Agribusiness Management · Foundations · also STAT-501, STAT-502, Statistics for Managers (MANAGE), Business Statistics & Analytics (IRMA)
The statistics every later subject leans on: describing data, probability, sampling, testing and relating variables, plus the mathematics economists use.
In the exams: agricultural master's entrance examinations, elements of statistics and tests of significance (heavy); agricultural master's entrance examinations, agribusiness quantitative units (present); quantitative sections of agricultural development bank officer recruitment and agricultural field officer recruitment in banking
Business statistics is the toolkit for turning numbers about farms, markets and firms into decisions. An agribusiness manager is surrounded by data: daily mandi arrivals and prices, crop-cutting yields, milk collection by village, dealer sales of seed and fertiliser, loan repayment records, rainfall. Statistics describes such data honestly (averages, spread, shape), puts a number on uncertainty (probability and probability distributions), explains how a small sample can speak for a large population (sampling), and gives rules for deciding whether an observed difference is real or chance (tests of significance). It then relates variables to each other (correlation and regression), tracks prices and quantities over time (index numbers) and separates trend from season in a series (time series). The last two units cover the mathematics economists use every day: derivatives for marginal quantities and optimisation, and matrices and integrals for systems of equations, input-output models and surplus measures.
Almost every later course leans on it: production economics on derivatives, marketing research on sampling and chi-square tests, price analysis on index numbers and seasonal indices, econometrics on regression.
It is examined worldwide in much the same way: ICAR AIEEA-PG lists elements of statistics and tests of significance; bank recruitment tests (NABARD, IBPS Agricultural Field Officer) and management entrance tests (CAT, XAT, GMAT, GRE) test quantitative aptitude and data interpretation; agribusiness Master's programmes abroad list introductory statistics as a prerequisite. Questions are mostly numerical, so each method here is taught with an agricultural example and worked arithmetic.
Research methodology and econometrics are separate subjects; this manual stops where a first statistics course stops.
Statistics begins with a clear idea of what was measured. A variable is a characteristic that changes from unit to unit. Variables are qualitative (categorical: crop grown, caste, district) or quantitative (numerical). Quantitative variables are discrete when they take countable values and continuous when any value in a range is possible. The scale of measurement decides which statistics are allowed: nominal data (categories with no order) support counts and the mode; ordinal data (ranks such as 'poor, fair, good') support the median and rank correlation; interval and ratio data support means, standard deviations and the full toolkit. Ratio data have a true zero, so statements such as 'twice the yield' make sense.
Secondary data (a census, a market bulletin) must be checked for definition, coverage and base period before use.
Raw data are summarised in a frequency distribution: the range of values is divided into classes and the number of observations in each class is counted. A cumulative frequency distribution counts observations up to each class boundary; plotted, it gives an ogive, from which the median and quartiles can be read.
The arithmetic mean is the sum of observations divided by their number. Seven plots of wheat yielding 32, 35, 38, 40, 45, 48 and 52 quintals per hectare have a total of 290 and a mean of 41.43 q/ha. For a frequency distribution the mean is the sum of f times x divided by the sum of f, where x is each class mid-point. The mean uses every observation but is pulled by extreme values.
Two algebraic properties are examined repeatedly. First, the sum of deviations of observations from their mean is zero. Second, the sum of squared deviations is smallest when measured from the mean.
A weighted mean gives each value a weight reflecting its importance. A farmer who sells 20 quintals at 2,000 rupees, 30 quintals at 2,200 and 50 quintals at 2,100 received an average price of (40,000 + 66,000 + 105,000) / 100 = 2,110 rupees per quintal, not the simple mean of the three prices (2,100).
The geometric mean is the n-th root of the product of n values. It is the right average for growth rates and ratios. If farm output grows by 10 per cent, then 20 per cent, then falls by 5 per cent, the growth factors are 1.10, 1.20 and 0.95; their product is 1.254 and its cube root is 1.0784, so the average annual growth is about 7.84 per cent, not the arithmetic 8.33 per cent. The harmonic mean is n divided by the sum of reciprocals. It is correct for averaging rates when the numerator quantity is fixed: a trader who spends the same amount of money on onions at 20 and at 25 rupees per kilogram paid an average of 2 / (1/20 + 1/25) = 22.22 rupees per kilogram. For positive values that are not all equal, arithmetic mean > geometric mean > harmonic mean, and for two values GM squared equals AM times HM.
The median is the middle value of ordered data: the value with half the observations below and half above. For n ordered observations it is the ((n+1)/2)-th value; when n is even it is the mean of the two middle values. For the wheat plots above the median is 40 q/ha. In a grouped distribution the median class is the one containing the (N/2)-th observation, and median = L + ((N/2 - cf) / f) times h, where L is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency and h its width. The median is not affected by extreme values and can be computed with open-ended classes, which makes it the standard average for incomes, landholdings and farm sizes, which are skewed. The sum of absolute deviations is smallest when measured from the median.
The mode is the most frequent value, the most common farm size or the most commonly bought pack size of seed. In grouped data, mode = L + ((f1 - f0) / (2f1 - f0 - f2)) times h, where f1 is the modal class frequency and f0, f2 those of the classes before and after. For moderately skewed data Karl Pearson's empirical relation holds approximately: mode = 3 median - 2 mean.
In a symmetric distribution mean, median and mode coincide; in a positively skewed distribution (long right tail, typical of landholdings) mean > median > mode.
