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Lesson 2 of 14 · 14 min

Expected value, variance and standard deviation

Expected value is your probability-weighted forecast; variance and standard deviation measure how far the actual outcome is likely to land from that forecast.

Video lesson · 7 min. Watch it first, then use the text below to go deeper.

In short

The expected value $E(X)$ multiplies each possible outcome by its probability and adds the products.
It is a forward-looking forecast (or the true population mean), unlike a sample mean, which equally weights past observations.
Variance is the probability-weighted average of squared deviations from $E(X)$ . It is never negative and is zero only when the outcome is certain.
Standard deviation is the positive square root of variance and is in the same units as the variable, so it is easier to interpret.
Always work in the same order: expected value, then variance, then standard deviation.

Expected value: the probability-weighted forecast

A random variable is a quantity whose outcome is uncertain: next year's EPS, a bond's recovery rate, a portfolio's return. When you can list the possible outcomes and attach a probability to each, you have a discrete probability distribution.

The expected value of a random variable X, written $E(X)$ , is the probability-weighted average of its possible outcomes. Multiply each outcome by its probability and add up the results. The more likely an outcome, the harder it pulls the forecast towards itself.

For this to work, the outcomes must be mutually exclusive (only one can happen) and exhaustive (one of them must happen). Their probabilities therefore sum to exactly 1. Checking that sum is the first thing to do with any distribution an exam question gives you.

The expected value does not have to be an outcome that can actually occur. A project that ends with an NPV of NOK 40m, NOK 10m or −NOK 30m can have an expected NPV of NOK 11m, even though NOK 11m is never the realised result. Think of it as the average you would get if the same situation could be repeated many times.

The Swedish bank's dividend distribution. The bars sum to 1, and weighting each outcome by its bar height gives an expected dividend of SEK 6.75, a value the dividend itself can never take.

Investors use expected values everywhere: to forecast EPS and other financial variables, to compare the rewards of competing investments, and to value assets. Whenever a question gives you probabilities and asks for a forecast or an expected figure, it wants a probability-weighted average.

Expected value versus the sample mean

Both condense a random variable into one central number, but they answer different questions. The expected value looks forward: it is a forecast, or the true mean of the whole population. A sample mean looks back: it is the average of a particular set of observations you have already collected.

The weights differ too. An expected value weights each outcome by its probability. A sample mean gives every observation the same weight, $1/n$ . An unlikely scenario gets little weight in an expected value, even though a single past occurrence of it counts fully in a historical average.

Forecast versus history
	Expected value	Sample mean
Looks	Forward (a forecast)	Back (historical data)
Describes	The population, or 'true', mean	One particular set of observations
Weights	Probability of each outcome	Equal weight $1/n$ per observation

Variance and standard deviation: risk around the forecast

A forecast alone is not enough, because the actual result will almost never equal it exactly. You also need to know how widely the possible outcomes are scattered around the forecast. That scatter is called dispersion, and for an investor it is a measure of risk.

Variance is the expected value of the squared deviation from the mean: $\sigma^2(X) = E\{[X - E(X)]^2\}$ , also written $\text{Var}(X)$ . In practice you take each outcome's distance from $E(X)$ , square it, weight it by that outcome's probability and add everything up.

Squaring does two jobs. It stops positive and negative deviations from cancelling out, and it penalises large misses more than small ones. Because variance is a sum of probability-weighted squares, it can never be negative.

A variance of zero means there is no dispersion: the outcome is certain and X is not really random at all. Any variance above zero means the outcomes are spread out, and a larger variance means more spread, all else equal.

Standard deviation $\sigma(X)$ is the positive square root of variance. Variance is in squared units (a return in percent gives a variance in percent squared), which is awkward to interpret. Standard deviation brings the measure back to the variable's own units, so a standard deviation of 6% can be set directly beside an expected return of 7%.

