Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A parameter describes a whole probability distribution or population. It is a single fixed value.
- A statistic is calculated from data (a sample or a simulation). It changes from sample to sample.
- The expected value (mean) of a discrete random variable is the probability-weighted average of its values.
- The standard deviation of a random variable measures the typical distance of its values from the mean, over the long run.
- The expected value does not have to be a possible value of X, and it is not the most likely value.
Key relationships
| Quantity | Symbol | How to find it | Interpretation |
|---|---|---|---|
| Mean (expected value) | μ or E(X) | Σ xᵢ · P(xᵢ) | long-run average value of X |
| Variance | σ² or V(X) | Σ (xᵢ − μ)² · P(xᵢ) | squared units; a step towards σ |
| Standard deviation | σ or SD(X) | √(variance) | typical long-run distance of X from μ |
| Check on μ | Σ (xᵢ − μ) · P(xᵢ) = 0 | deviations balance at the mean | |
| Sample mean from data | x̄ | Σ x ÷ n | a statistic that estimates μ |
Assumptions and conventions
- X is discrete: its possible values can be listed (countable), often a finite list.
- The probabilities must each be between 0 and 1 and add to 1. Check this first.
- No n − 1 and no division by a count: the probabilities do the weighting.
- On a calculator, use the probability list as frequencies and read σx, not Sx.
Mistakes to avoid
- Averaging the values without weighting by their probabilities.
- Calling E(X) “what will happen” on one trial. It is a long-run average.
- Forgetting the square root, so reporting the variance as σ.
- Using Sx or dividing by n − 1.
- Calling a simulated mean a parameter. It is a statistic; μ is the parameter.
- No context or units: name the random variable, the process (days, plays, customers) and the units.
Quick self-check
- X = 0 or 2, each with probability 0.5. Find μ and σ. (μ = 1, σ = 1)
- Y = 10 with probability 0.2 and 20 with probability 0.8 (minutes). Find μ and σ. (μ = 18 minutes; σ² = 64(0.2) + 4(0.8) = 16, so σ = 4 minutes)
- A simulation of 200 plays of a game gives a mean gain of $0.35. Is $0.35 a parameter or a statistic? (A statistic: another 200 plays would give a different value.)
Next: practice questions.