Sustainability & Environment (Agri-Food Systems) · Foundations · also Environmental Statistics, Mathematics Foundation (GATE ES Section 1), Statistical Approaches and Modelling in Environmental Science, Biostatistics for Ecologists
The mathematics and statistics needed for every numerical question in this track: calculus and differential equations for growth and decay, probability and risk, regression and hypothesis tests, and sampling designs for field data. Experimental design and survey sampling in depth live in the Agricultural Statistics track; this subject covers the environmental applications.
In the exams: GATE ES Section 1 (Mathematics Foundation); ICAR AICE subject 71 Unit 7; UGC NET Environmental Sciences Unit 9
Environmental science is quantitative. Pollutants decay and accumulate, populations grow, floods recur, samples vary and experiments must be designed so that differences can be trusted. This subject collects the mathematics and statistics that every numerical question in the sustainability track relies on: linear algebra, calculus, ordinary differential equations and Laplace transforms, descriptive statistics, probability and Bayes' theorem, probability distributions, risk and return periods, correlation and regression, hypothesis tests, sampling designs, experimental designs and diversity indices, each illustrated with environmental and agricultural examples.
Statistical theory, experimental design and survey sampling are developed in depth in the Agricultural Statistics track (subjects such as probability theory, hypothesis testing, linear models and regression, experimental design, factorial and split-plot designs, and sampling theory). This manual keeps to the environmental-science angle: what each method is for, how to compute it, and how to avoid common mistakes with field data. It complements the Environmental Monitoring subject, which covers measurement quality, and the Environmental Systems Modelling subject, which uses these tools to build models.
Graduate engineering entrance examinations in environmental science examine this material with short numerical questions: eigenvalues, derivatives and integrals, differential equations, Laplace transforms, probability, distributions, regression and test statistics. Lecturer-eligibility and doctoral fellowship examinations in environmental science, and professional engineering licensing examinations, test statistical approaches, sampling and experimental design, and diversity indices. Every worked example in this manual is computed, and formulas are stated in the form used.
A matrix is a rectangular array of numbers; environmental applications include systems of mass-balance equations, transition matrices for populations and land-use change, and multivariate data tables. Matrices of compatible sizes can be added and multiplied; the product AB is defined when the number of columns of A equals the number of rows of B, and in general AB ≠ BA. The transpose Aᵀ swaps rows and columns. The identity matrix I has ones on the diagonal and zeros elsewhere. The determinant of a square matrix indicates whether it is invertible: for a 2 × 2 matrix [[a, b], [c, d]], det = ad − bc. A matrix with zero determinant is singular and has no inverse. The inverse of a 2 × 2 matrix is (1/det) × [[d, −b], [−c, a]]. The rank of a matrix is the number of linearly independent rows or columns. A system of linear equations Ax = b has a unique solution when A is square and non-singular (x = A⁻¹b), no solution when the equations are inconsistent, and infinitely many solutions when they are dependent; the rank of A compared with the rank of the augmented matrix [A | b] decides which. Practical methods include Gaussian elimination and, for small systems, Cramer's rule, in which each unknown equals the ratio of the determinant of A with one column replaced by b to the determinant of A. Linear systems arise, for example, when mass balances for several connected lakes or reservoirs are solved simultaneously, or when mixing models estimate the proportions of several sources contributing to a sample from tracer concentrations.
An eigenvector v of a square matrix A is a non-zero vector that A only stretches: Av = λv, where the scalar λ is the corresponding eigenvalue. Eigenvalues are the roots of the characteristic equation det(A − λI) = 0. For the matrix [[2, 1], [1, 2]], the characteristic equation is (2 − λ)² − 1 = 0, giving eigenvalues 3 and 1. Two useful checks: the sum of the eigenvalues equals the trace (the sum of diagonal elements, here 4), and the product equals the determinant (here 3). A triangular matrix has its diagonal elements as eigenvalues, and a symmetric matrix has real eigenvalues. Eigenvalues have direct environmental meaning. In a Leslie matrix model of an age-structured population, the first row contains age-specific fecundities and the sub-diagonal contains survival probabilities; the dominant eigenvalue λ is the asymptotic population growth rate per time step (λ > 1 growth, λ < 1 decline), and the corresponding eigenvector gives the stable age distribution. For a two-stage matrix [[0, 4], [0.5, 0]], λ² = 4 × 0.5 = 2, so λ = √2 ≈ 1.41, a 41 per cent increase per step. Markov chain models of land-use change use transition probability matrices whose eigenvector for eigenvalue 1 gives the long-run (steady-state) land-use distribution. In multivariate statistics, principal component analysis finds the eigenvectors of the covariance or correlation matrix, ordering them by eigenvalue to reduce many correlated variables (such as water quality parameters) to a few components that explain most of the variance.
