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Interquartile range boxplot (with an interquartile range) and a probability density function (pdf) of a normal n (0,σ2) population in descriptive statistics, the interquartile range (iqr) is a measure of statistical dispersion, which is the spread of the data The sample minimum (smallest observation) the lower quartile or first quartile the median (the middle value) the upper quartile or third quartile the sample maximum (largest observation) in addition to the median of a single set of. [1] the iqr may also be called the midspread, middle 50%, fourth spread, or h‑spread.

The interquartile range (iqr), defined as the difference between the upper and lower quartiles ( ), may be used to characterize the data when there may be extremities that skew the data It consists of the five most important sample percentiles The interquartile range is a relatively robust statistic (also sometimes called resistance) compared to the range and standard deviation.

Common robust estimators one of the most common robust measures of scale is the interquartile range (iqr), the difference between the 75th percentile and the 25th percentile of a sample

Other trimmed ranges, such as the interdecile range (10% trimmed range) can also be used. The difference between upper and lower quartiles is also called the interquartile range, midspread or middle fifty → iqr = q3 − q1. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range For instance, when the variance of data in a set is large, the data is widely scattered

On the other hand, when the variance is small, the data in the set is clustered. In descriptive statistics, the range of a set of data is size of the narrowest interval which contains all the data It is calculated as the difference between the largest and smallest values (also known as the sample maximum and minimum) The cauchy distribution is the probability distribution with the following cumulative distribution function (cdf)

And the quantile function (inverse cdf) of the cauchy distribution is it follows that the first and third quartiles are , and hence the interquartile range is

For the standard distribution, the cumulative distribution function simplifies to arctangent function :

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