Variance and standard deviation · RiskAtoms Academy

By the end you will be able to

1Explain spread around the mean, and why deviations are squared before averaging.
2Calculate the variance as the probability-weighted average squared deviation, and the standard deviation as its square root.
3Use the scaling rules: Var(aX + b) = a²Var(X) and sd(aX) = |a| sd(X).
4Say when to divide by N (a full list of outcomes) and when by n − 1 (estimating from data).

Why you need it

Two investments can both average 1% a month. One barely moves; the other swings from −4% to +6%. The average cannot tell them apart. The standard deviation can, and in finance it has its own name: volatility.

These Core Path lessons use it:

L2.1.03: Risk and return
Risk is measured as the standard deviation of returns, set against the expected return.
L2.5.01: Measuring volatility
Volatility is the standard deviation of returns, estimated from data with n − 1 and scaled up to a year.

It builds on the mean from M-C02: Expectation, and leads to M-C06: The normal distribution, whose bell curve is set by a mean and a standard deviation.

The idea

How far from the average?

Here are six months of two illustrative return series. Both average 1% a month. But one is a gentle ride and the other a rollercoaster.

Steadymean 1% · sd 0.41%

Bumpymean 1% · sd 3.83%

Six months each, in per cent. Dashed line: the mean. Each stick is a deviation from the mean. Illustrative figures.

The difference is how far each month lands from the average: the sticks. Short sticks, low spread. Long sticks, high spread. To put one number on spread, we want a kind of average stick length. Each stick is a deviation, and the way we average them gives the variance and the standard deviation: about 0.41% for the steady series and 3.83% for the bumpy one.

The rules

We call the mean $\mu$ (the Greek letter “mu”), as in M-C02: Expectation, and use the outcomes $x_i$ and probabilities $p_i$ from there.

Scaling. What happens to the spread if you double everything, or add a fixed amount?

The rules, for reference

\operatorname{Var}(X) = \sigma^2 = \sum_i p_i\,(x_i - \mu)^2 \qquad \sigma = \sqrt{\operatorname{Var}(X)}

\operatorname{Var}(aX + b) = a^2\operatorname{Var}(X) \qquad \operatorname{sd}(aX + b) = |a|\,\operatorname{sd}(X)

A list of N equally likely values (the population formula): $\sigma^2 = \tfrac1N\sum (x_i-\mu)^2$ . Estimating from a sample of n observations: divide by n − 1 instead. See M-D01: Estimating mean and volatility.

Another route to the same number

Expanding the square gives a shortcut that is handy with a calculator: the mean of the squares minus the square of the mean.

\operatorname{Var}(X) = \mathbb{E}[X^2] - \mu^2 \qquad \text{die: } 15.1667 - 3.5^2 = 2.9167

Try it

Five points, each equally likely. Drag them and watch the squares. Things to try: load “One outlier” and see how much of the variance one far-away point supplies. Move a point twice as far from the mean: its square gets four times bigger. Then set a to 2, or to −2, and compare the spread.

Try itSpread explorer

Open the statistics calculator →

Load:

Drag a point along the line, or type its value (0 to 20).

Each coloured square has a side equal to one point's distance from the mean, so its area is that point's squared deviation. The variance is the average area. The dashed square has exactly that area: its side is the standard deviation σ.

Point x	x − mean	(x − mean)²	Share
	−3	9	18%
	−2	4	8%
	−1	1	2%
	0	0	0%
	6	36	72%
Total	0	50

Standard deviation σ

3.162

Mean μ11

Variance σ² (÷ 5)10

Biggest share72%

Now scale every point by a and shift it by b. The spread follows a; b only slides the picture.

Scale a

Shift b0

\operatorname{Var}(2X + 0) = 40 = (2)^2 \times 10 \qquad \operatorname{sd} = 6.325 = 2 \times 3.162

Worked examples

1. Variance of a two-outcome return

Step 1 of 4

1
The set-up
Next year an investment returns 10% with probability 0.6, or −5% with probability 0.4. Work in per cent throughout.

2. Five months of data: N or n − 1?

Step 1 of 4

1
The set-up
A fund returned 2%, −1%, 3%, 0%, 1% in five months. First treat these five as the whole list, each equally likely.

3. From per cent to euros

Step 1 of 4

1
The set-up
A holding worth €200,000 has a daily return X with a standard deviation of 1.5%. The day’s profit or loss in euros is €200,000 × X. How spread out is it?

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

Why are the deviations squared before they are averaged?

ATo make the answer biggerBBecause the plain deviations always average to zero, and squaring makes every one count as positiveCBecause squares are easier to addDTo get the answer in the same units as the data

Question 2 · objective 2

X is 0 or 10, each with probability 0.5. What is Var(X)?

Question 3 · objective 3

Var(X) = 4. What is Var(3X + 5)?

Question 4 · objective 3

A return X has a standard deviation of 2%. What is the standard deviation of −5X, in %?

%

Question 5 · objective 4

You have 60 monthly returns of a fund and want to estimate how volatile the fund is. What do you divide the sum of squared deviations by?

A60B59C61D60²

Glossary

Deviation: How far an outcome is from the mean: x − μ. Positive above the mean, negative below.
Variance: The probability-weighted average of the squared deviations from the mean: Var(X) = Σ pᵢ(xᵢ − μ)², also written σ². Its units are the data's units squared.
Standard deviation: The square root of the variance, σ. In the same units as the data; read it as a typical distance from the mean.
Population formula: Treating a list as the whole set of equally likely outcomes, so each has probability 1/N and you divide by N.
Sample (n − 1) formula: Treating a list as a few observations from a bigger process. To estimate that process's spread you divide by n − 1.
Volatility: In finance, the standard deviation of an asset's returns. Estimated from data with n − 1.