A derivative measures the instantaneous rate of change of a function: dy/dx = lim(h→0) [f(x + h) − f(x)]/h. Standard rules include d(xⁿ)/dx = nxⁿ⁻¹, d(eᵃˣ)/dx = aeᵃˣ, d(ln x)/dx = 1/x, d(sin x)/dx = cos x, the product rule (uv)′ = u′v + uv′, the quotient rule and the chain rule. Partial derivatives measure change with respect to one variable while others are held constant, as in the sensitivity of a model output to one parameter. Maxima and minima of a function occur where its first derivative is zero (stationary points); the second derivative distinguishes them: f″ < 0 indicates a maximum and f″ > 0 a minimum. Environmental examples are common. In the logistic growth model, the growth rate rN(1 − N/K) is maximised where its derivative with respect to N is zero, r(1 − 2N/K) = 0, giving N = K/2, the basis of maximum sustainable yield. In the Streeter–Phelps model of river oxygen sag, the critical point of maximum oxygen deficit occurs where the derivative of the deficit with respect to time is zero. In economics of pollution control, the optimal level of abatement is where marginal abatement cost equals marginal damage, found by setting a derivative to zero. Limits are used to find behaviour at extremes, such as the steady state of a system as time goes to infinity, and L'Hôpital's rule evaluates indeterminate forms such as 0/0.
Integration is the reverse of differentiation and computes accumulated quantities: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1), ∫(1/x) dx = ln|x| + C and ∫eᵃˣ dx = eᵃˣ/a + C. A definite integral gives the area under a curve between limits. In environmental work, integration converts rates to totals: the total pollutant load carried by a river over a storm is ∫Q(t)C(t) dt, the total runoff volume is the integral of the hydrograph, and the cumulative emissions over a period are the integral of the emission rate, which underlies the cumulative carbon budget concept in climate science. When data are discrete measurements, integrals are estimated numerically. The trapezoidal rule approximates the area under a curve by trapezoids: with equally spaced points at interval h, area ≈ (h/2)[y0 + 2(y1 + y2 + … + yn−1) + yn]. For flows of 2, 5, 7 and 4 m³/s measured one hour apart, the trapezoidal estimate is (1/2)(2 + 2 × 5 + 2 × 7 + 4) = 15 (m³/s)·h, or 54,000 m³. Simpson's rule, which fits parabolas to pairs of intervals, is more accurate for smooth curves and requires an even number of intervals. Numerical integration is also used to solve differential equations step by step, as in Euler's and Runge–Kutta methods used in environmental simulation models.
A differential equation relates a function to its derivatives. Many environmental processes are described by first-order ordinary differential equations (ODEs). First-order decay, dC/dt = −kC, describes radioactive decay, BOD exertion, die-off of bacteria and degradation of many pesticides; separating variables and integrating gives C = C0e^(−kt), with half-life t½ = ln 2/k. For k = 0.1 per day, a concentration of 100 falls to 100e^(−1) ≈ 36.8 after 10 days and halves in about 6.9 days. Exponential growth, dN/dt = rN, gives N = N0e^(rt), with doubling time ln 2/r. When there is a constant input as well as first-order loss, dC/dt = a − kC, as for a pollutant entering a well-mixed lake at a constant rate and decaying, the solution approaches a steady state C* = a/k: C(t) = a/k + (C0 − a/k)e^(−kt). With a = 5 units per day and k = 0.25 per day, the steady state is 20, and the system gets within about 5 per cent of it after three time constants (3/k = 12 days). Linear first-order equations of the form dy/dt + p(t)y = q(t) are solved with an integrating factor e^(∫p dt). The logistic equation dN/dt = rN(1 − N/K) is non-linear but separable, giving N(t) = K/[1 + ((K − N0)/N0)e^(−rt)]. Second-order linear ODEs with constant coefficients, ay″ + by′ + cy = 0, are solved from the characteristic equation am² + bm + c = 0: real distinct roots give exponential solutions, a repeated root gives (A + Bt)e^(mt), and complex roots give oscillating solutions, as in some predator–prey and vibration problems. Systems of ODEs, such as the Lotka–Volterra equations or multi-compartment models, are usually solved numerically.
