32  Measures of Distribution

Beyond center and spread, a dataset’s shape matters too. Two districts can have the same mean and standard deviation for yield, yet one has most farms clustered near the average with a few severe crop failures dragging the tail down, while the other has most farms modest but a handful of standout performers pulling the tail up. Skewness and kurtosis are the two measures that capture this shape — asymmetry and “tailedness,” respectively.

32.1 Skewness

Skewness measures the degree of asymmetry in a distribution. A distribution is symmetrical if it looks the same on both sides of its center.

  • Zero skewness: a perfectly symmetrical distribution.
  • Positive skewness: a longer tail stretching toward higher values (right-skewed).
  • Negative skewness: a longer tail stretching toward lower values (left-skewed).

Formula for skewness (matching what R’s moments::skewness() computes):

\[ Skewness = \frac{\frac{1}{N}\sum_{i=1}^{N} (X_i - \overline{X})^3}{\left(\frac{1}{N}\sum_{i=1}^{N} (X_i - \overline{X})^2\right)^{3/2}} \]

Where \(N\) is the number of observations, \(X_i\) is each individual observation, and \(\overline{X}\) is the mean.

  • Skewness > 0 → positively skewed (right-skewed)
  • Skewness = 0 → symmetric
  • Skewness < 0 → negatively skewed (left-skewed)

32.1.1 Three Yield Distributions Compared

The three datasets below all represent crop yield (tons/ha) across 23 farms in a district, but with different shapes: one roughly symmetric, one with a handful of crop failures pulling the left tail down, and one with a handful of standout high-yield farms pulling the right tail up.

Roughly Symmetric

Close to zero, confirming a roughly symmetric spread of yields around the average.

Left-Skewed (Crop Failures)

A clearly negative skewness — most farms performed well, but a handful of crop failures stretch the distribution’s tail toward the low end.

Right-Skewed (Standout Performers)

A clearly positive skewness — most farms yielded modestly, but a few standout farms (better seed, precision irrigation) stretch the tail toward the high end.

32.2 Kurtosis

Kurtosis measures the “tailedness” of a distribution — how much of the data sits in the tails versus the peak, relative to a normal distribution.

  • Mesokurtic (kurtosis ≈ 3): tail weight similar to a normal distribution.
  • Leptokurtic (kurtosis > 3): heavier tails and a sharper peak than normal — more extreme values (outliers) than a normal distribution would predict.
  • Platykurtic (kurtosis < 3): lighter tails and a flatter peak than normal — fewer extreme values.

Formula for kurtosis (matching what R’s moments::kurtosis() computes — note this is raw kurtosis, where a normal distribution scores 3, not excess kurtosis, where a normal distribution scores 0):

\[ Kurtosis = \frac{\frac{1}{N}\sum_{i=1}^{N} (X_i - \overline{X})^4}{\left(\frac{1}{N}\sum_{i=1}^{N} (X_i - \overline{X})^2\right)^{2}} \]

  • Kurtosis > 3 → leptokurtic (heavy tails)
  • Kurtosis = 3 → mesokurtic (normal-like)
  • Kurtosis < 3 → platykurtic (light tails, flatter)

32.2.1 Calculation in R

Kurtosis on the same three yield datasets used for skewness above:

The roughly symmetric dataset comes out below 3 (platykurtic — a fairly flat, tightly clustered distribution), while both the left- and right-skewed datasets come out well above 3 (leptokurtic) — the crop failures and standout performers that create the skew are exactly the kind of extreme values that drive kurtosis up.

32.2.2 Application

  • Risk assessment: skewness and kurtosis on a season’s yield or price data help a cooperative gauge how likely extreme outcomes — a bumper harvest, a severe shortfall — really are, beyond what the mean and standard deviation alone suggest.
  • Quality control: in a seed-processing or grading operation, these measures flag when a batch’s characteristics deviate from the expected shape, not just the expected average.
  • Climate analysis: rainfall and temperature records are rarely symmetric — skewness and kurtosis help characterize how often and how severely a region experiences extreme weather.

Summary

Concept Description
Measures of Distribution
Skewness Measures the asymmetry of a distribution: positive skew has a longer right tail, negative skew a longer left tail
Kurtosis Measures the tailedness of a distribution relative to normal (kurtosis = 3): leptokurtic (>3) has heavier tails, platykurtic (<3) has lighter tails
Application of Skewness and Kurtosis Used in risk assessment, quality control, and climate analysis to characterize extreme outcomes beyond the mean and standard deviation