You asked for it.....

"The quartiles allow for a more detailed description of a distribution than it is possible with only one measure of location, like the median, for example. Analogous to the latter the three quartiles are defined as those values which divide the distribution into four parts containing ideally all the same amount of data. Like the median, the quartiles can only be determined on at least an ordinal level of scale. Simply speaking, the lower, middle and upper quartile QR(x) are defined as those values being larger than 25%, 50% and 75% of the remaining data as well as smaller than 75%, 50% and 25% of the remaining data, resp. Thus, each quartile cuts the set of data into two sub-sets, which comprise proportions of x resp. 1-x of the remaining data, whereby x takes values of 0.25 (25%, first or lower quartile), 0.5 (50%, second or middle quartile, i.e. median) and 0.75 (75%, third or upper quartile). Thus, a proportions ratio pr = x / (1-x) can be defined, which ideally takes the values of 1/3, 1 and 3 for the lower, middle and upper quartile, resp.
When determining the quartiles, problems similar to those for the median will usually occur, because most probably the set won't contain a variable value that divides the distribution exactly into proportions of x and 1-x, resp. Then either that variable value can be taken as quartile which leads to proportions most closely to the required ones (this is the only possible strategy on an ordinal level of scale). Thus an error in the proportions has to be accepted but the quartile is realised by a measured value. Or an interpolation rule can be applied onto those two variable values leading to the two proportions closest to the required ones. Thus only a virtual value for the quartile is calculated but it has the advantage of diving the distribution exactly as it was required. The less data a set contains, the bigger these problems become.
The definition of the quartiles implies a generalisation to those proportions with no restriction on the value x, these are called quantiles."

Taken from: http://www.drgst.de/STAT/2_3-quartil-en.html

Translation: Quartiles, by definition, are 4 separate groups of quantifiable data; each with the same number of data points. To work out what the quartiles are, one lists the scores/values/data points in order, from top to bottom then splts the data set in half - and then the halves in half again.

In total the 4 quartiles constitute one's data population.

Despite the fact there's four quartiles (upper, upper middle, lower middle & lower), they are usually treated as three, upper (or top), middle and lower (or bottom).

This usage is valid for statistical purposes but grammatically inaccurate in that the "middle quartile" is not a quartile at all.

In this case, the upper quartile consists of the the highest 25% of scores, the lower quartile consists of the lowest 25% of scores and the middle quartile contains the remaining 50% of scores - those that fall 25% above and below the middle or median value.

phew

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