The hook
Surveying measurements live and die by the normal distribution. Random errors cluster around the mean, fall off symmetrically, and 68% land within ±1 standard deviation. Once you internalize this curve, error propagation, blunder detection, and confidence intervals all click.
Memorize these
Concepts that show up on the exam
Mean (μ, x̄)
Arithmetic average of the observations. Center of the distribution; least-squares estimate when all weights are equal.
Median
Middle value when sorted. For symmetric distributions equals the mean. More robust to outliers.
Mode
The most frequent value. For a continuous normal distribution it equals the mean; for a histogram, the tallest bar.
Standard deviation (σ, s)
Square root of the variance. The "1σ" interval contains 68% of normal observations. Use n−1 in the denominator for sample (s); n for population (σ).
Variance (σ²)
Average squared deviation from the mean. Adds linearly when independent random variables are summed (variances add, NOT standard deviations).
Confidence interval
Range expected to contain the true value with a stated probability. ±1σ ≈ 68%, ±1.96σ ≈ 95%, ±2.58σ ≈ 99%.
Hypothesis test
H₀ (null) vs. H₁ (alternative). Reject H₀ if test statistic is too extreme to have occurred by chance at chosen α (typically 0.05).
Linear regression
Fits a line ŷ = ax + b by minimizing Σ(yᵢ − ŷᵢ)². slope a = Σ((xᵢ − x̄)(yᵢ − ȳ)) / Σ(xᵢ − x̄)². intercept b = ȳ − a · x̄.
Keep these in muscle memory
Formulas to know cold
Sample mean
x̄ = (Σ xᵢ) / nSample standard deviation
s = √( Σ(xᵢ − x̄)² / (n − 1) )Use n−1 (Bessel correction) for SAMPLES of a larger population. Use n for the entire population.
Standard error of the mean
SE = s / √nThe mean of n observations is √n times more precise than a single observation.
95% confidence interval (large n)
CI = x̄ ± 1.96 · (s / √n)Try it before you peek
Worked example
The problem
A line is measured 5 times: 312.41, 312.45, 312.39, 312.43, 312.42 ft. Compute the mean, standard deviation, and the 95% confidence interval for the true line length.
Don't fall for these
What trips people up
Using σ when you should use s
Population σ uses n in the denominator; sample s uses n−1. With small n, the difference is significant. Use s unless you literally observed every member of the population.
Confusing σ and σ_mean
σ is the std dev of a single observation; σ/√n is the std dev of the MEAN of n observations. The mean is √n times more precise. Quoting σ when you need σ_mean overstates uncertainty.
Standard deviations don't add
VARIANCES add (when independent). σ² of (a + b) = σ²_a + σ²_b. Adding standard deviations directly gives a wrong (too-pessimistic) total.
Outliers vs. blunders
An observation more than 3σ from the mean has only a 0.3% chance of being a real random error. It\'s probably a blunder — investigate, don\'t just downweight.
Test yourself
How well did it stick?
A quick 5-question check on Probability and Statistics. See where you stand and what to review.