Two seed varieties can have the same average yield and very different risk. Dispersion measures how far observations lie from the centre, and in agribusiness it is the first measure of risk: variability of yield, of price, of milk supply to a plant, of repayment.
The range is the largest minus the smallest value; the coefficient of range is (L - S) / (L + S). The quartile deviation (semi-interquartile range) is (Q3 - Q1) / 2 and its coefficient is (Q3 - Q1) / (Q3 + Q1); it ignores the extreme quarters and suits skewed or open-ended data. The mean deviation is the average absolute deviation from the mean or median; it is least when taken about the median.
The standard deviation is the square root of the mean squared deviation from the mean. For a population of N values, variance = sum of (x - mean) squared / N. For a sample used to estimate a population, divide by n - 1 instead of n; this sample variance is an unbiased estimator of the population variance. For the seven wheat plots (32 to 52 q/ha, mean 41.43) the population standard deviation is 6.67 and the sample standard deviation is 7.21 q/ha.
The standard deviation is unaffected by a change of origin but changes with scale: if y = a + bx, then SD of y = |b| times SD of x and variance of y = b squared times variance of x. Adding a fixed transport charge to every price leaves the SD unchanged.
Absolute measures carry the units of the data, so they cannot compare series measured in different units or with very different means. The coefficient of variation (CV) is the standard deviation as a percentage of the mean: CV = (SD / mean) x 100. The seven wheat plots have a CV of 7.21 / 41.43 x 100 = 17.4 per cent using the sample SD.
The CV answers questions such as 'which variety gives the more stable yield?' If variety A averages 40 q/ha with SD 6 (CV 15 per cent) and variety B averages 50 q/ha with SD 8 (CV 16 per cent), B has the larger absolute spread and also slightly larger relative spread, so A is more consistent even though B yields more. It should be used only with ratio-scale data with a positive mean; it is meaningless for temperatures in Celsius or for series whose mean is near zero, such as profits that swing between loss and gain.
Skewness describes asymmetry. A positively skewed distribution has a long right tail (a few very large farms, a few very high prices after a shortage); a negatively skewed one has a long left tail (yields in a good year, where most farms do well but a few fail). Karl Pearson's coefficient of skewness is (mean - mode) / SD, or, using the empirical relation, 3(mean - median) / SD; it is zero for a symmetric distribution. Bowley's coefficient uses quartiles: (Q3 + Q1 - 2 median) / (Q3 - Q1), which lies between -1 and +1 and suits open-ended distributions. Moment-based skewness uses the third central moment: beta1 = mu3 squared / mu2 cubed, with the sign taken from mu3.
Kurtosis describes the peakedness of a distribution and the weight of its tails, relative to the normal. It is measured by beta2 = mu4 / mu2 squared. A normal distribution has beta2 = 3 (mesokurtic); beta2 > 3 is leptokurtic (sharper peak and fatter tails) and beta2 < 3 is platykurtic (flatter). Excess kurtosis, gamma2 = beta2 - 3, is zero for the normal. Daily commodity price changes are commonly leptokurtic: most days move little, but large jumps occur more often than a normal distribution would predict.
Probability measures uncertainty on a scale from 0 (impossible) to 1 (certain). A random experiment has outcomes that cannot be predicted exactly; the set of all outcomes is the sample space; an event is a subset of it.
There are three ways to assign probabilities. The classical (a priori) approach applies when outcomes are equally likely: P(A) = favourable outcomes / total outcomes, so P(sum of 7) = 6/36 = 1/6. The relative frequency approach uses long-run experience (18 droughts in 120 seasons gives 0.15). The subjective approach uses informed judgement.
Counting rules help with classical problems. The number of ordered arrangements of r items from n is nPr = n! / (n - r)!; the number of unordered selections is nCr = n! / (r! (n - r)!). A survey team choosing 3 villages out of 10 has 10C3 = 120 possible samples.
Events A and B are mutually exclusive if they cannot occur together (a loan is either repaid on time or not). They are independent if the occurrence of one does not change the probability of the other (rain in two distant, unrelated regions, perhaps). The two ideas are different: two mutually exclusive events with positive probabilities cannot be independent, because knowing one occurred tells you the other did not.
The complement rule: P(not A) = 1 - P(A). If each of three independent irrigation pumps fails on a given day with probability 0.1, the probability that at least one fails is 1 - 0.9 cubed = 1 - 0.729 = 0.271.
The addition rule: P(A or B) = P(A) + P(B) - P(A and B). If 40 per cent of farmers in a block use a pesticide, 30 per cent use drip irrigation and 12 per cent use both, then 58 per cent use at least one. For mutually exclusive events the last term is zero.
Conditional probability is the probability of A given that B has occurred: P(A | B) = P(A and B) / P(B). The multiplication rule rearranges this: P(A and B) = P(B) P(A | B). For independent events P(A | B) = P(A), so P(A and B) = P(A) P(B), which is also the test for independence: in the pesticide example, 0.4 x 0.3 = 0.12 equals P(both), so the two practices are statistically independent in that block.
Bayes' theorem reverses conditioning. Suppose a seed company buys a lot from three growers supplying 50, 30 and 20 per cent of its seed, with germination failure rates of 2, 3 and 5 per cent. The total probability of a failed packet is 0.5(0.02) + 0.3(0.03) + 0.2(0.05) = 0.010 + 0.009 + 0.010 = 0.029. Given a failed packet, the probability it came from the third grower is 0.010 / 0.029 = 0.345, much higher than its 20 per cent share.
