	CoVar₁₂=Σ(X₁X₂)/(N-1)	Var₂₂=Σ(X₂²)/(N-1)		1.75	.	.
x3	CoVar₁₃=Σ(X₁X₃)/(N-1)	CoVar₂₃=Σ(X₂X₃)/(N-1)	Var₃₃=Σ(X₃²)/(N-1)	.	.	.

Arrange results into a Correlation Matrix

Term				Result
	x1	x2	x3	x1	x2	x3
x1	r₁₁= Var₁₁/σ₁σ₁			1
x2	r₁₂= CoVar₁₂/σ₁σ₂	r₂₂= Var₂₂/σ₂σ₂
x3	r₁₃= CoVar₁₃/σ₁σ₃	r₂₃= CoVar₂₃/σ₂σ₃	r₃₃= Var₃₃/σ₃σ₃

Rerun all calculations from above with data converted to standard normal deviates (i.e., standardized score matrix) where you subtract the variable's mean from each value and divide it by the variable's standard deviation.

Variable	Individuals	Σ	Mean	σ²	σ
x1	-0.8808
x2
x3

- continue with calculations from section 1

Report results into a Variance-Covariance Matrix: What do you notice in comparison with the results of section 1? What explains the matching terms?

Variance-Covariance Matrix
	x1	x2	x3
x1
x2
x3

Repport Results for Correlation matrix: What do you notice in comparison with the results of section 1? What explains the matching terms?

Correlation Matrix
	x1	x2	x3
x1
x2
x3

Individuals

Rerun all calculations from above with data converted to ranked data. Correct for ties in the data by giving the tied values averaged ranks

Variable	Individuals			Σ	Mean	σ²	σ
x1	1.5	3	1.5
x2
x3

- continue with calculations from section 1

Report results into a Variance-Covariance Matrix: What do you notice in comparison with the results of section 1? What is different about this analysis? What are the data features that this analysis is more or less sensitive to? Compare with results from Spearmean's Rank Correlation

Variance-Covariance Matrix
	x1	x2	x3
x1
x2
x3

Report Results for Correlation matrix:

Correlation Matrix
	x1	x2	x3
x1
x2
x3

last modified: 1/25/13