What you will learn
- Calculate the standard deviation as the square root of the variance
- Explain why the square root brings the measure back to normal units
- Interpret standard deviation as the typical distance from the mean
- Use standard deviation as volatility to compare the risk of two assets
Variance solved the problem of measuring spread, but it left us with an awkward number in squared units. The standard deviation is the fix. It is just the square root of the variance, which undoes the squaring and returns the measure to the same units as the data. This one small step turns variance into the main risk number in finance: volatility.
Population standard deviation (Khan Academy)
Shows standard deviation as the square root of variance and why that returns spread to the original units. Watch this first.
From variance to standard deviation
Recall that variance is the average squared distance from the mean. Because we squared the deviations, the variance came out in squared units, like percent squared, which is impossible to picture. Taking the square root reverses that squaring and brings the number back to the original units. If returns are measured in percent, the standard deviation is also in percent, so you can read it as a typical distance from the mean. That interpretability is why the standard deviation, not the variance, is the number people actually quote.
Key terms
- Standard deviation
- The square root of the variance. Spread measured in the original units.
- Volatility
- The finance word for the standard deviation of returns. The standard measure of risk.
- Typical distance from the mean
- A plain-language reading of the standard deviation.
The formula in symbols
- Variance = Σ(x − mean)² / N
- √ = square root
The formula could not be simpler: the standard deviation is just the square root of the variance. Because variance is in squared units, taking the square root lands the measure back in the original units. The worked example below applies it to the variance of 5 from the previous lesson.
From variance to standard deviation
In the last lesson the numbers 2, 4, 6, 8 had a variance of 5. What is their standard deviation?
- Start from the variance. The variance is 5 (in squared units).
- Take the square root. The square root of 5 is about 2.24.
- Read it in normal units. The standard deviation is about 2.24, the typical distance of a value from the mean of 5.
Why it matters: Standard deviation is just the square root of variance. It measures roughly how far a typical value sits from the mean, in the same units as the data.
Find the volatility
A stock's yearly returns have a variance of 64 (in percent squared). What is its standard deviation, meaning its volatility, in percent?
What the number means
The standard deviation tells you roughly how far the values typically stray from the mean. A small standard deviation means the data clusters tightly around the center, so outcomes are predictable. A large standard deviation means the data is spread widely, so any single outcome is more uncertain. In finance, a stock with an 8 percent volatility keeps most of its returns fairly close to its average, while a stock with a 40 percent volatility swings far above and below its average and feels much riskier. When people say a stock is volatile, they mean its returns have a high standard deviation.
The mean is what you expect. The standard deviation is how far reality tends to wander from it, which is the plainest definition of risk.
Standard deviation formula, with a worked example (The Organic Chemistry Tutor)
A slow, beginner-friendly worked example tying variance and its square root together. Follow along with pen and paper.
Comparing risk with volatility
Here is where standard deviation earns its place in finance. Two stocks can have the exact same average return while having very different volatilities. Suppose Stock A and Stock B both average 8 percent per year. But Stock A has a volatility of 10 percent, so its returns mostly land between roughly negative 2 and 18 percent, while Stock B has a volatility of 30 percent, so its returns swing between roughly negative 22 and 38 percent. Same expected reward, very different risk. Standard deviation is what lets you see that difference in a single number, and it becomes one of the two axes, return and risk, on which every investment is plotted in portfolio theory.
Same return, which is riskier?
Stock A and Stock B both have an average return of 8 percent. Stock A has a standard deviation of 5 percent, and Stock B has a standard deviation of 25 percent. Which stock is riskier, and why?
Both average 8 percent, but Stock B's 25 percent standard deviation means its returns swing far more widely from that average than Stock A's 5 percent. Standard deviation captures the risk difference the average hides.Why take the square root?
Explain in a sentence or two why we bother taking the square root of the variance to get the standard deviation, instead of just using the variance directly.
Write an answer before comparing it with the model response.
Model answer
Variance is in squared units, like percent squared, which has no intuitive meaning, so it is hard to interpret or compare directly. Taking the square root undoes the squaring and returns the measure to the original units, like percent, so the standard deviation can be read as a typical distance from the mean. That makes it usable as a real-world measure of volatility and risk.
You now have the two core summaries of a single asset: the mean, or expected return, and the standard deviation, or volatility. The next big step is to describe how two assets move together, which is the foundation of diversification. That starts with covariance.