Expected value connects probability to decisions. If a crop earns a profit of 50,000 rupees in a good season (probability 0.6), 20,000 in a normal season (0.3) and loses 30,000 in a bad one (0.1), the expected profit is 30,000 + 6,000 - 3,000 = 33,000 rupees.
A random variable assigns a number to each outcome of an experiment. A discrete random variable has a probability mass function giving P(X = x) for each value; the probabilities sum to 1. Its mean (expected value) is E(X) = sum of x P(x) and its variance is E(X squared) - [E(X)] squared. For constants a and b, E(a + bX) = a + b E(X) and Var(a + bX) = b squared Var(X).
The binomial distribution counts successes in n independent trials, each with the same probability p of success. P(X = x) = nCx p^x (1 - p)^(n - x). Its mean is np and its variance np(1 - p), so the variance is always less than the mean. If a seed lot germinates with probability 0.9 and five seeds are sown, P(exactly 4 germinate) = 5 x 0.9^4 x 0.1 = 0.328. In a sample of 100 farmers where each adopts a practice with probability 0.2, the expected number of adopters is 20 with variance 16 and standard deviation 4.
The Poisson distribution describes the number of rare events in a fixed interval of time or space when events occur independently at a constant average rate m: P(X = x) = e^(-m) m^x / x!. Its mean and variance are both m. Pest counts per leaf, defects per tonne, breakdowns of a cold-storage compressor per month and farm accidents per season are typical. If insects average 2 per leaf, the chance of a clean leaf is e^(-2) = 0.135. The Poisson approximates the binomial when n is large and p small, with m = np; a common rule of thumb is n at least 20 and p at most 0.05. Clumped counts show variance above the mean (over-dispersion) and fit the Poisson badly.
The normal distribution is the bell-shaped continuous distribution defined by its mean mu and standard deviation sigma. It is symmetric, so mean, median and mode coincide; its skewness is zero and beta2 = 3. About 68.3 per cent of values lie within one SD of the mean, 95.4 per cent within two and 99.7 per cent within three, and exactly 95 per cent lie within 1.96 SD.
Any normal variable is converted to the standard normal (mean 0, SD 1) by z = (x - mu) / sigma, and probabilities are read from a table of the cumulative distribution Phi(z). If a herd's daily milk yield per cow is normal with mean 12 litres and SD 2 litres, the proportion of cows above 15 litres is P(z > 1.5) = 1 - 0.9332 = 0.0668, about 6.7 per cent. If plot yields are normal with mean 50 and SD 5 q/ha, the proportion between 45 and 60 is Phi(2) - Phi(-1) = 0.9772 - 0.1587 = 0.8185.
The normal distribution matters for three reasons. Many measurements (weights, heights, yields under uniform conditions) are roughly normal. The binomial and Poisson approach normality for large n or m, allowing normal approximations (continuity correction 0.5 for a discrete count). And sample means are approximately normal whatever the population (central limit theorem).
A census measures every unit; a sample measures some and generalises. Samples are cheaper, faster, can be more accurate than a hurried census, and are the only option when measurement destroys the unit (crop cutting, germination tests). The list sampled from is the sampling frame, such as a cooperative's member register; an outdated frame is a common source of bias.
In probability sampling every unit has a known, non-zero chance of selection, which allows sampling error to be measured. Simple random sampling (SRS) gives every possible sample of size n the same chance. Systematic sampling selects every k-th unit after a random start, where k = N / n; from a list of 1,200 farmers, a sample of 60 uses k = 20. It is easy in the field but can go wrong if the list has a periodic pattern that matches k.
Stratified sampling divides the population into homogeneous groups (strata) such as farm-size classes or agro-climatic zones and samples each separately. It improves precision when strata differ from each other and guarantees each group is represented. Under proportional allocation each stratum's sample is proportional to its size: with strata of 600, 300 and 100 farmers and n = 50, the samples are 30, 15 and 5. Under Neyman (optimum) allocation, n_h is proportional to N_h times S_h, the stratum size times its standard deviation, so more variable strata get more units.
Cluster sampling selects whole groups (villages, market yards) and measures all units in them. It saves travel but is usually less precise than SRS of equal size, because units within a cluster are alike. Multistage sampling selects in stages, for example districts, then villages within them, then farms, then plots within farms; large crop-yield surveys commonly use stratified multistage designs. Non-probability methods (convenience, judgement, quota) cannot have their sampling error estimated; research methodology covers them further.
A statistic such as the sample mean varies from sample to sample. Its probability distribution over all possible samples is its sampling distribution, and the standard deviation of that distribution is the standard error (SE). For the sample mean, SE = sigma / square root of n. With a population SD of 12 kg and a sample of 36, SE = 2 kg. When sampling without replacement from a finite population and n is more than about 5 per cent of N, multiply by the finite population correction, square root of (N - n)/(N - 1). For a sample proportion, SE = square root of p(1 - p)/n.
The central limit theorem states that, for a large enough random sample, the sample mean is approximately normal with mean mu and SE sigma / square root of n, whatever the shape of the population. The usual rule of thumb is n of 30 or more, larger for strongly skewed populations.
