Binomial, Poisson and normal distributions
The three key probability distributions: binomial, Poisson and normal, their formulas, mean and variance, z-scores and examples from seeds and pests.
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What is the chance that at least eight of ten seeds germinate? How many plants in a quadrat are likely to carry an aphid colony? What share of cows will produce more than 20 litres a day? Each of these questions is answered by a probability distribution: a rule that gives the probability of each possible outcome.
Three distributions cover most of agricultural statistics. The binomial handles counts of successes in a fixed number of trials. The Poisson handles counts of rare events. The normal handles continuous measurements and underlies most tests. Knowing which one fits the data decides which analysis is valid.
Some probability basics
- Probability ranges from 0 (impossible) to 1 (certain).
- Addition rule: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, the last term is zero.
- Multiplication rule: P(A and B) = P(A) × P(B) when A and B are independent.
- A random variable is discrete if it takes countable values, such as numbers of seeds or insects, and continuous if it takes any value in a range, such as yield, height or weight.
Binomial distribution
The binomial applies when there are n independent trials, each with only two outcomes (success or failure), and a constant probability p of success. Examples include germination of seeds, the presence or absence of disease in plants, and male or female calves.
The variance is always less than the mean, because q < 1. The distribution is symmetric when p = 0.5, and skewed otherwise.
Germination of ten seeds
A seed lot has a germination rate of 90% (p = 0.9). If ten seeds are sown in a pot, the probability that all ten germinate is 0.910 = 0.349. The probability of exactly nine is 10 × 0.99 × 0.1 = 0.387. So the probability of at least nine is 0.349 + 0.387 = 0.736. The expected number germinating is np = 9, and the variance is npq = 0.9. This is why nurseries sow extra seeds: even good seed will often leave a gap.
Poisson distribution
The Poisson describes the number of times a rare event happens in a fixed interval of time, space or volume, when events occur independently at a constant average rate λ. Examples include weed seedlings per quadrat, insect pests per leaf, bacterial colonies per plate, and accidents per month.
The Poisson is the limit of the binomial when n is large and p is small, with np = λ. A useful rule is that it is a good approximation when n ≥ 20 and p ≤ 0.05.
The variance to mean ratio
Because the mean equals the variance in a Poisson distribution, the variance to mean ratio tests how organisms are spread:
- s2 ⁄ x̄ ≈ 1 indicates a random distribution, which is Poisson.
- s2 ⁄ x̄ > 1 indicates an aggregated (clumped) distribution. This is typical of most insect pests and weeds, and is often described by the negative binomial distribution.
- s2 ⁄ x̄ < 1 indicates a regular (uniform) distribution.
This matters for pest sampling. Clumped pests need more samples to estimate density reliably, and they are often counted after a log(x + 1) transformation.
Aphid colonies per plant
In a field, aphid colonies average λ = 2 per plant and are randomly distributed. The probability that a plant has no colonies is e−2 = 0.135, and the probability it has at least one is 1 − 0.135 = 0.865. If a scout found a sample variance of 9 with the same mean, the ratio 9 ⁄ 2 = 4.5 would show strong aggregation, with many clean plants and a few heavily infested ones.
Normal distribution
The normal (Gaussian) distribution is the familiar bell-shaped curve. It is symmetric about the mean μ, and its spread is set by the standard deviation σ. Mean, median and mode are equal.
Key areas under the curve:
- μ ± 1σ contains 68.3% of values;
- μ ± 2σ contains 95.4% (and μ ± 1.96σ contains exactly 95%);
- μ ± 3σ contains 99.7%.
Yield, plant height, grain weight, milk yield and many other measurements are roughly normal. The central limit theorem explains why the normal is so important: means of reasonably large samples are approximately normal, whatever the shape of the original distribution. This is why tests on means, such as t, F and ANOVA, work widely.
Daily milk yield
Daily milk yields in a herd are normal, with μ = 14 L and σ = 3 L. The proportion of cows giving more than 20 L is found from z = (20 − 14) ⁄ 3 = 2.0. From z tables, P(Z > 2.0) = 0.023, about 2.3% of cows. The yield exceeded by the top 10% of cows is μ + 1.28σ = 14 + 3.84 = 17.8 L.
Choosing a distribution
| Situation | Distribution | Mean and variance |
|---|---|---|
| Successes in n trials (germination, infection) | Binomial | np; npq |
| Counts of rare, random events (weeds per quadrat) | Poisson | λ; λ |
| Clumped counts (most pests) | Negative binomial | variance > mean |
| Continuous measurements (yield, height) | Normal | μ; σ2 |
Data that do not fit the normal can be transformed before ANOVA. Common transformations are square root for counts, log for data with variance that grows with the mean, and arcsine (angular) for percentages.
Key points
- Binomial: P(x) = nCx px qn−x; mean np, variance npq; used for success and failure counts.
- Poisson: P(x) = e−λ λx ⁄ x!; mean = variance = λ; used for rare event counts.
- The variance to mean ratio is 1 for random, more than 1 for clumped, and less than 1 for regular spatial patterns.
- Normal: a symmetric bell curve; z = (x − μ) ⁄ σ; 68–95–99.7% of values lie within 1, 2 and 3σ.
- The central limit theorem makes sample means approximately normal.
- Use square root, log and arcsine transformations to make data suitable for ANOVA.
What the exams ask
- State the conditions, mean and variance for the binomial and Poisson distributions.
- Solve probability problems using the binomial, Poisson and normal distributions.
- Explain the variance to mean ratio for spatial distribution of pests.
- State the properties of the normal curve and the central limit theorem.
- Match transformations to types of data.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
When can I use the normal distribution for counts?
When counts are large. A binomial with np and nq both above about 5, or a Poisson with λ above about 10, is close to normal. For small counts or many zeros, use the exact discrete distribution, or transform the data before analysis.
Is yield always normally distributed?
Not exactly. Yields can be skewed by crop failures, and plot data can contain outliers. For tests on treatment means, approximate normality of the residuals is usually enough, thanks to the central limit theorem and the robustness of ANOVA.
What does "p = 0.05" have to do with the normal curve?
Many tests compare a statistic to a reference distribution. In the normal distribution, 5% of values lie beyond ±1.96σ. If a test statistic falls in that region, the result is called significant at the 5% level, because a value so extreme would be expected by chance only 5% of the time if the null hypothesis were true.
Sources
- Gupta, S. C. and Kapoor, V. K. Fundamentals of Mathematical Statistics, 12th edition. Sultan Chand, New Delhi, 2020.
- Snedecor, G. W. and Cochran, W. G. Statistical Methods, 8th edition. Iowa State University Press, 1989.
- Southwood, T. R. E. and Henderson, P. A. Ecological Methods, 3rd edition. Blackwell Science, 2000.
- Gomez, K. A. and Gomez, A. A. Statistical Procedures for Agricultural Research, 2nd edition. Wiley, 1984.
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