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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; the interquartile range is a relatively robust statistic (also sometimes called "resistance") compared to the range and standard deviation. There is also a ...
The IQR, mean, and standard deviation of a population P can be used in a simple test of whether or not P is normally distributed, or Gaussian. If P is normally distributed, then the standard score of the first quartile, z 1, is −0.67, and the standard score of the third quartile, z 3, is +0.67.
The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point.
Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value.. For example, robust estimators of scale are used to estimate the population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.
Tukey promoted the use of five number summary of numerical data—the two extremes (maximum and minimum), the median, and the quartiles—because these median and quartiles, being functions of the empirical distribution are defined for all distributions, unlike the mean and standard deviation; moreover, the quartiles and median are more robust ...
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 statistics, the quartile coefficient of dispersion is a descriptive statistic which measures dispersion and is used to make comparisons within and between data sets. Since it is based on quantile information, it is less sensitive to outliers than measures such as the coefficient of variation .
The addition of the deciles allow one to compute the interdecile range, which for a normal distribution can be scaled to give a reasonably efficient estimate of standard deviation, and the 10% midsummary, which when compared to the median gives an idea of the skewness in the tails.