VISUAL LESSON · 4 PICTURES

No Covariance Is Not Covariance Equal to Zero

Is covariance absent, or is it defined and numerically zero?

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BEFORE THE FIRST PICTURE

How this lesson unfolds.

Begin with Does covariance exist, or is its value zero? The sequence then connects One variable supports mean and variance and builds toward A second variable makes covariance meaningful.

Undefined and zero answer different questions. With only one variable the requested two-variable quantity is not defined; with two variables it is defined and may evaluate to zero. — Picture 1 of 4 · tap to inspect in full resolution

Does covariance exist, or is its value zero?

INTUITION

With only one variable the requested two-variable quantity is not defined; with two variables it is defined and may evaluate to zero.

Learning goalDistinguish core distinction without confusing undefined covariance, zero covariance and independence.
Helpful foundationsmean · variance · paired variables
Common trapZero covariance is a defined numerical result and does not generally imply independence; one-variable covariance is undefined.
Next connectionOne variable supports mean and variance

One variable supports mean and variance. Alice's attendance indicator has a mean and variance, but covariance needs a second variable and therefore is not defined in this one-variable setup. — Picture 2 of 4 · tap to inspect in full resolution

What can we measure when only Alice exists?

INTUITION

Alice's attendance indicator has a mean and variance, but covariance needs a second variable and therefore is not defined in this one-variable setup.

Learning goalDistinguish one variable case without confusing undefined covariance, zero covariance and independence.
Helpful foundationsmean · variance · paired variables
Common trapZero covariance is a defined numerical result and does not generally imply independence; one-variable covariance is undefined.
Next connectionVariance asks how one variable spreads

Variance asks how one variable spreads. Variance squares one variable's deviations from its own mean, so a single variable is sufficient; covariance instead multiplies deviations from two variables. — Picture 3 of 4 · tap to inspect in full resolution

Why does variance not need a partner?

INTUITION

Variance squares one variable's deviations from its own mean, so a single variable is sufficient; covariance instead multiplies deviations from two variables.

Learning goalDistinguish variance versus covariance without confusing undefined covariance, zero covariance and independence.
Helpful foundationsmean · variance · paired variables
Common trapZero covariance is a defined numerical result and does not generally imply independence; one-variable covariance is undefined.
Next connectionA second variable makes covariance meaningful

A second variable makes covariance meaningful. With paired variables covariance can be positive, negative, or zero; zero means signed linear contributions cancel, not necessarily independence. — Picture 4 of 4 · tap to inspect in full resolution

What changes when Bob joins Alice?

INTUITION

With paired variables covariance can be positive, negative, or zero; zero means signed linear contributions cancel, not necessarily independence.

Learning goalDistinguish two variable case without confusing undefined covariance, zero covariance and independence.
Helpful foundationsmean · variance · paired variables
Common trapZero covariance is a defined numerical result and does not generally imply independence; one-variable covariance is undefined.
Next connectionTest nonlinear dependence and independence separately