Measuring Risk | Econ Academy

Two gambles can have the same expected value but very different risk — the variability of outcomes. A gamble that always pays 100 dollars (no risk) and one that pays either 0 dollars or 200 dollars with equal probability both have $E (w) = 100$ dollars, but the second is riskier. We measure risk by looking at how far outcomes deviate from the expected value, using variance and standard deviation.

Key Term

Variance and Standard Deviation

Variance: $Var (w) = E [(w - E (w))^{2}] = \sum_{i} p_{i} \times (w_{i} - E (w))^{2}$

Standard deviation: $σ = Var (w)$

Higher variance/ $σ$ means more spread in outcomes — more risk. A certain outcome has $Var = 0$ and $σ = 0$ .

Variance measures the expected squared deviation from the mean; standard deviation is its square root

Risk is the deviation from the expected payoff. In economics, we care about risk because most people are risk-averse — they prefer a certain 100 dollars to a 50/50 gamble of 0 dollars or 200 dollars, even though both have $E (w) = 100$ dollars. Understanding how to quantify risk is essential for evaluating insurance, investment decisions, and any situation with uncertain outcomes.

Worked Example

Gamble A: win 50 dollars with $p = 0.5$ or 150 dollars with $p = 0.5$ . Gamble B: win 90 dollars with $p = 0.5$ or 110 dollars with $p = 0.5$ . Both have $E (w) = 100$ dollars. Which is riskier? Calculate the standard deviation of each.

Interactive — Same E(w), Different Spread

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Two gambles with the same expected value but different spreads.

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Why do we use variance (squared deviations) rather than just average deviation?

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Measuring risk:

Risk in economics refers to:

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