Mean, quartiles, déciles, quantiles
Median
Définition :
The median of statistical series is the value of the series that splits the population in two equal parts. In other terms, within a population, there is the same number of observations above the median than below the median.
Exemple : GMAT tests results
Let's look at the GMAT tests results of a class
in order to determine the median of the grades, we must first sort the grades (in ascending or descending order indifferently).
There are 29 grades, so the median will correspond to the 15th value in the sorted series. Indeed, there are 14 preceding values and 14 following values to the value 710, which is the median of the series.
Unfortunately, real life will not necessarily provide you with an odd number of observations. The generalisation of the definition of the median is as follows:
Let us have a statistical series with n values.
When n is odd, the median will have the value ranked 
in the sorted series.
When n is even, there is no value in the series corresponding exactly to the definition above. We can only refer to a median interval limited by the value ranked 
and the value ranked 
. The median interval is thus
.
Sometimes, it is possible to use the value 
as the median value (but do remember it is not part of your series).
Quantiles, quartiles, deciles
Définition : Quantiles
The notion of quantiles is very close to that of the median we defined in paragraph 1.3.
The purpose of the median is to split a population in two parts with the same number of items or individuals. Extending this idea, we now imagine values splitting a population in four equal parts.
Consider the following example where the grades (out of 50) of a exam are sorted in ascending order:
Exemple :
We saw the median would be 
.
By definition, the first quartile Q1 is the number such that one quarter of the population is ranked below this it, the second quartile Q2 is the number such that two quarters of the population are ranked below it, and the third quartile Q3 is the number such that three quarters of the population are ranked below it.
In our example we have 
.
We note that Q2 is by definition equal to the median.
In a similar manner, we define deciles as values that divide a population in ten equal parts and percentiles the values dividing a population in one hundred equal parts.
Generally speaking, kth order quantiles are the values dividing the data set in k equal parts.
The attentive reader will note that the fifth decile and the fiftieth percentile are equal to the median, and that the twenty-fifth and seventy-fifth percentiles are respectively equal to the first and third quartile.
We suggest you, as an exercise, to determine the deciles for the above series.
