31 Measures of Dispersion
Measures of dispersion describe the spread or variability within a dataset. Unlike measures of central tendency (mean, median, mode), which summarize data with a single central value, measures of dispersion reveal how much the data varies around that center — how “spread out” it is. Understanding variability matters as much as knowing the average: two districts can report the same mean yield while one has every farm clustered tightly around it and the other swings wildly from bumper harvests to near-total loss. The primary measures of dispersion are the range, interquartile range (IQR), variance, standard deviation, and mean absolute deviation (MAD).
The worked example below uses fertilizer application (kg) recorded across five farms: 90, 120, 150, 180, 210.
31.1 Range
The range is the simplest measure of dispersion: the difference between the maximum and minimum values.
Example: for the fertilizer dataset {90, 120, 150, 180, 210}, the range is \(210 - 90 = 120\) kg.
31.2 Interquartile Range (IQR)
The IQR measures the middle spread of the data — the central 50% of values — as the difference between the 75th percentile (Q3) and the 25th percentile (Q1). Because it ignores the extreme top and bottom quarters, it’s far less sensitive to outliers than the range.
Example: for {90, 120, 150, 180, 210}, Q1 is 120 and Q3 is 180, so the IQR is \(180 - 120 = 60\) kg.
31.3 Variance
Variance measures the average of the squared differences from the mean, capturing how far the data points typically deviate from it. The formula differs slightly between a full population and a sample.
- Population variance (\(\sigma^2\)): \(\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\)
- Sample variance (\(s^2\)): \(s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}\)
Example: for {90, 120, 150, 180, 210}, with a mean of 150, the sample variance is:
\[s^2 = \frac{(90-150)^2 + (120-150)^2 + (150-150)^2 + (180-150)^2 + (210-150)^2}{5-1} = \frac{9000}{4} = 2250\]
31.4 Standard Deviation
The standard deviation is the square root of the variance, expressed in the same units as the original data — kilograms of fertilizer, rather than kilograms squared. It’s the most commonly used measure of dispersion precisely because it’s easy to interpret.
- Population standard deviation (\(\sigma\)): \(\sigma = \sqrt{\sigma^2}\)
- Sample standard deviation (\(s\)): \(s = \sqrt{s^2}\)
Example: continuing from the variance example, the sample standard deviation of {90, 120, 150, 180, 210} is \(\sqrt{2250} \approx 47.43\) kg.
31.5 Mean Absolute Deviation (MAD)
Mean absolute deviation measures the average distance between each data point and the mean, ignoring direction (positive or negative). It’s a more intuitive, if less commonly used, alternative to variance and standard deviation.
Example: for {90, 120, 150, 180, 210} with a mean of 150, the MAD is:
\[MAD = \frac{|90-150| + |120-150| + |150-150| + |180-150| + |210-150|}{5} = \frac{180}{5} = 36 \text{ kg}\]
31.5.1 Calculation in R
A standard deviation of about 47 kg on a mean of 150 kg tells a cooperative that fertilizer use varies quite a bit from farm to farm — worth investigating before treating “150 kg” as a one-size-fits-all recommendation.
Summary
| Concept | Description |
|---|---|
| Measures of Dispersion | |
| Range | The difference between the maximum and minimum values; the simplest measure of spread |
| Interquartile Range (IQR) | The difference between the 75th and 25th percentiles, covering the central 50% of the data |
| Variance | The average of the squared deviations from the mean |
| Standard Deviation | The square root of the variance, expressed in the same units as the original data |
| Mean Absolute Deviation (MAD) | The average absolute distance of each point from the mean |