Quartiles

In descriptive statistics, the quartiles of a ranked set of data values are the three points that divide the data set into four equal groups, each group comprising a quarter of the data. A quartile is a type of quantile. The first quartile (Q₁) is defined as the middle number between the smallest number and the median of the data set. The second quartile (Q₂) is the median of the data. The third quartile (Q₃) is the middle value between the median and the highest value of the data set.

In applications of statistics such as epidemiology, sociology and finance, the quartiles of a ranked set of data values are the four subsets whose boundaries are the three quartile points. Thus an individual item might be described as being "in the upper quartile".

For discrete distributions, there is no universal agreement on selecting the quartile values.

This rule is employed by the TI-83 calculator boxplot and "1-Var Stats" functions.

The values found by this method are also known as "Tukey's hinges"; see also midhinge.

This always gives the arithmetic mean of Methods 1 and 2; it ensures that the median value is given its correct weight, and thus quartile values change as smoothly as possible as additional data points are added.

Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Ordered Data Set: 7, 15, 36, 39, 40, 41

As there are an even number of data points, all three methods give the same results.

There are methods by which to check for outliers in the discipline of statistics and statistical analysis. As is the basic idea of descriptive statistics, when encountering an outlier, we have to explain this value by further analysis of the cause or origin of the outlier. In cases of extreme observations, which are not an infrequent occurrence, the typical values must be analyzed. In the case of quartiles, the Interquartile Range (IQR) 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 mathematical method to check for outliers and determining "fences", upper and lower limits from which to check for outliers.

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