Probability, Statistics & Regression

Quick Answer: FE Environmental statistics focuses on describing data (mean, standard deviation), normal-distribution rules, linear regression, and basic inference. You will interpret lab results, compliance monitoring sets, and QA/QC charts — not derive proofs.

Environmental engineers collect enormous amounts of data: effluent BOD, ambient PM₂.₅, groundwater contaminant levels, and flow records. The FE exam tests whether you can summarize that data, spot outliers, and draw justified conclusions.

Descriptive

Statistics

Given (n) observations (x_1, x_2, \ldots, x_n):

Worked example: Five effluent TSS readings (mg/L): 12, 15, 14, 50, 13. Find mean and median.

Mean: (\bar{x} = (12+15+14+50+13)/5 = 104/5 = 20.8) mg/L

Median (sorted: 12, 13, 14, 15, 50): 14 mg/L

The single spike at 50 pulls the mean upward; the median better represents typical performance. On compliance questions, know whether the regulation references arithmetic mean, geometric mean (common for bacteria), or percentile (e.g., 90th percentile).

Normal

Distribution

Many environmental variables approximate a bell curve when measurement error dominates or when aggregating many independent sources. For a normal distribution with mean (\mu) and standard deviation (\sigma):

Worked example: Ambient CO mean = 2.0 ppm, σ = 0.4 ppm. What fraction exceeds 2.8 ppm?

2.8 = 2.0 + 2(0.4), so 2.8 is +2σ. About 95% fall within ±2σ, leaving ~2.5% in each tail above +2σ. Roughly 2.5% of readings exceed 2.8 ppm.

Standard normal tables or the Handbook's statistical tables convert z-scores (z = (x-\mu)/\sigma) to cumulative probabilities.

Linear

Regression

Fit a line (y = mx + b) to paired data (e.g., instrument reading versus true concentration). The slope (m) and intercept (b) minimize squared residuals. The correlation coefficient (r) ranges from −1 to +1:

• (r \approx +1): strong positive linear trend
• (r \approx 0): weak linear association
• (r \approx -1): strong negative trend

Worked example: Calibration points: (1, 2.1), (2, 4.0), (3, 5.9), (4, 8.2). A near-perfect line suggests slope ≈ 2 and intercept ≈ 0.1. If a sample reads 6.0 on the instrument, estimated true value ≈ 6.0 (after verifying calibration).

On the FE, you may compute slope from two handbook formulas or read coefficients from a calculator regression output.

Probability

Basics

For independent events (A) and (B):

[P(A \text{ and } B) = P(A) \times P(B)]

For mutually exclusive events:

[P(A \text{ or } B) = P(A) + P(B)]

Worked example: Two parallel pumps each have 0.95 reliability. What is the probability the system fails (both fail) if they are independent?

[P(\text{both fail}) = 0.05 \times 0.05 = 0.0025 = 0.25%]

If the system needs at least one operating, (P = 1 - 0.0025 = 99.75%).

Confidence Intervals and Hypothesis

Tests

A 95% confidence interval for the true mean (large (n), known σ) is:

[\bar{x} \pm z_{0.025},\frac{\sigma}{\sqrt{n}}]

For small samples with estimated (s), replace (z) with (t) from Student's t-distribution with (n-1) degrees of freedom.

Conceptual example: You measure lead in 10 soil samples; (\bar{x} = 48) mg/kg, (s = 6) mg/kg. The remediation standard is 50 mg/kg. A one-sample t-test asks whether the true mean significantly exceeds 50. If the 95% CI is 43–53 mg/kg, you cannot conclude exceedance at 95% confidence because 50 lies inside the interval.

Environmental FE items rarely require full five-step hypothesis tests but expect you to interpret p-values, confidence intervals, and control charts.

Quality Control

Charts

Shewhart control charts plot sample statistics versus time with centerline (\bar{x}) and control limits at (\bar{x} \pm 3s). A point outside limits suggests an assignable cause (process upset, lab error) rather than random variation.

Common Distributions in Environmental

Work

When data are log-normally distributed, the geometric mean is appropriate for reporting bacteria (fecal coliform) compliance.

Exam

Traps

• Using (n) instead of (n-1) in sample variance (Handbook tables usually clarify).
• Confusing standard error (s/\sqrt{n}) with standard deviation (s).
• Applying normal-distribution percentages to skewed data without checking.

Tip: Carry your calculator's statistics mode manual mentally — know how to enter data lists, run LinReg, and read r and r².

Statistics on the FE Environmental exam supports defensible engineering judgment: Is this effluent spike a violation or random noise? Does the regression support our calibration? Those questions appear throughout practice.

Regression Diagnostics

Check r² for explained variance. Residual plots revealing curvature suggest a nonlinear model — environmental data often need log transforms for skewed concentrations.

Reporting and Compliance Statistics

Permit limits often reference percentiles rather than means. A 90th-percentile stormwater benchmark means 90% of samples must fall below the threshold — different from an arithmetic average. When comparing a data set to a standard, confirm whether the rule uses instantaneous maximum, daily average, monthly geometric mean (common for bacteria), or annual mean. Mixing these definitions is a frequent FE trap.