The spread of a random variable

What you will be able to do

• find the variance and the standard deviation of a random variable from its table;
• use the shortcut $E(X^2) - \mu^2$;
• say which of the $n$ and $n - 1$ formulas a distribution uses, and why a simulated sample comes close.

Two bus routes

Two made-up bus routes go to the same school. The delay of each bus, in minutes, follows these tables.

Route A:

Route B:

Both have a mean delay of $3$ minutes. Check route B: $0 + 0.4 + 1.8 + 0.8 = 3$. Yet route A is always within a minute of $3$, while route B is often on time and sometimes $8$ minutes late. If you must not be late, the mean does not tell you enough.

Variance and standard deviation

For data, you measured spread by the distance of each value from the mean. A random variable works the same way, with each squared distance weighted by its probability. The variance is

The standard deviation is its square root, $\sigma$. It has the same unit as $X$. The variance has the unit squared.

For route A: $(2 - 3)^2 \times 0.25 + 0 + (4 - 3)^2 \times 0.25 = 0.5\ \text{min}^2$. So $\sigma \approx 0.707\ \text{min}$.

For route B: $9 \times 0.4 + 1 \times 0.2 + 9 \times 0.3 + 25 \times 0.1 = 9\ \text{min}^2$. So $\sigma = 3\ \text{min}$.

Route B spreads about four times as widely. That is the number a careful traveller wants.

A shortcut

Expanding the squares gives an equal formula that is often quicker:

For route B, $E(X^2) = 0 + 4 \times 0.2 + 36 \times 0.3 + 64 \times 0.1 = 18$. So $\sigma^2 = 18 - 3^2 = 9$, as before.

The $n$ formula or the $n - 1$ formula?

In the chapter on describing distributions you met two standard deviations of data. One divides by $n$ (the population formula). The other divides by $n - 1$ (the sample formula). The $\sigma$ of a random variable is the population kind: the probabilities already add up to $1$, so nothing more is divided.

A long simulation is a sample. Its standard deviation, by either formula, comes close to $\sigma$. With thousands of values the two formulas give almost the same number.

You should find that the standard deviation grows in step with $d$, but only with the square root of $w$. Doubling $d$ doubles $\sigma$. Doubling $w$ multiplies $\sigma$ by about $1.41$.

Changing the units

What if every bus on route B ran $5$ minutes later? Every value moves up by $5\ \text{min}$, so the mean becomes $8\ \text{min}$. The distances from the mean do not change, so $\sigma$ stays $3\ \text{min}$. What if we measure in seconds? Every value is multiplied by $60$. The mean becomes $180\ \text{s}$ and $\sigma$ becomes $180\ \text{s}$ too.

Try it in Python

The program finds the mean, the variance and the standard deviation of both routes with exact fractions. Then it simulates $5000$ days of route B and measures the spread of that sample both ways.

Your turn

The usual mistakes

• Forgetting to weight by the probabilities. Each squared distance is multiplied by its probability.
• Stopping at the variance. The standard deviation is the square root, and it has the unit of $X$.
• Writing $E(X^2)$ as $\mu^2$. They are different. Their difference is the variance.
• Thinking that adding a constant changes the spread. It moves every value together, so $\sigma$ stays the same.

Remember these

• $\sigma^2 = \sum (x - \mu)^2\,P(X = x) = E(X^2) - \mu^2$.
• $\sigma$ is in the unit of $X$, and $\sigma^2$ is in that unit squared.
• A distribution's $\sigma$ is the population kind. A long simulation's sample $\sigma$ comes close to it.
• Adding a constant leaves $\sigma$ alone. Multiplying every value by $a$ multiplies $\sigma$ by $\lvert a \rvert$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Route A and route B have the same mean delay. Which is more reliable, and which number shows it?

2. $X$ is $0$ or $2$, each with probability $0.5$. Find the mean, the variance and the standard deviation.

3. $X$ is a mass in grams. What is the unit of its variance?

4. Every delay on route A goes up by $4$ minutes. What happens to the mean and to the standard deviation?

5. Can a standard deviation be negative? When is it zero?

References

• OpenStax, Introductory Statistics, Chapter 4, section on the mean or expected value and the standard deviation.