Variance vs Standard Deviation: What's the Difference?
If you've ever calculated expected values in probability or analyzed outcomes in game theory, you've likely encountered both variance and standard deviation. While these two statistical measures are closely related—in fact, one is literally the square root of the other—understanding when and why to use each one can dramatically improve your analytical skills.
The Mathematical Relationship
Let's start with the fundamentals. In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation is obtained as the square root of the variance.
Think of it this way: if you're measuring how spread out your data is, variance gives you that measurement in squared units, while standard deviation converts it back to the original units of measurement. Standard deviation is measured in a unit similar to the units of the mean of data, whereas the variance is measured in squared units.
The formula is straightforward: Standard Deviation = √(Variance), or conversely, Variance = (Standard Deviation)².
Why the Difference Matters in Expected Value Calculations
When you're calculating expected values in probability theory, both measures serve distinct purposes. The variance of a random variable X, with mean EX=μ_X, is defined as the average value of (X-μ_X)². This squared deviation makes variance particularly useful for certain mathematical operations.
An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. This property becomes invaluable when you're working with complex probability distributions or multiple random variables simultaneously.
However, there's a practical limitation. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion. If you're measuring outcomes in dollars, variance will be in dollars squared—not exactly intuitive for interpretation.
Applications in Game Theory
In game theory, understanding variance and standard deviation becomes critical when analyzing strategies and outcomes. The expected value of a game of chance is the average net gain or loss that we would expect per game if we played the game many times. But expected value alone doesn't tell the whole story—you need to understand the variability of outcomes.
Consider a simple gambling scenario: Two games might have the same expected value of $25, but vastly different risk profiles. Game A might consistently pay out amounts close to $25, while Game B might alternate between $0 and $100. The variance and standard deviation reveal this crucial difference in risk.
Expected value is important in game theory because it allows players to make informed decisions based on the likelihood of different outcomes and the potential gains or losses associated with each outcome. Adding variance analysis to your toolkit helps you understand not just the average outcome, but how volatile those outcomes might be.
The Poker Player's Perspective
Professional poker players understand this distinction intuitively. In poker, players use expected value to determine whether to call or fold based on the potential outcomes of each move. But they also need to consider variance—the natural fluctuations in results over time.
A high-variance playing style might have a positive expected value but require a larger bankroll to weather the inevitable losing streaks. The standard deviation helps quantify this risk in terms players can understand: dollars won or lost.
Practical Calculation Examples
Let's work through a concrete example. To find the expected value or long term average, μ, simply multiply each value of the random variable by its probability and add the products.
Suppose you're analyzing a carnival game where you can win $10 with probability 0.1, $2 with probability 0.4, or $0 with probability 0.5. The expected value would be: (0.1 × $10) + (0.4 × $2) + (0.5 × $0) = $1.80.
To find the variance, you'd calculate the squared deviation of each outcome from this expected value, multiply by its probability, and sum them up. The standard deviation would then be the square root of that variance, giving you a measure of typical deviation in dollars—much easier to interpret than variance in dollars-squared.
When to Use Which Measure
The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers.
Here's a practical guide:
- Use variance when: You're performing mathematical operations on multiple random variables, working with theoretical proofs, or need to combine different sources of variability
- Use standard deviation when: You need to communicate results to others, want intuitive interpretation in original units, or are comparing variability across different datasets
- Use both when: You're conducting comprehensive statistical analysis or making investment decisions where you need both mathematical precision and practical interpretation
Real-World Applications
In the financial sector, standard deviation and variance are fundamental tools for risk assessment and portfolio management. Standard deviation is widely used to measure the volatility of investment returns. A higher standard deviation indicated greater price fluctuations and, consequently, higher risk.
Manufacturers use standard deviation to monitor product consistency and establish acceptable ranges for variation. If a factory produces bolts that should be 10mm in diameter, they might specify that any bolt within 1 standard deviation of the mean is acceptable, immediately rejecting those beyond 2 standard deviations.
For more detailed information on probability theory concepts, check out the comprehensive resources at Wikipedia's variance article and ProbabilityCourse.com.
Key Takeaways
Understanding the difference between variance and standard deviation isn't just academic—it's essential for anyone working with probability, expected values, or game theory. Variance provides mathematical elegance and ease of manipulation, while standard deviation offers intuitive interpretation and practical usability.
Remember: variance squares the units of measurement, making it perfect for theoretical work but challenging for practical interpretation. Standard deviation brings those units back to earth, letting you communicate risk and variability in terms everyone can understand. Master both, and you'll have the complete toolkit for analyzing uncertain outcomes—whether you're calculating expected values, optimizing game theory strategies, or simply trying to understand the risks inherent in any probabilistic situation.
The next time you're faced with a decision under uncertainty, don't just calculate the expected value. Look at the variance and standard deviation too. They might reveal that your "safe" bet is riskier than you thought, or that a seemingly volatile option is actually more stable than it appears. That's the power of understanding these fundamental statistical measures.