The Laplace transform converts a function of time f(t) into a function of a complex variable s: F(s) = ∫₀^∞ e^(−st) f(t) dt. It turns linear differential equations with initial conditions into algebraic equations, which are solved for F(s) and then converted back by the inverse transform using tables and partial fractions. Standard transforms include L{1} = 1/s, L{t} = 1/s², L{tⁿ} = n!/sⁿ⁺¹, L{e^(at)} = 1/(s − a), L{sin at} = a/(s² + a²) and L{cos at} = s/(s² + a²). Key properties are linearity; the derivative property, L{f′(t)} = sF(s) − f(0), and L{f″(t)} = s²F(s) − sf(0) − f′(0); and the shift property, L{e^(at)f(t)} = F(s − a). The unit step and impulse (Dirac delta, L{δ(t)} = 1) functions represent sudden or instantaneous inputs, such as a pollutant spill. As an example, solving dC/dt + kC = 0 with C(0) = C0: transforming gives sC(s) − C0 + kC(s) = 0, so C(s) = C0/(s + k), whose inverse is C0e^(−kt). Laplace methods are used for reactor and river models with step or pulse inputs, for groundwater and transport equations, and for systems with several linked compartments. In practice, environmental modellers mostly use numerical solvers, but Laplace transforms remain a standard GATE topic and help in understanding system responses.
Descriptive statistics summarise data. Measures of central tendency include the arithmetic mean (sum ÷ number), the median (middle value when ordered, robust to outliers), the mode (most frequent value), the geometric mean (the nth root of the product of n values, or the antilog of the mean of logarithms, appropriate for multiplicative or log-normally distributed data such as bacterial counts and many pollutant concentrations) and the harmonic mean (used for rates, such as average speeds). For the values 10, 100 and 1000, the arithmetic mean is 370 but the geometric mean is 100, which better represents data spanning orders of magnitude; regulations for coliforms in recreational waters often use geometric means. Measures of dispersion include the range, the interquartile range (IQR, the difference between the 75th and 25th percentiles, robust to outliers), the variance and the standard deviation. The sample variance s² = Σ(xi − x̄)²/(n − 1), using n − 1 (Bessel's correction) to give an unbiased estimate of the population variance; the standard deviation s is its square root. The coefficient of variation CV = s/x̄ × 100 per cent compares variability between variables with different units or means. For the values 4, 8, 6, 5 and 7, the mean is 6, the sample variance is 2.5, the standard deviation about 1.58 and the CV about 26 per cent. Adding an outlier such as 30 to the data changes the mean greatly but the median only slightly (from 6 to 6.5), illustrating the robustness of the median.
Skewness describes asymmetry. In a positively (right) skewed distribution, common for environmental concentrations, rainfall amounts and species abundances, a long tail extends to high values and mean > median > mode; in a negatively skewed distribution the reverse holds. Pearson's coefficient of skewness, 3(mean − median)/standard deviation, is a simple measure. Kurtosis describes the weight of the tails relative to a normal distribution: leptokurtic distributions have heavy tails and a sharp peak (excess kurtosis > 0), platykurtic distributions have light tails (excess kurtosis < 0), and the normal distribution is mesokurtic (kurtosis 3, excess 0). Exploratory data analysis comes before formal tests. Histograms, box-and-whisker plots (showing median, quartiles and potential outliers beyond 1.5 × IQR from the quartiles), scatter plots and time-series plots reveal skewness, outliers, trends and relationships. Skewed data are often log-transformed to make them more symmetric and to stabilise variance before parametric analysis. Censored data, values below a detection limit, require special handling (such as substitution methods with caution, maximum likelihood or non-parametric methods) rather than replacing them with zero. Percentiles are often used in environmental standards and reporting: for example, compliance may be judged on the 98th percentile of daily values or the 90th percentile of water quality samples, which describes typical high values while limiting the influence of single extreme readings.
Probability measures the likelihood of an event, from 0 (impossible) to 1 (certain). For any event A, P(not A) = 1 − P(A). The addition rule gives P(A or B) = P(A) + P(B) − P(A and B), which reduces to P(A) + P(B) for mutually exclusive events. The multiplication rule gives P(A and B) = P(A) × P(B | A), where P(B | A) is the conditional probability of B given A; for independent events, P(A and B) = P(A) × P(B). For example, if the probability of a drought year is 0.2 in each of two independent years, the probability of droughts in both years is 0.04 and of at least one drought is 1 − 0.8² = 0.36. Counting rules help compute probabilities: the number of ways to arrange n objects is n!; the number of permutations of r objects from n is n!/(n − r)!; and the number of combinations (unordered selections) is nCr = n!/[r!(n − r)!]. Combinations underlie the binomial distribution in the next chapter. The law of total probability states that if events B1, B2, … partition the possibilities, P(A) = Σ P(A | Bi)P(Bi). Probability trees are a helpful way to organise such calculations.