The same results fix sample size. To estimate a mean within a margin of error E with confidence given by z, n = (z sigma / E) squared; with z = 1.96, sigma = 15 and E = 3, n = 96.04, rounded up to 97. For a proportion, n = z squared p(1 - p) / E squared; if about 40 per cent of farmers are expected to use crop insurance and the margin is 5 percentage points at 95 per cent confidence, n = 3.8416 x 0.24 / 0.0025 = 368.8, rounded up to 369. When p is unknown, p = 0.5 gives the largest, safest n (385 for a 5-point margin).
Sampling error arises only because a sample was taken and shrinks as n grows. Non-sampling errors (frame errors, non-response, bad questions, enumerator and recall bias) do not shrink with n and often exceed sampling error in farm surveys.
Estimation uses a sample to say something about a population parameter. A point estimate is a single number (the sample mean yield of 41.4 q/ha); an interval estimate gives a range with a stated confidence level. A good estimator is unbiased, consistent, efficient and sufficient. The sample mean is unbiased for the population mean; the sample variance with divisor n - 1 is unbiased for the population variance.
A 95 per cent confidence interval for a mean with known sigma, or with large n, is sample mean plus or minus 1.96 SE. If 64 milk samples show a mean fat content of 4.2 per cent with SD 0.4, SE = 0.05 and the interval is 4.2 plus or minus 0.098, that is 4.10 to 4.30 per cent. For small samples from a normal population with unknown sigma, replace 1.96 by the t value with n - 1 degrees of freedom. For a proportion, use p plus or minus z times square root of p(1 - p)/n.
The confidence level describes the method, not the particular interval: if the procedure were repeated many times, about 95 per cent of the intervals would contain the true mean. It is not correct to say there is a 95 per cent probability that this particular fixed interval contains the true mean, in the classical framework.
A test of significance asks whether a sample result is consistent with a stated claim about the population or whether it is too unusual to be explained by chance. The null hypothesis H0 is the claim of no effect or no difference (the new variety's mean yield equals 50 q/ha); the alternative H1 is what we accept if H0 is rejected. H1 may be two-tailed (the mean differs from 50) or one-tailed (the mean exceeds 50); the choice must be made before seeing the data, from the business question.
The level of significance alpha is the probability of rejecting H0 when it is true, usually 5 or 1 per cent. The critical value divides the rejection region from the acceptance region: for a z test at 5 per cent it is 1.96 two-tailed and 1.645 one-tailed; at 1 per cent, 2.576 two-tailed and 2.326 one-tailed. The p-value is the probability, if H0 were true, of a result at least as extreme as the one observed; reject H0 when p is below alpha.
Two errors are possible. A Type I error rejects a true H0 (concluding a fertiliser works when it does not); its probability is alpha. A Type II error fails to reject a false H0 (missing a real improvement); its probability is beta. The power of a test is 1 - beta, the probability of detecting a real effect. For a given sample size, lowering alpha raises beta; only a larger sample, or a less variable measurement, reduces both.
When the population SD is known, or the sample is large (commonly n of 30 or more), the test statistic for a mean is z = (sample mean - mu0) / (sigma / square root of n). A seed company claims a hybrid yields 50 q/ha; 36 demonstration plots average 52 q/ha, with a known SD of 6. Then SE = 1 and z = 2.0. Since 2.0 exceeds the two-tailed 5 per cent critical value of 1.96, the claim that the mean is exactly 50 is rejected; the data suggest a higher mean.
For the difference between two means from large independent samples, z = (mean1 - mean2) / square root of (s1 squared/n1 + s2 squared/n2). For a single proportion, z = (p - P0) / square root of (P0(1 - P0)/n), using the hypothesised P0 in the standard error. If an extension agency claims 60 per cent adoption and a sample of 200 farmers shows 132 adopters (66 per cent), SE = square root of (0.6 x 0.4 / 200) = 0.0346 and z = 1.73; this exceeds the one-tailed 5 per cent value of 1.645 if the question was whether adoption is higher than 60 per cent.
When sigma is unknown and the sample is small, the standardised mean follows Student's t distribution with n - 1 degrees of freedom, provided the population is roughly normal. The t distribution is symmetric like the normal but has fatter tails; as the degrees of freedom increase it approaches the normal. Its critical values are larger than z values: for 15 degrees of freedom the two-tailed 5 per cent value is 2.131, compared with 1.96.
One-sample t: t = (sample mean - mu0) / (s / square root of n). Sixteen cows fed a new ration average 27 litres with s = 4; against a standard of 25 litres, t = 2 / 1 = 2.0, below 2.131, so the difference is not significant at 5 per cent (two-tailed).
Two-sample (independent) t test assumes equal population variances and uses the pooled variance sp squared = [(n1 - 1)s1 squared + (n2 - 1)s2 squared] / (n1 + n2 - 2), with t = (mean1 - mean2) / square root of (sp squared (1/n1 + 1/n2)) and n1 + n2 - 2 degrees of freedom.
The paired t test applies when the same units are measured twice (yield before and after training, on the same farms) or units are matched. Work with the differences d: t = mean d / (s_d / square root of n), with n - 1 degrees of freedom. Pairing removes farm-to-farm variation and so is more powerful than treating the two sets as independent; using an independent-samples test on paired data is a classic error.
The chi-square statistic compares observed frequencies O with expected frequencies E: chi-square = sum of (O - E) squared / E.
