L3-07 · Expectation and variance
Calculate a distribution's mean and variance without treating the mean as a certain outcome.
Level 3 · High-school bridge
Objective. Calculate a distribution’s mean and variance without treating the mean as a certain outcome.
Before this lesson. L2-04
Hands-on lab. Chance and repeated trials
Learn¶
A probability distribution assigns probabilities to possible outcomes. Its expectation is the probability-weighted average. For a finite set, multiply each outcome by its probability, then add. The probabilities must be nonnegative and sum to one. A mean can differ from every individual outcome.
Variance measures the average squared distance from the expectation. Squaring prevents positive and negative deviations from canceling. The standard deviation is its square root. Variance has squared outcome units. The standard deviation has the same units as the original measured outcome.
Repeated observations help estimate these quantities. Independent observations have a simpler error calculation than dependent observations. If measurements share drift or another cause, more samples do not necessarily reduce error as quickly as the independent formula predicts. Record the sampling assumptions before reporting uncertainty.
Practice¶
Solution
The expectation is (-1)(1/2) + (1)(1/2) = 0. Both squared outcomes equal one. Thus E[X²] = 1 and the variance is 1 - 0² = 1.
Check your understanding¶
Which quantity always uses squared outcome units?
A. The expectation
B. The standard deviation
C. The variance
Show the answer
C. The variance
Variance averages squared deviations. Taking its square root produces standard deviation, which returns to the original outcome units.
Sources¶
This lesson draws on Illowsky & Dean, 2023.
Next: L3-08 · Entropy and uncertainty
- Illowsky, B., & Dean, S. (2023). Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction