Variance

Definition

Variance measures how spread out values are around the mean.

Basic question:

How far do values usually sit from the average?

Variance = average squared distance from the mean.

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Why Mean Is Not Enough

Two datasets can have the same mean but different spread.

Example:

A: 4, 5, 6
B: 0, 5, 10

Both have mean:

5

But B is more spread out.

Mean tells center; variance tells spread.

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How Variance Is Computed

Basic steps:

1. find the mean
2. measure each value's distance from the mean
3. square those distances
4. average the squared distances

Squaring prevents positive and negative distances from canceling.

Variance uses squared deviations from the mean.

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Simple Example

Values:

4, 5, 6

Mean:

5

Distances from mean:

-1, 0, 1

Squared distances:

1, 0, 1

Average squared distance:

(1 + 0 + 1) / 3 = 2/3

The variance is 2/3.

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Low Variance

Low variance means values are close to the mean.

Example:

9, 10, 11

The values cluster tightly.

Low variance means low spread.

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High Variance

High variance means values are far from the mean.

Example:

0, 10, 20

The values are spread out.

High variance means high spread.

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Why Variance Matters For Learning

Variance helps describe uncertainty and stability.

Examples:

high variance predictions -> unstable outputs
low variance errors -> consistent behavior
high variance data -> broad range of examples

Variance tells how much values fluctuate around their center.

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Final Summary

• Variance measures spread around the mean.
• Mean tells center; variance tells dispersion.
• Variance averages squared distances from the mean.
• Low variance means values are close together.
• High variance means values are spread out.
• Learning uses variance to reason about uncertainty, stability, and data spread.