Bayes' theorem reverses conditional probabilities: P(B | A) = P(A | B) P(B) ÷ P(A). It is the basis of interpreting diagnostic tests and of Bayesian statistics, which update prior beliefs with data to obtain posterior probabilities. Consider a rapid field test for a contaminant that is present in 1 per cent of wells. The test detects the contaminant when present 95 per cent of the time (sensitivity) and gives a negative result when it is absent 90 per cent of the time (specificity, so the false positive rate is 10 per cent). If a well tests positive, the probability that it is really contaminated is P = (0.95 × 0.01) ÷ (0.95 × 0.01 + 0.10 × 0.99) = 0.0095/0.1085 ≈ 0.088, only about 9 per cent. Because the contaminant is rare, most positive results are false positives, so confirmatory testing is essential. This 'base rate' effect is often misjudged and matters for screening programmes, pest and disease surveillance, and early warning systems. Bayesian methods are increasingly used in environmental science: to calibrate models (estimating posterior distributions of parameters), to combine prior knowledge with limited data in species distribution and stock assessment models, in Bayesian networks for decision support (such as risk of nutrient pollution under different management), and in source apportionment mixing models. The Bayesian inference subject of the Agricultural Statistics track develops the theory.
The binomial distribution describes the number of successes X in n independent trials, each with probability p of success: P(X = k) = nCk p^k (1 − p)^(n − k), with mean np and variance np(1 − p). It applies to presence–absence data, such as the number of wells exceeding a standard out of n sampled, or seed germination. For n = 5 and p = 0.2, P(X = 2) = 10 × 0.2² × 0.8³ ≈ 0.205. The Poisson distribution describes counts of rare, independent events in a fixed interval of time or space: P(X = k) = e^(−λ) λ^k ÷ k!, with mean and variance both equal to λ. It models counts of organisms per quadrat when distribution is random, counts of rare floods or accidents per year, and colonies on a plate. For λ = 2, P(X = 0) = e^(−2) ≈ 0.135 and P(X = 1) ≈ 0.271. The Poisson approximates the binomial when n is large and p small (λ = np). The ratio of variance to mean indicates spatial pattern in ecology: about 1 for random distribution, greater than 1 for clumped (aggregated) distribution, common in nature and better modelled by the negative binomial distribution, and less than 1 for uniform (regular) distribution.
The normal (Gaussian) distribution is symmetric and bell-shaped, defined by its mean µ and standard deviation σ. About 68 per cent of values lie within ±1σ of the mean, 95 per cent within ±1.96σ (often rounded to 2σ) and 99.7 per cent within ±3σ. Any normal variable is converted to the standard normal Z = (X − µ)/σ, with mean 0 and standard deviation 1, whose cumulative probabilities are tabulated: P(Z < 1.96) = 0.975 and P(Z < 1.645) = 0.95. The central limit theorem explains why the normal distribution is so important: the distribution of sample means approaches normality as sample size increases, whatever the distribution of the underlying data, with standard error σ/√n. Many environmental variables are positive and right-skewed, such as pollutant concentrations, hydraulic conductivity and particle sizes. They are often well described by the log-normal distribution, in which the logarithm of the variable is normally distributed; its median is the geometric mean, and the mean exceeds the median. The exponential distribution describes waiting times between random events, such as times between storms, with mean 1/λ. Extreme-value distributions, such as the Gumbel distribution, describe annual maxima of floods, rainfall or wind and are used in design, as the next chapter explains. The gamma distribution is used for rainfall amounts and is the basis of the Standardized Precipitation Index. Choosing a distribution should be guided by the nature of the variable and checked with plots (such as normal probability or Q–Q plots) and goodness-of-fit tests, because wrong assumptions about distributions lead to wrong probabilities of exceeding standards.
