Is Variance and Standard Deviation the Same? Key Differences

Understanding the Fundamental Difference

If you've ever worked with probability theory or expected value calculations, you've likely encountered both variance and standard deviation. While these two statistical measures are intimately related, they are definitely not the same thing—and understanding their differences is crucial for anyone serious about game theory, poker strategy, or mathematical decision-making.

The formula is easy: it is the square root of the Variance. That's the core relationship in a nutshell. Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both metrics measure how spread out data points are from their average, but they do so in fundamentally different ways that matter for practical applications.

What Variance Actually Measures

The average of the squared differences from the Mean. That's what variance represents. When calculating variance, you take each data point, subtract the mean, square that difference, and then average all those squared values. This squaring process is what makes variance so powerful mathematically, but it also creates a practical problem: Variance is expressed in much larger units (e.g., meters squared).

In probability theory and expected value calculations, variance plays a critical role. Plays a fundamental role in statistical tests like ANOVA and regression analysis. The squared nature of variance makes it especially useful for mathematical operations where you need to combine multiple sources of uncertainty or perform algebraic manipulations.

Why We Square the Differences

You might wonder why we bother squaring the differences at all. The answer is elegant: squaring eliminates negative values and gives more weight to outliers. Variance gives more weight to outliers due to the squaring of differences. This sensitivity makes variance particularly valuable when you need to identify extreme variations in your data—whether that's analyzing quality control in manufacturing or calculating risk in financial portfolios.

Standard Deviation: The Practical Metric

Here's where standard deviation becomes invaluable: Standard deviation is expressed in the same units as the original values (e.g., minutes or meters). This makes it far easier to interpret and communicate. If you're measuring poker results in dollars, your standard deviation will also be in dollars—not "dollars squared," which would be meaningless.

Standard deviation is often easier to explain to non-technical stakeholders. In game theory applications, this clarity is essential. When you're calculating expected value for a poker decision or evaluating risk in a strategic game, you need metrics you can actually understand and act upon.

The Empirical Rule in Action

Standard deviation gives us the famous "68-95-99.7 rule." Around 68% of values are within 1 standard deviation of the mean. Around 95% of values are within 2 standard deviations of the mean. Around 99.7% of values are within 3 standard deviations of the mean. This practical framework helps you understand the probability of different outcomes in any scenario involving uncertainty.

Expected Value Calculations and Variance

When calculating expected value—the weighted average of all possible outcomes—variance and standard deviation tell you how much uncertainty surrounds that expectation. For any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value.

Consider a simple dice roll example. When you roll a die, each outcome (1 through 6) has an equal 1/6 chance. Therefore, the expected value of rolling a die is 3.5. When you roll a die many times, the average will converge on this value. But individual rolls will vary—that's where understanding variance becomes critical.

Real-World EV Application

Let's look at a practical example. Imagine a game show where you have a 0.5 chance to win $100, a 0.4 chance to win $500, and a 0.1 probability to lose $100 (negative because you lose money). E(X) = 100 * 0.5 + 500 * 0.4 – 100 * 0.1 = $240. Hence, the expected value for that game is $240. Knowing your expected value is $240 tells you the long-term average, but variance and standard deviation tell you how wild the short-term swings might be.

Game Theory and Poker Applications

In poker and other strategic games, understanding the relationship between variance and standard deviation becomes essential for bankroll management and strategic planning. The truth is that variance in poker is far beyond what humans are capable of truly conceptualizing. If you've ever played around with a poker variance calculator, you'd find that it takes tens of thousands of hands before you can see a statistically significant edge materialize.

Standard deviation in poker is a value expressed in bb/100 (i.e. a winrate) that helps us understand how "swingy" our poker game is. A typical value for standard deviation is usually around 80bb/100. This metric helps players understand whether their current results reflect skill or simply short-term luck.

Managing Strategic Uncertainty

Such deviation from the real result and the expected value is called variance. In other words, the variance is a good or a bad luck from the math standpoint. For strategic decision-making in games, understanding both metrics helps you separate skill from luck and make better long-term choices.

Which Metric Should You Use?

The choice between variance and standard deviation depends on your specific application. Standard deviation is expressed in the same units as your data, making it more intuitive for direct interpretation. Use standard deviation when you need to communicate results or make practical decisions about risk.

However, variance is important in statistical tests. When performing formal mathematical analysis, combining multiple probability distributions, or conducting hypothesis testing, variance is often the required metric.

Key Takeaways for Practical Application

Remember these essential points: variance and standard deviation are related but distinct measures. Variance is equal to the average squared deviations from the mean, while standard deviation is the number's square root. Also, the standard deviation is a square root of variance. Both tell you about spread and uncertainty, but standard deviation does so in units you can actually understand and apply.

For anyone working with expected value calculations—whether in poker strategy, game theory applications, or general probability problems—mastering both concepts gives you a more complete picture of risk and uncertainty. Standard deviation helps you understand the practical range of outcomes, while variance provides the mathematical foundation for more advanced statistical analysis.

The bottom line? While they're mathematically linked, variance and standard deviation serve different purposes. Use standard deviation for intuitive understanding and communication. Use variance for mathematical operations and formal statistical tests. Master both, and you'll have a powerful toolkit for any scenario involving probability, expected value, or strategic decision-making under uncertainty.