Interquartile Range of Distributions
The interquartile range of a continuous distribution can be calculated by integrating the probability density function (which yields the cumulative distribution function — any other means of calculating the CDF will also work). The lower quartile, Q1, is a number such that integral of the PDF from -∞ to Q1 equals 0.25, while the upper quartile, Q3, is such a number that the integral from -∞ to Q3 equals 0.75; in terms of the CDF, the quartiles can be defined as follows:
where CDF−1 is the quantile function.
The interquartile range and median of some common distributions are shown below
| Distribution | Median | IQR |
|---|---|---|
| Normal | μ | 2 Φ−1(0.75) ≈ 1.349 |
| Laplace | μ | 2b ln(2) |
| Cauchy | μ |
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