The risk puzzle

Two mutual funds both averaged 8% annual returns over 10 years. Yet one investor made steady gains while the other had a rollercoaster ride. What's different?

What do you think?

Fund A returned exactly 8% every year. Fund B returned alternating +28% and -12%. Both average 8%. Which is riskier?

The mean tells you the center. Variance tells you how far outcomes typically stray from that center.

Variance: measuring spread

Variance

The variance of a random variable $X$ is: $\text{Var}(X) = E[(X - \mu)^2] = E[X^2] - (E[X])^2$ where $\mu = E[X]$ . It measures the average squared distance from the mean.

Why square the deviations? Because positive and negative deviations would cancel. Squaring ensures all deviations count.

The second formula — $E[X^2] - (E[X])^2$ — is almost always easier to compute.

What do you think?

If X is constant (always equals 5), what is Var(X)?

Enter a whole number

Standard deviation

Standard Deviation

The standard deviation is the square root of variance: $\text{SD}(X) = \sigma = \sqrt{\text{Var}(X)}$ It has the same units as $X$ , making it more interpretable than variance.

If $X$ is measured in dollars, $\text{Var}(X)$ is in "dollars squared" (meaningless!), but $\text{SD}(X)$ is back in dollars.

Drag and explore

Drag points away from the mean to see how variance responds. Notice that outliers have a disproportionate effect because of the squaring:

The Spread Stretch

Drag points left/right to change their values. Watch variance respond.

Mean (μ)

6.17

Variance (σ²)

3.97

Std Dev (σ)

1.99

Points

Squared deviations from mean

(-4.2)² = 17.4(-2.2)² = 4.7(-1.2)² = 1.4(-1.2)² = 1.4(-0.2)² = 0.0(-0.2)² = 0.0(-0.2)² = 0.0(+0.8)² = 0.7(+0.8)² = 0.7(+1.8)² = 3.4(+1.8)² = 3.4(+3.8)² = 14.7

Sum = 47.7 ÷ 12 = 3.97

Key properties of variance

Variance Rules

For constants $a$ and $b$ : $\text{Var}(aX + b) = a^2 \cdot \text{Var}(X)$ Shifting by $b$ doesn't change the spread. Scaling by $a$ multiplies variance by $a^2$ .

Shifting does not affect variance. Adding a constant to every outcome moves the center but doesn't change how spread out the data is.

Operation	Effect on Mean	Effect on Variance
$X + c$	$\mu + c$	No change
$aX$	$a\mu$	$a^2 \sigma^2$
$aX + b$	$a\mu + b$	$a^2 \sigma^2$

Experiment with scaling and shifting to build intuition for why only $a$ (not $b$ ) affects variance:

Variance Scaling Lab

Transform Y = aX + b where X is a fair die roll

Scale: a = 1.0: 1.0

-33

Shift: b = 0: 0

-1010

X (original) Y = 1.0X + 0

E[X]

3.50

E[Y] = aE[X]+b

3.50

Var(X)

2.92

Var(Y) = a²Var(X)

2.92

Y = X (no transformation). Try changing a or b!

Variance of a sum

For independent random variables:

$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \quad \text{(if } X \perp Y\text{)}$

Unlike linearity of expectation, this requires independence! For dependent variables: $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$ .

Worked example: dice

For a single fair die roll $X$ :

Variance of a fair die

E[X] = \frac{1+2+3+4+5+6}{6} = 3.5

Mean

Step 1 of 4

Practice problems

X has E[X] = 10 and E[X²] = 120. What is Var(X)? (whole number)

If Var(X) = 9, what is Var(3X)? (whole number)

If Var(X) = 9, what is Var(X + 100)? (whole number)

Summary

Concept	Formula
Variance	$\text{Var}(X) = E[X^2] - (E[X])^2$
Standard deviation	$\sigma = \sqrt{\text{Var}(X)}$
Scaling rule	$\text{Var}(aX + b) = a^2 \text{Var}(X)$
Sum (independent)	$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$

Variance captures risk, volatility, and uncertainty. When two distributions have the same mean, variance is what separates them.

Test your understanding

A coin flip: X = 1 (heads) or 0 (tails), each with probability 1/2. What is Var(X)? (decimal, e.g. 0.42)

You flip 100 independent fair coins. What is the variance of the total number of heads? (whole number)

Temperature is converted via F = 1.8C + 32. If Var(C) = 4, what is Var(F)? (decimal, e.g. 0.42)

What's next

Variance tells us how spread out a distribution is, but not which direction. Next we'll explore skewness (asymmetry) and kurtosis (tail heaviness).