Semiquartile distance

The use of the mean with either the SD or SEM implies, however, that we have reason to believe that the sample of data being summarized are from a population that is at least approximately normally distributed. If this is not the case, then we should rather use a set of statistical descriptions which do not require a normal distribution. These are the median, for location, and the semiquartile distance, for a measure of dispersion. These somewhat less familiar parameters are characterized as follows. 

Semiquartile Distance. When all the data in a group are ranked, a quartile of the data contains one ordered quarter of the values. Typically, we are most interested in the borders of the middle two quartiles Qx and Q3, which together represent the semiquartile distance and which contain the median as their center. Given that there are N values in an ordered group of data, the upper limit of the jih quartile ( >-) may be computed as being equal to the [(jN — l)/4th] value. Once we have used this formula to calculate the upper limits of Qx and Q3, we can then compute the semiquartile distance (which is also called the quartile deviation, and as such is abbreviated as QD) with the formula QD = (Q3 — Q )/2. 

The fourth and twelfth values in this data set are 4 and 7, respectively. The semiquartile distance can then be calculated as... 

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