Probability, Statistics, and Uncertainty

Descriptive Statistics

FE statistics starts with summarizing data. The mean x̄ = (Σxᵢ)/n is the arithmetic average; the median is the middle value (average of the two middle values for even n); the mode is the most frequent value. Spread is measured by variance and standard deviation.

For a sample, σ = √[Σ(xᵢ − x̄)²/(n − 1)]—note the n − 1 denominator (Bessel's correction). For a full population, divide by N instead. Mislabeling sample vs. population is a frequent FE error, so read whether the data set is "a sample of" or "all" items.

Worked Example — Mean and SD

Data: 2, 4, 4, 6. Mean = 16/4 = 4. Deviations: −2, 0, 0, 2; squared: 4, 0, 0, 4; sum = 8. Sample variance = 8/(4 − 1) = 2.67, so s = √2.67 ≈ 1.63. The population SD would use 8/4 = 2 → σ = 1.41.

Probability Distributions

Select the model from the problem's structure:

Binomial mean = np, variance = np(1−p). Poisson mean = variance = λ. The normal distribution is standardized with the z-score z = (x − μ)/σ, then read from a z-table.

Empirical Rule

For a normal distribution, about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. So if part lengths are normal with μ = 50 mm and σ = 2 mm, roughly 95% lie between 46 and 54 mm. This rule answers many tolerance and yield questions without a z-table.

Confidence Intervals, Regression, and Uncertainty

A confidence interval (CI) brackets a population mean from sample data. With known σ, the interval is x̄ ± z·(σ/√n), where z = 1.645 (90%), 1.96 (95%), or 2.576 (99%). The term σ/√n is the standard error—it shrinks as sample size grows, so larger samples give tighter intervals. When σ is unknown and n is small, the t-distribution replaces z.

Linear regression fits y = a + bx by least squares. FE questions focus on interpreting the slope b (change in y per unit x), the intercept a, and the correlation coefficient r (−1 ≤ r ≤ 1), where r near ±1 means a strong linear fit and r near 0 means none. Predict only within the observed x-range; extrapolation beyond the data is unreliable.

Uncertainty Propagation

Measurement questions distinguish accuracy (closeness to true value), precision (repeatability), bias (systematic offset), and resolution (smallest readable increment). When quantities add or subtract, their absolute uncertainties add: u_total = √(u₁² + u₂²). When they multiply or divide, their relative (percent) uncertainties combine: (u/Q)² = (u₁/x₁)² + (u₂/x₂)². Choosing the wrong rule—absolute when relative is needed—is the classic error, so first identify whether the measured quantities are being added or multiplied.

Basic Probability Rules and Counting

Underpinning the distributions are a few rules the FE expects you to apply directly. The probability of an event is between 0 and 1. For the union of two events, P(A or B) = P(A) + P(B) − P(A and B); the subtraction prevents double-counting the overlap. For independent events, the multiplication rule gives P(A and B) = P(A)·P(B)—for example, two independent components each 0.9 reliable give a series reliability of 0.81. The complement rule, P(not A) = 1 − P(A), is the fast way to handle "at least one" problems: P(at least one failure) = 1 − P(no failures).

Conditional probability is P(A | B) = P(A and B)/P(B). Counting uses permutations (order matters) P(n, r) = n!/(n − r)! and combinations (order does not) C(n, r) = n!/[r!(n − r)!]; the combination formula is exactly the coefficient in the binomial distribution.

Worked Example — Binomial

A machine produces 5% defective parts. In a sample of 4, the probability of exactly one defect is C(4,1)(0.05)¹(0.95)³ = 4(0.05)(0.857) = 0.171. Here n = 4 trials are fixed and p = 0.05 is constant, which is precisely when the binomial model applies.

Reading the Problem for the Right Tool

Most FE statistics errors are tool-selection errors, not arithmetic. Use these cues:

Match the cue, write the formula, and substitute. Because the FE gives only about three minutes per question, the time saved by recognizing the model immediately is often the difference between finishing the section and guessing on the last items.

Finally, keep the expected value in mind for decision and reliability items: E(X) = Σ xᵢ·P(xᵢ) is the long-run average outcome, and it is the basis for comparing risky alternatives by their mean payoff or cost.