Step 1: compute $E(X)$ .
Step 2: for each outcome, find the deviation $X_i - E(X)$ and square it.
Step 3: multiply each squared deviation by its probability and add them up. This is $\sigma^2(X)$ .
Step 4: take the square root to get $\sigma(X)$ .

Putting the numbers to work

Expected value and standard deviation together describe the trade-off investors care about: reward and risk. When two investments offer about the same expected return, the one with the lower standard deviation delivers that forecast with less uncertainty, so a risk-averse investor prefers it.

Standard deviation also tells you how much faith to put in a forecast. An expected EPS of EUR 2.50 with a standard deviation of EUR 0.05 is a tight forecast. The same expected EPS with a standard deviation of EUR 0.60 is a rough guess.

On the BA II Plus, the data worksheet does the whole calculation. Enter each outcome as X and its probability, written as a whole-number percentage, as the frequency Y. Then read the population standard deviation $\sigma_x$ , not the sample figure $S_x$ .

Fund A (E = 6.8%) and Fund B (E = 6.7%) have almost the same forecast, but A's outcomes are spread from −2% to 14% (σ = 6.27%) while B's stay between 4% and 9% (σ = 1.95%).

Why the population figure? The n − 1 in a sample standard deviation corrects for estimating the mean from data. A probability distribution is already the full model of the outcomes, so there is nothing to correct.

Formulas

Expected value: $E(X) = P(X_1)X_1 + P(X_2)X_2 + \dots + P(X_n)X_n = \sum_{i=1}^{n} P(X_i)\,X_i$
$X_i$ is one of the n possible outcomes and $P(X_i)$ its probability. The probabilities must sum to 1.
Variance: $\sigma^2(X) = E\{[X - E(X)]^2\} = \sum_{i=1}^{n} P(X_i)\,[X_i - E(X)]^2$
Probability-weighted average of squared deviations from the expected value. Units are the variable's units squared.
Standard deviation: $\sigma(X) = \sqrt{\sigma^2(X)}$
The positive square root of variance, in the same units as X.

Worked examples

Example 1 Expected dividend and its dispersion

A Swedish bank will pay a dividend per share of SEK 8.00 (probability 0.20), SEK 7.00 (0.50) or SEK 5.50 (0.30). What are the expected dividend, its variance and its standard deviation?

Example 2 Same expected return, very different risk

Two EUR funds face the same scenarios: boom (probability 0.30), normal (0.40) and slump (0.30). Fund A returns 14%, 8% and −2%; Fund B returns 9%, 7% and 4%. Which fund should a risk-averse investor prefer?

Example 3 Finding a missing probability, then the risk

A Norwegian oil project ends with an NPV of NOK 40m, NOK 10m or −NOK 30m. The middle outcome has probability 0.50 and the expected NPV is NOK 11m. What are the other two probabilities, and what is the standard deviation of NPV?

On the exam

Expect a direct calculation of expected value, variance or standard deviation from a small probability table, with distractors that use a simple average or stop at the variance. Concept items ask what a zero variance means, why standard deviation is easier to interpret, or how an expected value differs from a sample mean.

Taking a simple average of the outcomes instead of weighting each by its probability.
Reporting the variance when the question asks for the standard deviation, or the reverse.
Using $S_x$ instead of $\sigma_x$ on the calculator; a probability distribution needs the population figure.
Measuring deviations from zero or from the most likely outcome instead of from $E(X)$ .

Check yourself

Q1A UK insurer's EPS will be GBP 3.00, 2.40 or 1.80 with probabilities 0.2, 0.5 and 0.3. Expected EPS is closest to:
Q2A stock returns 10% with probability 0.6 and −5% with probability 0.4. The standard deviation of its return is closest to:
Q3Which statement about a random variable whose variance equals zero is most accurate?
Q4Compared with a sample mean, an expected value is best described as:
Q5Investments X and Y both have an expected return of 7%. X has a standard deviation of 4% and Y of 9%. Which statement is most accurate?

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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