Goodness of fit tests whether data follow a stated distribution. A retailer offers four pack sizes of seed and wants to know whether buyers prefer any; among 100 buyers the choices are 30, 20, 25 and 25. Under 'no preference' each expected frequency is 25, so chi-square = (25 + 25 + 0 + 0)/25 = 2.0 with 4 - 1 = 3 degrees of freedom. The 5 per cent critical value for 3 df is 7.815, so there is no evidence of preference. Degrees of freedom are the number of classes minus 1 minus the number of parameters estimated from the data (for example, minus one more if a Poisson mean had to be estimated).
The test of independence uses a contingency table. The expected frequency of each cell is (row total x column total) / grand total, and the degrees of freedom are (rows - 1)(columns - 1); a 2 x 2 table has 1 df, with 5 per cent critical value 3.841. The test shows association, not its direction or cause.
Conditions matter: observations must be independent counts (not percentages), and expected frequencies should generally be at least 5. Small expected cells are pooled with neighbours; for a 2 x 2 table with small expected counts, Yates' continuity correction or an exact test is used.
The F distribution is the distribution of the ratio of two independent variance estimates. To test whether two normal populations have equal variances (whether two procurement centres have equally variable moisture content, for example), compute F = larger sample variance / smaller sample variance with (n1 - 1, n2 - 1) degrees of freedom, and compare with the table value. If s1 squared = 25 from 11 samples and s2 squared = 10 from 9 samples, F = 2.5 against a 5 per cent critical value F(10, 8) of 3.35, so the variances are not significantly different.
Analysis of variance (ANOVA), developed by R. A. Fisher for agricultural field experiments, tests whether several means are equal at once. The total variation (total sum of squares) is split into variation between treatment means (treatment sum of squares) and variation within treatments (error sum of squares). Each is divided by its degrees of freedom to give a mean square: for k treatments and N observations, treatment df = k - 1, error df = N - k, total df = N - 1. The test statistic is F = treatment mean square / error mean square. If three fertiliser schedules are each tried on four plots with treatment SS = 60 and error SS = 90, the mean squares are 30 (2 df) and 10 (9 df), F = 3.0, and the 5 per cent critical value F(2, 9) is 4.26, so the schedules do not differ significantly.
Running many separate t tests instead of one ANOVA inflates the chance of a false positive. A significant F says only that not all means are equal; post-hoc comparisons (such as the least significant difference) identify which. ANOVA assumes independent observations, normal errors and equal variances.
Correlation measures the strength and direction of the linear relationship between two quantitative variables. Always draw a scatter diagram first; a curve or one outlier can make r misleading.
Covariance measures joint variation: cov(X, Y) = sum of (x - mean x)(y - mean y) / n. Its sign shows direction, but its size depends on units. Karl Pearson's correlation coefficient standardises it: r = cov(X, Y) / (SD of X x SD of Y), equivalently r = Sxy / square root of (Sxx Syy). For five seasons with rainfall of 600, 700, 800, 900 and 1,000 mm and yields of 18, 21, 22, 25 and 29 q/ha, Sxy = 2,600, Sxx = 100,000 and Syy = 70, giving r = 2,600 / square root of 7,000,000 = 0.983, a very strong positive relationship.
Properties: r lies between -1 and +1; r = +1 or -1 means all points lie on a straight line; r = 0 means no linear relationship (there could still be a strong curved one, such as yield rising and then falling with fertiliser dose). r is a pure number, unaffected by changes of origin and scale, except that a negative scale factor on one variable reverses its sign.
The coefficient of determination, r squared, is the proportion of the variation in one variable explained by its linear relation with the other. An r of 0.8 means r squared = 0.64: 64 per cent of the variation in yield is associated with rainfall, and 36 per cent is not.
A sample correlation can arise by chance. To test H0: population correlation = 0, use t = r square root of (n - 2) / square root of (1 - r squared) with n - 2 degrees of freedom. With r = 0.5 from 18 observations, t = 0.5 x 4 / 0.866 = 2.31, which exceeds the 5 per cent two-tailed value of 2.120 for 16 df, so the correlation is significant. The same r from 8 observations would not be.
Spearman's rank correlation coefficient is used for ordinal data (rankings of market attractiveness by two experts, or farmers' ranking of constraints) and for quantitative data that are skewed or have outliers. Rank each variable, compute the difference in ranks d for each unit, and use rho = 1 - 6 sum d squared / (n(n squared - 1)). Rho also lies between -1 and +1.
Partial correlation measures the association of two variables after removing the linear effect of a third: r12.3 = (r12 - r13 r23) / square root of ((1 - r13 squared)(1 - r23 squared)). Multiple correlation R measures how well one variable is predicted by a linear combination of several others. Both lead into multiple regression.
Correlation is not causation. Fertiliser and tractor sales rise together because both follow farm incomes. A high correlation between two trending time series (nonsense or spurious correlation) is especially common, because any two series that grow over time will correlate. Before acting on a correlation, ask what mechanism could produce it, whether a third variable drives both, and whether the direction of influence could run the other way.
Regression describes how a dependent variable Y changes with an explanatory variable X. The simple linear regression of Y on X is Y = a + bX, where b, the regression coefficient or slope, is the expected change in Y for a one-unit change in X, and a, the intercept, is the value of Y when X is zero.
