Probability, Statistics, and Uncertainty

Start with the random variable

Before using any probability formula, name the random variable. Is it a count of defects, a continuous measurement, a yes-or-no event, or a value predicted from a trend line? That decision usually selects the model. Counts in a fixed number of trials suggest binomial. Rare events over a fixed interval suggest Poisson. Measurements affected by many small independent effects often use a normal model. Uniform distributions appear when every value in an interval is equally likely.

Descriptive statistics under pressure

Mean is the balance point of the data. Median is the middle value after sorting. Standard deviation measures spread. The sample standard deviation divides squared deviations by n - 1, while the population standard deviation divides by n. If an FE question specifies sample data, use the sample version unless the stem clearly says the entire population is given.

Do not compute more than the question asks. If a set is symmetric, the mean and median may be identified without full arithmetic. If answer choices are far apart, a rough estimate can eliminate options before you use the calculator.

Distributions and expected value

Expected value is the weighted average of possible outcomes. For a discrete table, multiply each outcome by its probability and add. For a binomial random variable, the expected count is np. For a Poisson random variable, both mean and variance equal lambda. These are fast points when the stem asks for expected defects, expected failures, or average arrivals.

Normal problems often use a z score: z = (x - mean)/standard deviation. If the value is one standard deviation from the mean, the empirical rule may be enough. Roughly 68% lies within plus or minus one standard deviation, 95% within two, and 99.7% within three. If the problem gives a z table or asks for a percentile, use the table rather than a memorized shortcut.

Uncertainty and measurement language

Uncertainty questions connect statistics to instrumentation. Accuracy means closeness to the true value. Precision means repeatability. Bias is systematic offset. Resolution is the smallest change a device displays. A sensor can be precise but inaccurate if it repeats the wrong value.

For independent additive quantities, absolute uncertainties combine by root-sum-square: sigma_z = sqrt(sigma_x^2 + sigma_y^2). For products and quotients, relative uncertainties combine by root-sum-square. This is why multiplying two measurements with 3% and 4% independent uncertainty gives about 5% relative uncertainty, not 7%, when random errors are assumed.

For FE Mechanical, tie uncertainty back to decisions. A test result near an allowable limit may need margin. A regression trend should not be extrapolated far outside the data. A measurement with coarse resolution may not justify a highly precise reported answer. The statistically correct answer is often the one that respects what the data can and cannot support.