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Probability, Statistics & Regression

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FE Environmental Study Guide Exam

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1 Introduction, Mathematics & Ethics

Exam Format, NCEES CBT Logistics & Blueprint Map Calculus & Differential Equations Probability, Statistics & Regression Numerical Methods & Engineering Math Mathematics Applications for Environmental FE

2 Ethics, Economics & Materials

NCEES Model Rules & Engineering Ethics Professional Practice, Liability & Public Safety Engineering Economics: Time Value of Money Benefit-Cost, Replacement & Inflation Analysis Materials: Concrete, Corrosion & Polymers

3 Environmental Science & Chemistry

Geosynthetics & Environmental Materials Environmental Chemistry & Stoichiometry Water Chemistry: pH, Alkalinity & Hardness Microbiology, BOD Kinetics & Monod Growth Henry's Law & Phase Equilibrium for Air Quality

4 Risk Assessment & Water Resources

Environmental Science Applications EPA Four-Step Risk Assessment Paradigm Exposure, Dose, Hazard Quotient & Cancer Risk Hydrology & Rainfall-Runoff Analysis Open-Channel Flow & Manning Equation

5 Groundwater & Water Treatment

Groundwater: Darcy, Theis & Cooper-Jacob Streeter-Phelps & Water-Quality Modeling Water Treatment: Sedimentation & Filtration Disinfection, CT Rule & Chlorination Wastewater: Activated Sludge & BOD Removal

6 Advanced Treatment & Air Quality

Advanced Treatment: MBR, GAC & RO Sludge Handling & Water Reuse NAAQS, Criteria Pollutants, HAPs & AQI Gaussian Plume, Pasquill Stability & Effective Stack Height AP-42 Factors & Air Pollution Control Technologies

7 Air Permitting & Solid/Hazardous Waste

PSD, NSR, BACT, LAER, Title V & Offsets RCRA Lists, Characteristics & Generator Categories Subtitle D Landfill Design, Liners & Leachate CERCLA, UST, TCLP & Land Disposal Restrictions MSW Collection, Transfer, Composting, WTE & Recycling

8 Sustainability, EHS & Exam Day

LCA Stages, Carbon Footprint & Scope 1/2/3 Emissions LEED Credits, ISO 14001 EMS & Triple Bottom Line PELs, TLVs, Hierarchy of Controls & Confined Spaces EPCRA Reporting, TRI Form R, DOT Hazmat & RCRA Manifests Exam Format, NCEES CBT Logistics & Blueprint Map

Probability, Statistics & Regression

Key Takeaways Explain This Topic AI

Mean, median, and mode describe central tendency; standard deviation and variance quantify spread around the mean.

Normal distribution properties let you estimate fractions within ±1σ (≈68%), ±2σ (≈95%), and ±3σ (≈99.7%) of the mean.

Linear regression fits y = mx + b to data; correlation coefficient r indicates strength and direction of linear association.

Sample mean x̄ and sample standard deviation s use n−1 in the denominator for unbiased variance estimation.

Confidence intervals and hypothesis testing appear conceptually; know when a t-test or z-test applies to environmental data sets.

Last updated: July 2026

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 , … , x n x_1, x_2, \ldots, x_n x 1 ​ , x 2 ​ , … , x n ​ :

  • Statistic Formula Use
  • Sample mean x ˉ = 1 n ∑ x i \bar{x} = \frac{1}{n}\sum x_i x ˉ = n 1 ​ ∑ x i ​ Central value

Sample variance s 2 = 1 n − 1 ∑ ( x i − x ˉ ) 2 s^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2 s 2 = n − 1 1 ​ ∑ ( x i ​ − x ˉ ) 2 Spread squared

  • Sample std. dev. s = s 2 s = \sqrt{s^2} s = s 2 ​ Spread in same units as data
  • Median Middle value when sorted Robust to outliers
  • Mode Most frequent value Categorical or binned data

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

Mean: x ˉ = ( 12 + 15 + 14 + 50 + 13 ) / 5 = 104 / 5 = 20.8 \bar{x} = (12+15+14+50+13)/5 = 104/5 = 20.8 x ˉ = ( 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 μ \mu μ and standard deviation σ \sigma σ :

  • Range Approximate fraction of data
  • μ ± 1 σ \mu \pm 1\sigma μ ± 1 σ 68%
  • μ ± 2 σ \mu \pm 2\sigma μ ± 2 σ 95%
  • μ ± 3 σ \mu \pm 3\sigma μ ± 3 σ 99.7%

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 − μ ) / σ z = (x-\mu)/\sigma z = ( x − μ ) / σ 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 ≈ + 1 r \approx +1 r ≈ + 1 : strong positive linear trend
  • r ≈ 0 r \approx 0 r ≈ 0 : weak linear association
  • r ≈ − 1 r \approx -1 r ≈ − 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 and B ) = P ( A ) × P ( B ) P(A \text{ and } B) = P(A) \times P(B) P ( A and B ) = P ( A ) × P ( B )

For mutually exclusive events: P ( A or B ) = P ( A ) + P ( B ) P(A \text{ or } B) = P(A) + P(B) P ( A 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 ( both fail ) = 0.05 × 0.05 = 0.0025 = 0.25 % P(\text{both fail}) = 0.05 \times 0.05 = 0.0025 = 0.25\% P ( both fail ) = 0.05 × 0.05 = 0.0025 = 0.25%

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

  • Confidence Intervals and Hypothesis
  • Tests

A 95% confidence interval for the true mean (large (n), known σ) is: x ˉ ± z 0.025 σ n \bar{x} \pm z_{0.025}\,\frac{\sigma}{\sqrt{n}} x ˉ ± z 0.025 ​ n ​ σ ​

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; x ˉ = 48 \bar{x} = 48 x ˉ = 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 x ˉ \bar{x} x ˉ and control limits at x ˉ ± 3 s \bar{x} \pm 3s x ˉ ± 3 s . A point outside limits suggests an assignable cause (process upset, lab error) rather than random variation.

  • Common Distributions in Environmental
  • Work
  • Distribution Typical application
  • Normal Measurement error, aggregated emissions
  • Log-normal Concentrations that cannot go negative, skewed right
  • Poisson Rare event counts (e.g., violations per year)
  • Binomial Pass/fail of n identical trials

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 / n s/\sqrt{n} s / 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.