The method of least squares chooses a and b to make the sum of squared vertical deviations (residuals) of the observed Y from the line as small as possible. The results are b = Sxy / Sxx and a = mean Y - b mean X. The fitted line always passes through the point of means. For the rainfall-yield data (means 800 mm and 23 q/ha, Sxy = 2,600, Sxx = 100,000), b = 0.026 q/ha per mm and a = 23 - 0.026 x 800 = 2.2. The predicted yield at 850 mm is 2.2 + 0.026 x 850 = 24.3 q/ha. The residual for an observation is actual minus predicted; residuals sum to zero.
There are two regression lines. The regression of Y on X (used to predict Y from X) minimises vertical deviations; the regression of X on Y (used to predict X from Y) minimises horizontal deviations. Their slopes are byx = r (SD of Y / SD of X) and bxy = r (SD of X / SD of Y). Hence r squared = byx x bxy, and r = square root of (byx x bxy) with the common sign of the two coefficients. Both coefficients have the same sign, and they cannot both exceed 1 in absolute value. The two lines intersect at the means; they coincide only when r = +1 or -1, and are perpendicular when r = 0.
The total variation in Y (total sum of squares, Syy) splits into the part explained by the regression (regression sum of squares, b Sxy) and the unexplained residual sum of squares. The coefficient of determination R squared = explained / total; in simple regression it equals r squared. The standard error of estimate, the square root of residual sum of squares / (n - 2), is the typical size of a prediction error in the units of Y.
To test whether the slope differs from zero, t = b / SE(b), where SE(b) = standard error of estimate / square root of Sxx, with n - 2 degrees of freedom.
Transformed variables widen the range of useful models. Regressing log Y on log X gives a constant-elasticity model in which b is the elasticity of Y with respect to X.
Prediction within the range of the observed X is interpolation and is reasonably safe; prediction far outside it is extrapolation and can be badly wrong, because the relationship may bend. A fitted regression shows association in the data used; it does not prove that changing X will change Y.
An index number measures the relative change in a variable or group of variables between a base period and a current period, expressed with the base as 100.
A simple aggregative price index is sum of current prices / sum of base prices x 100. It is easy but is dominated by items with high unit prices and changes if units change. The simple average of price relatives (each p1/p0 x 100, then averaged) treats every item equally; using the geometric mean of relatives is better behaved. Weighted indices reflect importance.
The Laspeyres price index weights by base-period quantities: sum p1 q0 / sum p0 q0 x 100. It answers: what would the base-period basket cost today? The Paasche index weights by current quantities: sum p1 q1 / sum p0 q1 x 100. Consider three farm products with base prices 20, 30 and 50, current prices 25, 33 and 60, base quantities 10, 5 and 2 and current quantities 8, 6 and 2. Laspeyres = 535 / 450 x 100 = 118.9; Paasche = 518 / 440 x 100 = 117.7. Fisher's ideal index is the geometric mean of the two: square root of (118.9 x 117.7) = 118.3.
Tests of adequacy check whether a formula behaves consistently. The time reversal test requires that the index for period 1 on base 0, multiplied by the index for period 0 on base 1, equals 1 (with indices as ratios). The factor reversal test requires that the price index times the quantity index equals the value ratio sum p1 q1 / sum p0 q0. The circular test requires that chaining indices over successive periods gives the same result as a direct comparison. Fisher's index satisfies the time reversal and factor reversal tests but not the circular test, hence 'ideal'. Laspeyres and Paasche satisfy neither reversal test. The simple geometric mean of relatives and fixed-weight aggregative indices satisfy the circular test.
When consumers substitute away from goods whose prices rise, a fixed base-period basket overstates the rise in the cost of living; so the Laspeyres index tends to have an upward bias and the Paasche a downward bias. Check the current base year and weights of any official index before using it, as they are revised from time to time.
Several operations recur. Base shifting: to move an index to a new base, divide each value by the index of the new base period and multiply by 100; an index of 150 in the current year and 125 in the new base year becomes 120. Deflating converts nominal values to real values: real = nominal / price index x 100. A farm wage of 30,000 rupees when the relevant consumer price index is 150 is worth 20,000 in base-period rupees. The farmers' terms of trade (parity) index is the index of prices received divided by the index of prices paid, times 100; values below 100 mean what farmers sell has become cheaper relative to what they buy since the base period.
A time series is a sequence of observations ordered in time. Classical analysis sees it as a combination of four components. Secular trend (T) is the long-run direction, driven by population, technology and area changes. Seasonal variation (S) is the regular pattern within a year, repeated every year, such as harvest-time price gluts. Cyclical variation (C) is the longer, irregular up-and-down wave of more than a year, such as business cycles or livestock price cycles driven by herd-building lags. Irregular or random variation (I) is what remains: floods, strikes, a sudden export ban.
In the additive model, Y = T + S + C + I, the components are measured in the units of the data and the seasonal swing is constant in size. In the multiplicative model, Y = T x S x C x I, trend is in units and the other components are ratios (index numbers around 100 per cent), so the seasonal swing grows with the level of the series. Agricultural prices and quantities usually suit the multiplicative model; a logarithmic transform turns it into an additive one.
Moving averages smooth out short fluctuations. A three-year moving average replaces each value by the mean of it and its two neighbours; for production of 120, 126, 135, 138, 147, 150 and 159 thousand tonnes over seven years, the three-year averages are 127, 133, 140, 145 and 152, centred on years 2 to 6. A moving average whose period equals the length of a regular cycle removes that cycle completely. When the period is even (4 quarters, 12 months), the average falls between two time points and must be centred by averaging successive pairs of moving averages.
