The mean of a random variable, also called its expected value, is the long-run average of its values over many repetitions. Its standard deviation measures how far the values typically fall from that mean. Edit a distribution, run it thousands of times, and see how unit changes and combining variables affect the mean and standard deviation.
Each value is weighted by its probability.
Converting units or computing profit from a count is a linear transformation. The mean gets the full rule; the standard deviation is only multiplied by |b|.
Means always add or subtract. If X and Y are independent, their variances add, whether you add or subtract the variables. Standard deviations never add directly.
E(X) = Σ x·P(x) is what the average of X settles toward over many, many repetitions. It does not have to be a possible value: the expected number of heads in 3 flips is 1.5.
σX measures how far the values of X typically fall from μX, in the units of X. Find the variance first, then take the square root.
As the number of independent trials grows, the average of the observed values gets closer and closer to μ. In the short run, averages can bounce around a lot.
Adding a shifts the mean by a but leaves the spread alone. Multiplying by b multiplies the mean by b and the standard deviation by |b|. The shape does not change.
For independent X and Y, Var(X + Y) = Var(X − Y) = σX² + σY². Subtracting does not cancel variability; a difference is still uncertain in both directions.
μX and σX describe the whole probability model, so they are parameters. The mean and SD of simulated or observed data are statistics that estimate them.
How the mean and standard deviation respond to each change.
| Situation | Mean | Standard deviation | Condition |
|---|---|---|---|
| Y = X + a | μX + a | σX (no change) | None |
| Y = bX | b · μX | |b| · σX | None |
| Y = a + bX | a + b · μX | |b| · σX | None |
| X + Y | μX + μY | √(σX² + σY²) | SD rule needs X, Y independent |
| X − Y | μX − μY | √(σX² + σY²) | SD rule needs X, Y independent |
| X₁ + X₂ + X₃ (3 independent copies) | 3μX | √3 · σX | Independent |
| 3X (one value tripled) | 3μX | 3σX | None |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 2.9. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.