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 |