Hydrological and climatic design uses the concepts of exceedance probability and return period. If an event, such as a flood of a given size, has a probability p of being equalled or exceeded in any year, its return period (recurrence interval) is T = 1/p: a 100-year flood has a 1 per cent chance of occurring in any year. The return period is an average interval, not a schedule; a 100-year flood can occur in consecutive years. Assuming independent years, the probability that an event of return period T occurs at least once in n years, called the risk, is R = 1 − (1 − 1/T)ⁿ. For a 100-year flood over 50 years, R = 1 − 0.99⁵⁰ ≈ 0.39, a 39 per cent chance. Reliability is the complement, 1 − R. Design standards are often expressed as acceptable risk over a structure's life: to keep risk to 10 per cent over 50 years, the required return period satisfies 1 − (1 − 1/T)⁵⁰ = 0.1, giving T ≈ 475 years, the classic basis of some seismic design maps. Climate change makes these probabilities non-stationary, as the adaptation subject notes, so return periods estimated from past records may underestimate future risk. Empirical exceedance probabilities are estimated from ranked annual maxima with plotting positions. The Weibull formula P = m/(n + 1), where m is the rank in descending order and n the number of years, gives the exceedance probability, and T = (n + 1)/m. For the largest of 30 annual floods, T ≈ 31 years.
To estimate events rarer than those observed, distributions are fitted to annual maxima. The Gumbel (extreme value type I) distribution is widely used for floods and rainfall. In the frequency factor method, the magnitude for return period T is xT = x̄ + KT × s, where x̄ and s are the mean and standard deviation of the annual maxima and, for a long record, KT = −(√6/π)[0.5772 + ln(ln(T/(T − 1)))]. For T = 100 years, KT ≈ 3.14; with a mean annual maximum flood of 500 m³/s and standard deviation of 150 m³/s, the 100-year flood is about 500 + 3.14 × 150 ≈ 970 m³/s. Other distributions used include the log-Pearson type III (standard for floods in the United States), generalised extreme value and generalised Pareto (peaks over threshold). Short records make estimates of rare events highly uncertain, and confidence intervals should be reported. Environmental risk assessment more broadly combines hazard identification, exposure assessment, dose–response (or concentration–response) assessment and risk characterisation, as in human health risk assessment of contaminants. For non-carcinogens, the hazard quotient HQ = exposure dose ÷ reference dose, with HQ > 1 indicating potential concern; for carcinogens, excess lifetime cancer risk = dose × slope factor, often compared with targets such as 10⁻⁶ to 10⁻⁴. Ecological risk assessment compares predicted environmental concentrations with predicted no-effect concentrations (PEC/PNEC). Engineering reliability uses failure probabilities and fault trees; for components in series, system reliability is the product of component reliabilities, and for parallel (redundant) components the system fails only if all fail. Risk is often expressed as probability × consequence, and risk matrices rank risks for management.
Correlation measures the strength and direction of a linear relationship between two variables. Pearson's correlation coefficient r = Sxy ÷ √(Sxx × Syy), where Sxy = Σ(x − x̄)(y − ȳ), Sxx = Σ(x − x̄)² and Syy = Σ(y − ȳ)², ranges from −1 (perfect negative) through 0 (no linear relationship) to +1 (perfect positive). Spearman's rank correlation applies the same idea to ranks and is robust to outliers and non-linear monotonic relationships; Kendall's tau is another rank-based measure. Correlation does not imply causation: two variables may be correlated because both depend on a third (confounding), because of a time trend, or by chance, particularly when many variables are screened. In environmental data, spatial and temporal autocorrelation (similarity of nearby or successive observations) violates independence assumptions and inflates apparent significance unless accounted for. The significance of r is tested with t = r√(n − 2)/√(1 − r²) with n − 2 degrees of freedom.
Simple linear regression fits the line y = a + bx that minimises the sum of squared vertical deviations (least squares). The slope b = Sxy/Sxx and the intercept a = ȳ − b x̄. For x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5: x̄ = 3, ȳ = 4, Sxy = 6, Sxx = 10 and Syy = 6, so b = 0.6, a = 2.2, and the fitted line is y = 2.2 + 0.6x; r = 6/√60 ≈ 0.77 and the coefficient of determination r² = 0.60, meaning the line explains 60 per cent of the variance in y. Regression assumes a linear relationship, independent errors, constant error variance (homoscedasticity) and, for inference, normally distributed errors. Residual plots check these assumptions; transformations (such as logarithms for exponential relationships or power laws) and weighted regression address violations. Predictions should not be extrapolated far beyond the range of the data. Regression is used to build calibration curves, rating curves relating river stage to discharge (often power functions fitted in log space), sediment rating curves, empirical models of crop yield against weather, and allometric equations relating tree biomass to diameter. Multiple regression relates y to several predictors: y = b0 + b1x1 + b2x2 + … + e. Each coefficient measures the effect of its predictor holding the others constant. Problems include multicollinearity (highly correlated predictors, which make coefficients unstable), overfitting with too many predictors, and confounding. Adjusted R², which penalises extra predictors, and information criteria such as AIC help select models. Analysis of variance tests the overall significance of the regression with an F statistic. Generalised linear models extend regression to counts (Poisson regression) and presence–absence data (logistic regression), widely used in species distribution and epidemiological studies, as developed in the Agricultural Statistics track's linear models and categorical data subjects.