The method of least squares fits a trend equation. For a linear trend Y = a + bt, coding time so that the values of t sum to zero (for 7 years, t = -3 to +3) simplifies the equations to a = sum Y / n and b = sum tY / sum t squared. For the production series, a = 975 / 7 = 139.29 and b = 177 / 28 = 6.32, so production rises by about 6.3 thousand tonnes per year, and the forecast for the year after the last (t = 4) is 139.29 + 6.32 x 4 = 164.6. With an even number of years, t is coded in half-units (-5, -3, -1, 1, 3, 5), so b is the change per half-year and must be doubled for an annual slope.
Where growth is proportional rather than constant, fit an exponential trend Y = A B^t by regressing log Y on t. If ln Y = a + bt, the compound growth rate is (e^b - 1) x 100 per cent; with b = 0.03 the rate is 3.05 per cent a year.
A seasonal index shows the typical level of each season as a percentage of the annual average; the indices for four quarters average 100 (they sum to 400), and twelve monthly indices sum to 1,200. The simple average method averages each season across years and expresses each season's average as a percentage of the grand average. It ignores trend and is suitable only when trend is weak. The ratio-to-moving-average method first estimates trend-cycle by a centred 12-month or 4-quarter moving average, divides each actual value by it to obtain ratios containing only S and I, averages the ratios for each season across years to remove I, and adjusts them to total 1,200 or 400.
Deseasonalised (seasonally adjusted) data remove the seasonal effect so that underlying movement can be seen: deseasonalised value = actual / seasonal index x 100. A cooperative's milk procurement of 1,200 tonnes in a flush quarter with a seasonal index of 120 is 1,000 tonnes after adjustment; if the same quarter last year adjusted to 950, the underlying growth is real, not seasonal.
A simple classical forecast multiplies the trend value for a future period by that period's seasonal index / 100. If the trend for next quarter is 900 tonnes and its seasonal index is 85, the forecast is 765 tonnes.
Simple exponential smoothing updates a forecast as new data arrive: new forecast = alpha x latest actual + (1 - alpha) x previous forecast, with alpha between 0 and 1. A high alpha reacts quickly to change; a low alpha smooths heavily. With alpha = 0.3, a previous forecast of 500 and an actual of 560, the next forecast is 518. Stationarity, ARIMA models and cointegration are covered in econometrics.
The derivative dy/dx measures the rate at which y changes as x changes by a small amount; geometrically, it is the slope of the curve. In economics every 'marginal' concept is a derivative. Marginal cost is dTC/dQ, marginal revenue is dTR/dQ, marginal physical product is dY/dX.
The rules needed for most problems are few. The derivative of a constant is zero; of x^n it is n x^(n - 1); of a constant times a function it is the constant times the derivative; of a sum it is the sum of derivatives. The product rule: d(uv)/dx = u dv/dx + v du/dx. The quotient rule: d(u/v)/dx = (v du/dx - u dv/dx) / v squared. The chain rule: dy/dx = dy/du x du/dx.
With total cost TC = 100 + 5Q + 0.5Q squared, fixed cost is 100 and marginal cost is 5 + Q, so the tenth unit costs about 15. Average cost is TC/Q.
Elasticity measures proportional response: point elasticity of demand = (dQ/dP) x (P/Q). For the demand curve Q = 100 - 2P at P = 20, Q = 60 and elasticity = -2 x 20/60 = -0.67, inelastic. The logarithmic derivative gives elasticity directly: elasticity = d ln Q / d ln P, which is why coefficients of log-log equations are elasticities.
For functions of several variables, partial derivatives hold the other variables constant. For the Cobb-Douglas production function Y = A L^alpha K^beta, the marginal product of labour is alpha Y / L and of capital beta Y / K, the output elasticities are alpha and beta, and returns to scale are constant, increasing or decreasing as alpha + beta equals, exceeds or falls short of 1.
To find a maximum or minimum of y = f(x), set the first derivative to zero (first-order condition) and check the second derivative (second-order condition): negative at the point means a maximum, positive a minimum. For y = x^3 - 6x^2 + 9x, dy/dx = 3x^2 - 12x + 9 = 0 gives x = 1 and x = 3; the second derivative 6x - 12 is -6 at x = 1 (maximum) and +6 at x = 3 (minimum).
Profit maximisation sets marginal revenue equal to marginal cost. In production economics the profit-maximising input level sets the value of the marginal product equal to the input price, or equivalently MPP = input price / output price. If Y = 20X - 0.5X^2 (Y in kg of grain, X in kg of nitrogen), MPP = 20 - X. With nitrogen at 20 rupees per kg and grain at 10 rupees per kg, the price ratio is 2, so the economic optimum is X = 18 kg, below the technical maximum of X = 20 where MPP = 0.
Constrained optimisation uses the Lagrange method. To maximise f(x, y) subject to g(x, y) = c, form L = f(x, y) + lambda (c - g(x, y)) and set its partial derivatives with respect to x, y and lambda to zero. To maximise utility U = xy with income 80, price of x = 2 and price of y = 4, the conditions give y = 2 lambda, x = 4 lambda and 2x + 4y = 80, so x = 20, y = 10 and lambda = 5. The multiplier lambda measures how much the objective rises when the constraint is relaxed by one unit (here, the marginal utility of income).
