Interquartile range
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In descriptive statistics, the interquartile range (IQR) is the range between the third and first quartiles and is a measure of statistical dispersion. The interquartile range is a more stable statistic than the (total) range, and is often preferred to the latter statistic.
Since 25% of the data are less than or equal to the first quartile and 25% are greater than or equal to the third quartile, the IQR is expected to include about half of the data. The length of the IQR should be measured in the same units as the data.
One should note that, in ungrouped data(like in the example below), Q2 should be the median of the data. Following the Q2 (Q3 or Q4) the equation should be as such: **Q2x1.5** for Q3 and **Q3x0.5** for Q2.
Interquartile range is used to build Box plots, that can give a simple graphical representation of a probability distribution.
[edit] Example
i x[i]
1 102
2 104
3 105 ---- the first quartile, Q1 = 105
4 107
5 108
6 109 ---- the second quartile, Q2 or median = 109
7 110
8 112
9 115 ---- the third quartile, Q3 = 115
10 115
11 118
From this table, the length of the interquartile range is 115 - 105 = 10.
The median is the corresponding measure of central tendency.
[edit] Interquartile range of distributions
The interquartile range of a continuous distribution can be calculated by integrating the pdf. The lower quartile, a, is the integral from minus infinity to a that equals 0.25, while the upper quartile, b, is the integral from b to infinity that equals 0.75.
[insert equations here]
The interquartile range and median of some common distributions are shown below
| Distribution | Median | IQR |
|---|---|---|
| Normal | μ | 2Φ-1(0.75)≈ 1.349 |
| Laplace | μ | |
| Cauchy | μ |
