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Statistics and research methods

Mean, median, mode and dispersion

Measures of central tendency and dispersion: mean, median, mode, range, variance, standard deviation, CV and standard error, with a crop yield example.

Exam essential5 min read
On this page
  1. Measures of central tendency
  2. Measures of dispersion
  3. Shape: skewness and kurtosis
  4. Key points
  5. What the exams ask
  6. Questions people ask
  7. Sources
  8. Related on Laisha
  9. Discussion

Ten farmers in a village report their wheat yields. Before any test or model, two questions come first: what is a typical value, and how much do the values vary? The first is answered by a measure of central tendency: mean, median or mode. The second is answered by a measure of dispersion: range, variance, standard deviation or coefficient of variation. Every statistical analysis in agriculture, from a variety trial to a crop-cutting survey, is built on these quantities.

A positively (right) skewed distribution, such as farm sizes or household incomes: the mode lies at the peak, the median splits the area in half, and the mean is pulled furthest toward the long right tail, so mode is less than median, which is less than mean. In a symmetric distribution all three coincide modemedianmeanlong right tail pulls the mean up value (for example farm size, ha) Positive skewmode < median < mean Symmetricmean = median = mode Rule of thumbmode ≈ 3 median − 2 mean
In a right-skewed distribution such as farm size, the mean is pulled toward the long tail, so mode < median < mean. In a symmetric distribution they coincide.Diagram: Laisha, original.

Measures of central tendency

Arithmetic mean

The mean is the sum of the values divided by their number.

Arithmetic mean
Ungrouped data: x̄ = Σx ⁄ n
Grouped data (frequency f, class mid-point x): x̄ = Σfx ⁄ Σf
Weighted mean: x̄w = Σwx ⁄ Σw

The mean uses every observation and is the basis of most statistical theory. Its weakness is that it is sensitive to extreme values. One very large farm can raise the mean farm size far above what most farmers own.

Two properties come up often: the sum of deviations from the mean is zero, Σ(x − x̄) = 0, and the sum of squared deviations is smallest when taken from the mean.

Median

The median is the middle value when the data are arranged in order. For an odd number n, it is the ((n + 1) ⁄ 2)th value; for even n, it is the average of the two middle values. For grouped data:

Median of grouped data
Median = L + [(n⁄2 − cf) ⁄ f] × h
L = lower limit of the median class; cf = cumulative frequency before it; f = its frequency; h = class width

The median is not affected by extreme values, so it is preferred for skewed data such as income, landholding and prices. Quartiles, deciles and percentiles extend the same idea.

Mode

The mode is the most frequent value, or the peak of the distribution. For grouped data:

Mode of grouped data
Mode = L + [(f1 − f0) ⁄ (2f1 − f0 − f2)] × h
f1 = frequency of the modal class; f0 and f2 = frequencies of the classes before and after it

The mode is the only average that works for categorical data, such as the most common variety grown in a district. A distribution may be bimodal or have no clear mode.

Other means

  • The geometric mean, (x1 × x2 × … × xn)1⁄n, is used for growth rates, ratios and index numbers. Compound annual growth rates in agricultural production are geometric means.
  • The harmonic mean, n ⁄ Σ(1⁄x), is used for averaging rates such as speeds, or prices when equal amounts are spent.
  • For any set of positive values, AM ≥ GM ≥ HM.

For a moderately skewed distribution, Karl Pearson's empirical relation is mode ≈ 3 median − 2 mean.

Measures of dispersion

Two varieties may give the same mean yield, but if one varies widely from field to field it is riskier for farmers. Dispersion measures that spread.

  • The range is the maximum minus the minimum. It is simple but depends only on two values.
  • The quartile deviation is (Q3 − Q1) ⁄ 2. It is not affected by extremes.
  • The mean deviation is Σ|x − x̄| ⁄ n.
  • The variance and standard deviation are the most widely used measures:
Variance and standard deviation (sample)
s2 = Σ(x − x̄)2 ⁄ (n − 1) = [Σx2 − (Σx)2 ⁄ n] ⁄ (n − 1)
s = √s2

The divisor n − 1 (degrees of freedom) makes s2 an unbiased estimate of the population variance σ2.