A matrix is a rectangular array of numbers; an m x n matrix has m rows and n columns. The product AB exists only when the number of columns of A equals the number of rows of B, and an (m x n) matrix times an (n x p) matrix gives an (m x p) matrix. Matrix multiplication is generally not commutative: AB need not equal BA.
The determinant of a 2 x 2 matrix with rows (a, b) and (c, d) is ad - bc. A square matrix with a zero determinant is singular: it has no inverse, and its rows are linearly dependent. The rank of a matrix is the number of linearly independent rows (or columns). The inverse of a 2 x 2 matrix is (1 / (ad - bc)) times the matrix with rows (d, -b) and (-c, a); A times its inverse is I.
Systems of linear equations are written AX = B and solved by X = A^(-1) B when A is non-singular, or by Cramer's rule, where each unknown equals the determinant of A with the corresponding column replaced by B, divided by the determinant of A. A market model with demand Qd = 100 - 2P and supply Qs = 20 + 2P gives the equations Q + 2P = 100 and Q - 2P = 20; the determinant is -4, and Cramer's rule gives P = 20 and Q = 60.
The Leontief input-output model is the classic application. If A is the matrix of technical coefficients (the input from sector i needed per rupee of output of sector j) and D the vector of final demand, gross outputs X satisfy X = AX + D, so X = (I - A)^(-1) D. The Leontief inverse gives the total (direct plus indirect) output each sector needs per unit of final demand.
Integration reverses differentiation. The indefinite integral of x^n is x^(n + 1) / (n + 1) plus a constant, for n not equal to -1; the integral of 1/x is ln x; the integral of e^x is e^x. The constant of integration is fixed by extra information. Because marginal cost is the derivative of total cost, total cost is the integral of marginal cost plus fixed cost: if MC = 2Q + 10 and fixed cost is 50, TC = Q^2 + 10Q + 50, and at Q = 10 total cost is 250. Likewise total revenue is the integral of marginal revenue, with zero constant because revenue is zero when nothing is sold.
A definite integral from a to b is the area under the curve between those limits: F(b) - F(a). The integral of 3x^2 from 0 to 2 is 8.
Consumer surplus is the area between the demand curve and the market price, up to the quantity bought. With demand P = 50 - 2Q and a price of 30, quantity is 10, the area under the demand curve from 0 to 10 is 500 - 100 = 400, spending is 300, and consumer surplus is 100. Producer surplus is the area between the price and the supply curve. These areas measure the welfare effects of a price support or tariff.
For continuous random variables, probabilities are areas under the density function, so the normal, t, chi-square and F tables of earlier chapters are tabulated integrals. The total area under any probability density is 1.
A student uses this subject in every numerical course that follows. In the first term it means computing means, SDs and CVs of yields and prices by hand and in a spreadsheet, running t and chi-square tests on survey data for a term paper, and fitting a trend and compound growth rate to area and production data from official statistics. For entrance and recruitment tests it means speed on standard formulas and knowing the stated critical values: 1.96, 1.645, 3.841. The habit to build early is to state H0 and the test before calculating, and to check units at each step.
Five years into a job at an agri-input company, a bank, a processor or an FPO, the manager uses statistics weekly. Sales managers compare dealer performance with CVs and seasonal indices, and forecast demand with trend times season. Procurement staff deseasonalise arrivals, deflate prices to real terms and track a terms-of-trade index. Credit officers estimate default probabilities by segment and design a stratified sample for field verification. Quality teams test whether a new supplier's seed germination differs from the standard. The practitioner also reads consultants' reports critically: what was the sample, was the difference significant and large enough to matter, is a correlation being sold as causation.
A senior manager with 25 years' experience rarely computes by hand but decides on the basis of numbers others produce. The questions they ask are statistical: is this year's yield rise trend, season or a good monsoon; is the sample behind this market study representative of our customers; what is the margin of error on this survey; is the claimed benefit of a new practice significant and what is its range; how volatile is this price relative to its mean and what does that imply for hedging and storage. They set the organisation's standards for forecasting, sampling for audits and procurement, and for how results are reported to boards and regulators.
A researcher with a PhD and 25 years' work designs sampling schemes for large farm surveys, chooses between index formulas for a new price series, and supervises students' tests and models. Their concerns are the ones a first course only flags: non-sampling error in farm surveys, which often exceeds sampling error; the substitution bias of fixed-weight indices; seasonality that shifts with changing cropping patterns; multiple testing across dozens of outcome variables; and the gap between statistical and economic significance. They also teach the mathematics that production economics and econometrics need, and review papers where the statistics, not the economics, is usually the weak point.
This manual covers a first course in statistics and the mathematics economists use, applied to agribusiness examples. It does not cover non-parametric tests beyond rank correlation and chi-square (sign, Mann-Whitney, Kruskal-Wallis), experimental designs beyond one-way ANOVA, Bayesian inference, multiple regression and its diagnostics (see econometrics), ARIMA and other time-series econometrics (see econometrics), multivariate methods and questionnaire design (see research methodology), or statistical software. Statistical tables are not reproduced; the critical values quoted are standard table values and questions state the ones they need. Official index numbers are described only in general terms because their base years and weights are revised; check the current release before use.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed with a script; critical values needed are stated in the question.
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