Dispersion (also called variability, spread or scatter) describes how far the values of a data set are from each other and from their centre. Two classes can have the same average mark while one has marks clustered between 60 and 70 and the other ranges from 20 to 100; only a measure of dispersion tells them apart.
There are several measures, each with its own strengths. The range is the simplest but depends on just two values. The interquartile range and quartile deviation use the middle half of the data and resist outliers. The variance and standard deviation use every observation and are the basis of most statistical methods; the mean absolute deviation is easier to explain but less common in formulas. The coefficient of variation expresses the standard deviation as a percentage of the mean so that data on different scales can be compared.
Choose sample or population: samples divide the squared deviations by n − 1, populations by N. The quartile method can also be changed.
How to use the Dispersion Calculator
- Data set
- Data are a
- Quartile method
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How it works
The data are sorted and summarised: extremes give the range, quartiles give the IQR and quartile deviation, deviations from the mean give the variance, standard deviation and mean absolute deviation, and SD ÷ mean gives the coefficient of variation.
Worked example
Eight daily commute times in minutes: 12, 15, 15, 18, 21, 24, 26, 30. Mean = 161 ÷ 8 = 20.125, median = 19.5, mode = 15. Range = 30 − 12 = 18. Q1 = 15 and Q3 = 25, so IQR = 10 and the quartile deviation is 5. Σ(x − x̄)² = 270.875; as a sample s² = 270.875 ÷ 7 = 38.70 and s = 6.22 minutes (as a population σ² = 33.86 and σ = 5.82). The mean absolute deviation is 5.125 and CV = 30.9%.
Measures of dispersion compared
| Measure | Formula | Units | Robust to outliers |
|---|---|---|---|
| Range | max − min | data units | No |
| IQR | Q3 − Q1 | data units | Yes |
| Quartile deviation | (Q3 − Q1) ÷ 2 | data units | Yes |
| Mean absolute deviation | Σ|x − x̄| ÷ n | data units | Partly |
| Variance | Σ(x − x̄)² ÷ (n − 1) | squared units | No |
| Standard deviation | √variance | data units | No |
| Coefficient of variation | s ÷ x̄ | none (%) | No |
Formula
Range = max − min · IQR = Q3 − Q1 · QD = IQR ÷ 2 · s² = Σ(x − x̄)² ÷ (n − 1) or σ² = Σ(x − μ)² ÷ N · CV = s ÷ x̄
Worked example
Inputs — Data set: 12, 15, 15, 18, 21, 24, 26, 30; Data are a: Sample (divide by n − 1); Quartile method: Median of halves, median excluded (TI-84, most textbooks).
Results — Standard deviation (s or σ) = 6.2206; Variance (s² for a sample, σ² for a population) = 38.6964; Range = 18; Interquartile range (IQR) = 10; Quartile deviation (IQR ÷ 2) = 5; Mean absolute deviation = 5.125.
Key terms
- Dispersion
- The spread or variability of data.
- Variance
- Average squared deviation from the mean.
- Standard deviation
- Square root of the variance, in the original units.
- Quartile deviation
- (Q3 − Q1) ÷ 2.
- Coefficient of variation
- Standard deviation divided by the mean.
Frequently asked questions
What is dispersion in statistics?
How spread out the data are. Common measures are the range, IQR, variance, standard deviation and mean absolute deviation.
Which measure of dispersion is best?
The standard deviation for roughly symmetric data without outliers; the IQR for skewed data or data with outliers.
What is the quartile deviation?
Half the interquartile range. For 12, 15, 15, 18, 21, 24, 26, 30 it is (25 − 15)/2 = 5.
How do I get the standard deviation from the variance?
Take the square root. A variance of 182.2 gives a standard deviation of √182.2 ≈ 13.5.
Why do sample and population variance differ?
The sample variance divides by n − 1 to correct the bias from estimating the mean; the population variance divides by N. For the example they are 38.70 and 33.86.
What does a coefficient of variation of 31% mean?
The standard deviation is about 31% of the mean – a moderate amount of variation for most measurements.
Can dispersion be zero?
Yes, only when every value is the same.