A hypothesis test evaluates whether data are consistent with a null hypothesis (H0), usually of no effect or no difference, against an alternative (H1). A test statistic is computed and compared with its sampling distribution under H0; the p-value is the probability, if H0 were true, of obtaining a result at least as extreme as that observed. If p is below the chosen significance level α (commonly 0.05), H0 is rejected. A type I error is rejecting a true H0 (probability α); a type II error is failing to reject a false H0 (probability β); power, 1 − β, increases with sample size, effect size and α and decreases with variability. Two-tailed tests detect differences in either direction; one-tailed tests detect a difference in one specified direction. Statistical significance is not the same as practical importance: with very large samples tiny, unimportant differences become significant, while with small samples important differences may not be detected. Reporting effect sizes and confidence intervals alongside p-values is better practice, and when many tests are performed, corrections such as Bonferroni adjust for the increased chance of false positives.
The t-test compares means when the population standard deviation is unknown. A one-sample t-test compares a sample mean with a reference value: t = (x̄ − µ0)/(s/√n), with n − 1 degrees of freedom. For a sample of 16 river samples with mean 52 mg/L and standard deviation 4 mg/L tested against a standard of 50 mg/L, t = 2/(4/4) = 2.0 with 15 degrees of freedom; the two-tailed critical value at α = 0.05 is about 2.13, so the difference is not significant at that level. The two-sample t-test compares two independent groups (pooled or Welch versions for equal or unequal variances), and the paired t-test compares paired observations, such as upstream and downstream samples taken on the same days, using the differences. The F-test compares variances: F = s1²/s2² (larger over smaller), with degrees of freedom n1 − 1 and n2 − 1; for variances of 16 and 4, F = 4. Analysis of variance (ANOVA) uses F ratios of between-group to within-group mean squares to test whether several means differ, the basis of experimental design analysis. The chi-square (χ²) test compares observed frequencies with expected frequencies: χ² = Σ(O − E)²/E. In a goodness-of-fit test, observed counts are compared with those expected under a hypothesis, with degrees of freedom equal to the number of categories minus one minus the number of estimated parameters; for observed counts of 30 and 20 against expected 25 and 25, χ² = 1 + 1 = 2.0 with 1 degree of freedom, below the critical value of 3.84 at α = 0.05. In a contingency table test of independence, such as whether pest presence is associated with field management, degrees of freedom are (rows − 1)(columns − 1), and expected counts are (row total × column total)/grand total. Expected counts should generally be at least 5. When data violate normality and cannot be transformed, non-parametric tests are used: the Mann–Whitney U test (two independent groups), the Wilcoxon signed-rank test (paired data), the Kruskal–Wallis test (several groups) and Spearman's correlation.
Sampling designs determine how representative and precise environmental estimates are; the monitoring subject covers practical sampling and preservation, and the sampling theory subjects of the Agricultural Statistics track develop the theory. In simple random sampling, every unit has an equal chance of selection; the sample mean is an unbiased estimate of the population mean, and its standard error is s/√n (multiplied by the finite population correction √(1 − n/N) when a large fraction of the population is sampled). For a standard deviation of 12 and 36 samples, the standard error is 2. Stratified random sampling divides the population into non-overlapping strata that are internally more homogeneous, such as soil types, land uses or river reaches, and samples each randomly. It improves precision when strata differ and guarantees coverage of all strata. Samples can be allocated proportionally to stratum size or, more efficiently, by Neyman (optimum) allocation, in which the number of samples in stratum h is proportional to Nh × Sh, the product of stratum size and standard deviation, so that more variable strata receive more samples. The stratified estimate of the mean is the weighted average of stratum means. Systematic sampling selects units at regular intervals in space or time, such as a grid of soil cores or samples every week. It is simple, gives even coverage and is efficient for mapping, but it can be biased if the interval coincides with a periodic pattern, such as sampling always on the same weekday or along crop rows. Cluster sampling selects groups of units (such as villages or fields) and samples within them; it is cheaper but less precise than simple random sampling of the same size because units within clusters tend to be similar. Two-stage sampling combines cluster and random selection. Adaptive cluster sampling adds neighbouring units when a rare feature is found, useful for patchy species or contamination hotspots.
