Expected Value, Variance, and Standard Deviation

Updated for the 2026-2027 CFA® Level I curriculum.

An investment may produce several possible returns, each with a different probability. Expected value summarizes the return at the center of that distribution, while variance and standard deviation show how widely the possible results are spread around it.

For CFA Level I, you should be comfortable moving through the full calculation in order: expected return first, variance second, and standard deviation last. You should also be able to interpret the three measures together in an investment problem.

Key Takeaways About Expected Value and Variance

• The probabilities of all possible outcomes must sum to .

• Expected value is calculated before variance because each deviation is measured from it.

• Variance uses probability-weighted squared deviations.

• Squaring prevents positive and negative deviations from cancelling each other out.

• Standard deviation is the square root of variance.

• Expected return and standard deviation should be interpreted together.

• A higher standard deviation indicates greater variability, but it does not automatically make an investment unattractive.

• Expected return represents an average across possible outcomes rather than a guaranteed result.

• The alternative variance formula can reduce the amount of arithmetic required.

What You Need to Know for CFA Level I

For CFA Level I, focus on both the sequence of the calculation and the meaning of the result.

You should be able to:

• Confirm that the outcome probabilities form a valid probability distribution.

• Calculate the expected value of a discrete random variable.

• Calculate variance using probability-weighted squared deviations.

• Apply the alternative variance formula.

• Convert variance into standard deviation.

• Keep return units consistent throughout the calculation.

• Compare investments with different expected returns and standard deviations.

• Explain why expected return is not necessarily the most likely outcome.

• Recognize whether an answer choice reports expected value, variance, or standard deviation.

What Is Expected Value?

Expected value is the probability-weighted average of every possible outcome of a random variable.

In an investment problem, the random variable is often a return. Each possible return is multiplied by its probability, and the resulting products are added together.

Suppose an asset can earn 15%, 6%, or . A simple average would treat those outcomes as equally likely. Expected value incorporates the actual probability assigned to each one.

The result describes the long-run average outcome implied by the probability distribution. It may not be one of the returns that can actually occur.

For example, an expected return of 7.5% does not mean:

• The investment will earn 7.5%.

• A 7.5% return is the most likely outcome.

• The investor is guaranteed to earn 7.5%.

It means the probability-weighted center of the possible returns is 7.5%.

What Is Variance?

Variance measures how widely possible outcomes are distributed around the expected value.

The calculation begins with the difference between each possible return and the expected return. These deviations are then squared and weighted by their respective probabilities.

Squaring the deviations has two effects:

1. Positive and negative deviations cannot cancel each other out.

2. Outcomes farther from the expected value contribute more heavily to the result.

Variance is expressed in squared units. When returns are entered as decimals, the variance is expressed in decimal units squared. This makes variance useful in formulas but less intuitive to interpret directly.

A larger variance indicates that the possible outcomes are spread more widely around the expected value. A smaller variance indicates that they are clustered more closely around it.

What Is Standard Deviation?

Standard deviation is the square root of variance.

Taking the square root returns the result to the same units as the original observations. If returns are expressed as percentages, standard deviation can also be reported as a percentage.

This makes standard deviation easier to compare with expected return. For example:

• Expected return: 8%

• Standard deviation: 4%

Both figures are expressed in return terms, so the analyst can evaluate the expected reward alongside the variability surrounding it.

Standard deviation measures variability in both directions. An unusually high positive return increases standard deviation just as an unusually negative return does.

It therefore measures total variability rather than downside risk alone.

Expected Value vs Variance vs Standard Deviation

These measures are designed to be read together. Expected value describes the return being considered, while standard deviation adds context about the uncertainty surrounding it.

How to Calculate Expected Value, Variance, and Standard Deviation

Use the following sequence in a discrete investment-return problem.

Step 1: Check the Probabilities

The probabilities of all possible outcomes should sum to .

Step 2: Calculate the Expected Value

Multiply each possible outcome by its probability and add the results.

Step 3: Calculate Each Deviation

Subtract the expected value from every possible outcome.

Step 4: Square the Deviations

Squaring removes the signs and gives greater weight to larger deviations.

Step 5: Weight the Squared Deviations

Multiply each squared deviation by the probability of its outcome.

Step 6: Add the Weighted Terms

The sum is the variance.

Step 7: Take the Square Root

The square root of variance is the standard deviation.

Expected Value, Variance, and Standard Deviation Formulas

The following blocks can be pasted directly into Notion block equations.

Expected Value Formula

Where:

• = expected value of random variable

• = possible outcome

• = probability of outcome

• = number of possible outcomes

The probability notation may also be written as :

Variance Formula

Where:

• = variance of

• = deviation of outcome from the expected value

• = probability assigned to outcome

Alternative Variance Formula

The expected value of the squared outcomes is:

The direct and alternative variance formulas produce the same result. The alternative formula may be quicker when the squared outcomes are easy to calculate.

Standard Deviation Formula

Standard deviation may also be written using the symbol :

Keep the Units Consistent

Convert percentage returns into decimals before beginning the calculation.

For example:

• 20% becomes 0.20

• 9% becomes 0.09

• 8% becomes 0.08

Keep the returns in decimal form through the variance calculation. Convert the final standard deviation back into a percentage for interpretation.

Worked Investment Example

An asset has three possible economic outcomes next year:

The probabilities form a valid distribution:

Step 1: Calculate the Expected Return

Convert the returns to decimals and weight each one by its probability:

The asset’s expected return is 7.50%.

Step 2: Calculate the Deviations

Subtract the expected return of 0.075 from each possible return:

Step 3: Calculate the Variance

Square each deviation, multiply it by its probability, and add the results:

Step 4: Check the Variance With the Alternative Formula

First, calculate :

Then subtract the squared expected return:

Both methods give the same variance.

Step 5: Calculate the Standard Deviation

The asset has:

• An expected return of 7.50%

• A standard deviation of approximately 10.01%

The standard deviation is larger than the expected return, which indicates a wide spread of possible outcomes around the probability-weighted average.

This result does not determine whether the investment is suitable. An investor would consider the expected return and variability alongside risk tolerance, objectives, and available alternatives.

How to Compare Expected Return and Standard Deviation

Expected return and standard deviation provide different parts of the investment picture.

Consider two investments:

Investment B offers the higher expected return, but its possible outcomes are spread much more widely around that expectation.

Investment A offers a lower expected return with more stable possible outcomes.

The figures alone do not identify the better investment. The decision depends on how the investor values additional expected return relative to the additional variability.

For comparisons that explicitly measure risk relative to expected return, you may also encounter the coefficient of variation in a separate dispersion context.

Common Exam Traps

• Using equal weights instead of the stated probabilities.

• Beginning the variance calculation before finding the expected value.

• Failing to confirm that the probabilities sum to .

• Forgetting to square the deviations.

• Squaring the probabilities instead of the deviations.

• Taking the square root before the variance calculation is complete.

• Mixing percentages and decimals in the same calculation.

• Reporting variance when the question asks for standard deviation.

• Reporting the expected return when the question asks for a risk measure.

• Assuming expected return is the most likely outcome.

• Interpreting a higher standard deviation as automatically making an investment worse.

• Using a historical sample-variance denominator such as in a discrete probability distribution problem.

Practice Question

A stock has the following possible returns next year:

What is the standard deviation of the stock’s return?

1. 6.20%

2. 7.88%

3. 62.16%

FAQs About Expected Value and Variance