**1.0 MEASURES OF VARIATION BASED ON QUANTILES
(REVIEW)**

Quantiles (quartiles, deciles, and percentiles) are measures of variation based on division
of a set of observations (arranged in order by size) into equal intervals and stating the value of observation at the end
of the given interval. Quantiles have an intuitive appeal.

Quartiles are based on dividing observations into 4 equal intervals. Deciles are based
10, quartiles on 4, and percentiles on 100 intervals.

The inter-quartile range, Q_{3 }- Q_{1}, and the semi interquartile range,
˝ (Q_{3 }- Q_{1}) have the advantages of being simple, intuitive, related to the median, and less sensitive
to extreme values.

Quartiles have the disadvantages of being unstable for small samples and not allowing further
mathematical manipulation.

Deciles are rarely used.

Percentiles, also called centile scores, are a form of cumulative frequency and can be
read off a cumulative frequency curve. They are direct and very intelligible.

The 2.5^{th} percentile corresponds to mean - 2SD. The 16^{th} percentile
corresponds to mean - 1SD. The 50^{th} percentile corresponds to mean + 0 SD. The 84^{th} percentile corresponds
to mean + 1SD. The 97.5^{th} percentile corresponds to mean + 2SD. The percentile rank indicates the percentage of
the observations exceeded by the observation of interest. The percentile range gives the difference between the values of
any two centiles.

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**2.0 THE RANGE**

The full range is based on extreme values. It is defined by giving the minimum
and maximum values or by giving the difference between the maximum and the minimum values.

The modified range is determined after eliminating the top 10% and bottom 10%
of observations.

The range has several advantages: it is a simple measure, intuitive, easy to
compute, and useful for preliminary or rough work.

The disadvantages of the range are: it is affected by extreme values, it is
sensitive to sampling fluctuations, and it has no further mathematical manipulation.

**3.0 NUMERICAL RANK**

The numerical rank expresses the observation's position in counting when the
observations are arranged in order of magnitude from the best to the worst.

**4.0 PERCENTILE RANK**

The percentile rank indicates the percentage of the observations exceeded by
the observation of interest.