Variance, standard deviation, and coefficient of variation

Exam P – Probability · SOA (Society of Actuaries) · 5 min read

Welcome to the World of "Spread"!

In our previous lessons, we learned about the Expected Value (or mean), which tells us the "center" of a random variable. But knowing the center is only half the story! Imagine two archers: one hits the bullseye every time, and the other misses wildly to the left and then wildly to the right. On average, they both hit the center—but their performance is very different!

In this chapter, we will learn how to measure how "spread out" or "risky" a random variable is. This is crucial for actuaries because, in insurance, uncertainty (spread) is what we are actually pricing. Don't worry if these formulas look a bit intimidating at first; we will break them down step-by-step!

1. Variance: The Measure of "Wiggle Room"

The Variance of a random variable, denoted as $ Var(X) $ or $ \sigma^2 $, measures the average squared distance from the mean. It tells us how much the outcomes "wiggle" around the average.

The Conceptual Formula

$ Var(X) = E[(X - \mu)^2] $
Where $ \mu $ is the mean $ E[X] $.

Wait, why do we square it?

Great question! If we just took the average distance from the mean, the positive differences (values above average) and negative differences (values below average) would cancel each other out, and we'd end up with zero. Squaring the numbers makes everything positive so we can see the total magnitude of the spread.

The "Exam Favorite" Shortcut Formula

In the heat of Exam P, you will almost always use this version. It is much faster to calculate!
$ Var(X) = E[X^2] - (E[X])^2 \生存$

Memory Trick: Think "The Mean of the Squares minus the Square of the Mean."

How to calculate $ E[X^2] $:

Discrete: $ \sum x^2 \cdot P(X=x) $
Continuous: $ \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx $

Quick Review: To find variance, find the "second moment" $ E[X^2] $, find the "first moment" (mean) and square it, then subtract!

2. Standard Deviation: Bringing it Back to Reality

Variance is great, but it has a weird quirk: it's in squared units. If $ X $ represents dollars, the variance is in "dollars squared." What does that even mean? To get back to units we understand, we use the Standard Deviation ($ \sigma $).

Formula: $ \sigma = \sqrt{Var(X)} $

Key Point: Standard deviation can never be negative. Because it is the square root of a squared average, it will always be $ \geq 0 $. If you calculate a negative standard deviation on the exam, pause and check your subtraction in the variance formula!

Analogy: The Pizza Delivery

If a pizza shop says their average delivery time is 30 minutes with a standard deviation of 5 minutes, you can reasonably expect your pizza in 25–35 minutes. If the standard deviation was 20 minutes, you might get it in 10 minutes or 50 minutes—that's a lot more uncertainty!

3. The Coefficient of Variation (CV)

Sometimes, we want to compare the spread of two different things. Is a $100 standard deviation "big"? It depends! If you're insuring a $1,000,000 house, $100 is tiny. If you're insuring a $200 bicycle, $100 is huge!

The Coefficient of Variation expresses the standard deviation as a percentage of the mean. It is a measure of relative risk.

Formula: $ CV = \frac{\sigma}{E[X]} $

Note: This is only used when the mean $ E[X] $ is not zero.

4. Rules for Transforming Variance

Exam P loves to ask what happens to the variance when you change the random variable (like adding a fee or applying a multiplier). These rules are different from the rules for the mean!

Rule A: Adding a Constant

$ Var(X + b) = Var(X) $
Why? If you give every student in a class 5 extra points, the average goes up, but the spread (the distance between the students) stays exactly the same. Adding a constant does not change the variance.

Rule B: Multiplying by a Constant

$ Var(aX) = a^2 \cdot Var(X) $
Common Mistake Alert! Students often forget to square the constant. If you double the size of every claim ($ 2X $), the variance increases by a factor of 4 ($ 2^2 $).

Combining them:

$ Var(aX + b) = a^2 \cdot Var(X) $

Quick Example:

If $ Var(X) = 10 $, what is $ Var(3X + 5) $?
1. Ignore the $ +5 $ (it doesn't affect spread).
2. Square the $ 3 $, which is $ 9 $.
3. $ 9 \cdot 10 = 90 $.

5. Summary and Key Takeaways

Did you know? Variance is often called the "second central moment." It is one of the most important concepts in all of actuarial science!

Key Takeaways:

Variance ($ \sigma^2 $) measures squared spread: $ E[X^2] - (E[X])^2 $.
Standard Deviation ($ \sigma $) is the square root of variance and is in the original units.
Coefficient of Variation is $ \sigma / \mu $, used for relative risk.
Constants: Adding a constant ($ +b $) changes nothing; multiplying by $ a $ scales the variance by $ a^2 $.

Common Pitfall to Avoid:
Always remember that $ (E[X])^2 $ is not the same as $ E[X^2] $. For example, if $ X $ is a fair die roll, $ E[X] = 3.5 $, so $ (E[X])^2 = 12.25 $. However, $ E[X^2] = (1^2+2^2+3^2+4^2+5^2+6^2)/6 = 15.16 $. They are very different numbers!

Keep practicing! Calculating variance becomes second nature once you've done it a dozen times. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does the Variance ($ Var(X) $) of a random variable represent?

It measures the average squared distance from the mean, quantifying the "spread" or "wiggle room" of outcomes around the center.

Why are differences from the mean squared when calculating the variance?

Squaring ensures all differences are positive so they do not cancel each other out, allowing us to measure the total magnitude of the spread.

What is the "Exam Favorite" shortcut formula for calculating Variance?

$ Var(X) = E[X^2] - (E[X])^2 $

How is the second moment ($ E[X^2] $) calculated for discrete and continuous variables?

For discrete: $ \sum x^2 \cdot P(X=x) $; for continuous: $ \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx $.

How is the Standard Deviation ($ \sigma $) related to Variance?

$ \sigma = \sqrt{Var(X)} $

What is a fundamental property regarding the sign of the standard deviation?

The standard deviation is always non-negative ($ \sigma \geq 0 $).

What is the formula for the Coefficient of Variation (CV)?

$ CV = \frac{\sigma}{E[X]} $

How does adding a constant $ b $ to a random variable affect its variance?

It does not change the variance: $ Var(X + b) = Var(X) $.

How does multiplying a random variable by a constant $ a $ affect its variance?

The variance is multiplied by the square of the constant: $ Var(aX) = a^2 \cdot Var(X) $.

What is the combined rule for the variance of a linear transformation $ Var(aX + b) $?

$ Var(aX + b) = a^2 \cdot Var(X) $

Why is it important to distinguish between $ E[X^2] $ and $ (E[X])^2 $?

They are fundamentally different values; the first is the average of squares, while the second is the square of the average.

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