Relative measures

The standard deviation is in the units of the data, so it cannot compare variation in yield (t per ha) with variation in plant height (cm). The coefficient of variation solves this:

Coefficient of variation and standard error
CV (%) = (s ⁄ x̄) × 100
Standard error of the mean: SE = s ⁄ √n

In field experiments, the CV of the error is a quality check. Values below about 10–15% for yield usually indicate a well-conducted trial; much higher values suggest poor uniformity or management.

The standard error measures how precisely the sample mean estimates the population mean. It falls as the sample grows, while s does not.

Worked example

Yields from eight plots

Plot yields of a wheat variety (t per ha) are 4.2, 4.8, 5.0, 4.5, 5.3, 4.6, 4.9 and 4.7.

  • Mean = 38.0 ⁄ 8 = 4.75. In order, the values are 4.2, 4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.3, so the median is (4.7 + 4.8) ⁄ 2 = 4.75. The data are roughly symmetric.
  • Range = 5.3 − 4.2 = 1.1.
  • Σx2 = 181.28 and (Σx)2 ⁄ n = 1,444 ⁄ 8 = 180.5, so s2 = (181.28 − 180.5) ⁄ 7 = 0.111 and s = 0.334 t per ha.
  • CV = 0.334 ⁄ 4.75 × 100 = 7.0%, which is good precision.
  • SE = 0.334 ⁄ √8 = 0.118 t per ha.

Shape: skewness and kurtosis

  • Skewness measures asymmetry. With positive skew, the long tail is to the right and mean > median > mode; this is typical of farm size, rainfall in dry areas and pest counts. With negative skew, the tail is to the left. Pearson's coefficient is Sk = (mean − mode) ⁄ s, or 3(mean − median) ⁄ s.
  • Kurtosis measures peakedness or heavy tails relative to the normal curve: leptokurtic distributions are peaked with heavy tails, mesokurtic distributions are normal, and platykurtic distributions are flat.

Key points

  • Mean = Σx ⁄ n. It uses all the data but is sensitive to outliers. The median is the middle value and is robust to outliers. The mode is the most frequent value and works for categorical data.
  • AM ≥ GM ≥ HM; the GM is used for growth rates; mode ≈ 3 median − 2 mean.
  • Sample variance s2 = Σ(x − x̄)2 ⁄ (n − 1); s is its square root.
  • CV = s ⁄ x̄ × 100 compares relative variation; SE = s ⁄ √n measures the precision of the mean.
  • With positive skew, mean > median > mode.

What the exams ask

  • Calculate the mean, median and mode from grouped data.
  • Calculate the variance, standard deviation, CV and standard error.
  • State when the median is preferred to the mean.
  • Explain the relation between AM, GM and HM, and Pearson's empirical relation.
  • Interpret skewness and kurtosis.

Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.

Questions people ask

Why divide by n − 1 instead of n?

The deviations are measured from the sample mean, which is itself calculated from the same data, so they are on average slightly smaller than deviations from the true population mean. Dividing by n − 1 corrects this bias. One degree of freedom is "used up" in estimating the mean.

Which average should I report for farm incomes?

Usually the median, because a few large incomes pull the mean upward. Reporting both is often best: the gap between them tells readers how skewed the distribution is.

What is a good CV for a field experiment?

It depends on the trait and the crop. For grain yield, CVs below about 10% are excellent and 10–20% are acceptable. Traits such as pest counts or disease scores naturally vary more and may have higher CVs.

Sources

  1. Gupta, S. C. and Kapoor, V. K. Fundamentals of Mathematical Statistics, 12th edition. Sultan Chand, New Delhi, 2020.
  2. Gomez, K. A. and Gomez, A. A. Statistical Procedures for Agricultural Research, 2nd edition. Wiley, 1984.
  3. Snedecor, G. W. and Cochran, W. G. Statistical Methods, 8th edition. Iowa State University Press, 1989.
  4. Rangaswamy, R. A Textbook of Agricultural Statistics, 2nd edition. New Age International, 2010.

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