Environmental variables are often spatially autocorrelated: nearby samples are more alike than distant ones. Geostatistics quantifies this with the semivariogram, which plots half the average squared difference between pairs of samples against their separation distance; its range indicates the distance beyond which samples are effectively independent, and its nugget represents small-scale variability and measurement error. Kriging uses the semivariogram to interpolate values at unsampled locations with estimates of uncertainty, widely used to map soil properties, groundwater levels and pollutant concentrations. Grid spacing for mapping should be less than the range of spatial dependence. Sample size planning, as in the monitoring subject, uses n = (zσ/E)² for estimating a mean, with adjustments for stratification and design effects. For detecting a change between two periods or sites, power analysis determines the samples needed to detect a specified difference with acceptable error rates. For proportions, such as the fraction of wells exceeding a standard, n = z²p(1 − p)/E². Practical constraints, such as access, cost, safety and seasonality, often shape designs, and judgemental sampling is used for targeted investigations, such as finding the source of a spill. However, statistical inference about a whole area requires probability-based sampling. Documenting the design, including how units were selected, is essential for valid analysis and for repeating surveys to detect change over time.
Experiments test the effects of treatments, such as fertiliser rates, tillage methods or pollutant concentrations, on responses. R.A. Fisher, working at Rothamsted in the 1920s, set out three principles of experimental design: replication (repeating each treatment to estimate error and increase precision), randomisation (allocating treatments to units by chance to avoid bias and validate statistical tests) and local control (blocking or grouping similar units to reduce error). The Agricultural Statistics track develops design and analysis in depth, including factorial, incomplete block and response surface designs; here the essentials are summarised. The completely randomised design (CRD) assigns treatments randomly to all experimental units. It is simple and flexible and suits homogeneous conditions, such as laboratory or greenhouse experiments. With t treatments and r replicates, the analysis of variance has t − 1 treatment degrees of freedom and t(r − 1) error degrees of freedom; for 5 treatments with 4 replicates, error df = 15. The randomised complete block design (RBD or RCBD) groups units into blocks that are internally homogeneous, such as strips of a field along a fertility or slope gradient, and randomises all treatments within each block. Block differences are removed from the error, increasing precision. The ANOVA partitions variation into blocks (r − 1 df), treatments (t − 1 df) and error ((t − 1)(r − 1) df); for 5 treatments in 4 blocks, error df = 12. RBD is the most common design in field experiments.
The Latin square design controls two sources of variation simultaneously, such as gradients in two directions across a field. With t treatments it uses t rows and t columns, each treatment appearing once in each row and each column, so there are t² plots and t replications. Error degrees of freedom are (t − 1)(t − 2); for a 4 × 4 Latin square, error df = 6. Its main limitation is that the number of replicates must equal the number of treatments, so it suits about 4–8 treatments. The split-plot design is used for factorial experiments when one factor requires larger plots than the other, for example irrigation methods or tillage (main plots) combined with crop varieties or fertiliser rates (sub-plots). Main-plot treatments are randomised to large plots within blocks, and sub-plot treatments are randomised within each main plot. The analysis has two error terms: main-plot error (usually larger, with fewer degrees of freedom) for testing the main-plot factor, and sub-plot error for testing the sub-plot factor and the interaction, which are estimated more precisely. Factorial experiments in general test the main effects of several factors and their interactions, which are often of greatest interest, such as whether a variety's response to nitrogen depends on irrigation. After a significant ANOVA F test, treatment means are compared with methods such as the least significant difference (LSD = t × √(2 × error mean square/r)), Tukey's honestly significant difference, or Duncan's multiple range test, or with planned contrasts and, for quantitative treatments such as fertiliser rates, regression of response on rate. The coefficient of variation of the experiment, √(error mean square)/grand mean × 100, indicates its precision; field experiments typically have CVs of about 10–20 per cent. Pseudoreplication, treating repeated measurements on the same unit or subsamples as independent replicates, is a common error in environmental and ecological experiments that inflates apparent significance.
Diversity indices summarise the number of species and the distribution of individuals among them, as the ecology subject introduces. Species richness S is the number of species, but it depends strongly on sample size, so richness indices adjust for the number of individuals N: Margalef's index d = (S − 1)/ln N and Menhinick's index S/√N. For 20 species among 500 individuals, Margalef's d = 19/ln 500 ≈ 3.06 and Menhinick's index ≈ 0.89. Rarefaction estimates expected richness for a standard sample size, and estimators such as Chao1 estimate total richness including undetected species from the numbers of rare species. The Shannon index H′ = −Σ pi ln pi, where pi is the proportion of individuals in species i, combines richness and evenness; it increases with both and is sensitive to rare species. Its maximum, for S equally abundant species, is ln S. Pielou's evenness J′ = H′/ln S ranges from 0 to 1. Some studies use log base 2 or 10, so the base must be stated when comparing values. Simpson's index D = Σ pi² (or Σ ni(ni − 1)/[N(N − 1)] for finite samples) is the probability that two randomly chosen individuals belong to the same species; it is a dominance measure, sensitive to common species. Diversity is expressed as 1 − D (Gini–Simpson) or 1/D (inverse Simpson). The Berger–Parker dominance index is the proportion of the most abundant species, Nmax/N; if the commonest species has 200 of 500 individuals, d = 0.4.
Different indices weight rare and common species differently, so rankings of communities can change with the index used. Hill numbers unify them as the 'effective number of species' of order q: q = 0 gives richness, q = 1 gives the exponential of Shannon's index (e^H′), and q = 2 gives inverse Simpson (1/D). Expressing diversity as effective numbers makes values intuitive and comparable: a community with e^H′ = 10 is as diverse as one with 10 equally common species. Alpha, beta and gamma diversity describe diversity within sites, turnover between sites and total regional diversity; similarity indices such as Jaccard and Sørensen (presence–absence) and Bray–Curtis (abundance-based) quantify differences in composition and are the basis of ordination and cluster analysis of communities. These are developed in the ecology and biodiversity subjects. Diversity indices are used in environmental assessment to compare sites upstream and downstream of discharges, to monitor restoration, to evaluate the biodiversity effects of farming systems (for example field margins versus crop interiors) and in biotic indices for water quality. Interpretation needs care: indices depend on sampling effort, taxonomic resolution and the definition of the community sampled; a high index does not necessarily mean a healthy or natural community (disturbed sites can be diverse with invasive or generalist species); and statistical comparisons need replicate samples or resampling methods such as bootstrapping to estimate confidence intervals. Combining indices with information on species identity, such as the presence of sensitive or indicator species, gives a fuller picture.
A student uses this subject to master the calculation patterns behind GATE Section 1 and the numerical parts of every other subject: eigenvalues and the trace–determinant check, maxima by derivatives, trapezoidal integration, decay and steady-state ODE solutions, Laplace transform pairs, mean, variance and CV, Bayes' theorem with base rates, binomial and Poisson probabilities, z values, return-period risk and Gumbel estimates, least-squares lines and r², t, F and χ² statistics, ANOVA degrees of freedom and diversity indices. The commonest errors are using n instead of n − 1, misreading return periods as schedules, and treating significance as importance.
A five-year practitioner, such as an environmental data analyst, a monitoring officer or a research assistant in an agricultural experiment station, uses the subject to design sampling and field experiments with adequate replication and randomisation, summarise skewed data with geometric means and percentiles, test compliance with standards, fit rating and calibration curves, compute flood and drought risks for design, and report results with confidence intervals and appropriate tests.
A senior professional with 25 years' experience, such as a head of a monitoring network, a research programme leader or a regulator, uses quantitative reasoning to judge the strength of evidence in reports, to set sampling and design standards, to choose acceptable risk levels for infrastructure and standards, to avoid decisions based on spurious correlations or underpowered studies, and to communicate uncertainty to decision-makers.
A researcher with a PhD and 25 years' experience applies and develops advanced methods (mixed models, generalised additive models, Bayesian hierarchical models, extreme value statistics under non-stationarity, geostatistics and machine learning) to environmental and agricultural data, supervises students' statistical design, and reviews the quantitative rigour of research and assessments.
This manual covers the methods at the level of GATE and university courses; it does not derive the theory of estimation, the full analysis of factorial, incomplete block and response surface designs, multivariate statistics, time-series analysis, mixed models or machine learning, which are developed in the Agricultural Statistics track. Critical values are quoted only where needed; statistical tables or software should be used in practice. Extreme value methods are illustrated with the Gumbel frequency factor for long records; small-sample corrections and other distributions are only named. Risk assessment parameters such as reference doses and slope factors are chemical-specific and not tabulated.
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