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# Quartile

In descriptive statistics, a quartile is one of the three values which divide the sorted data set into four equal parts.

Thus:

• first quartile = lower quartile = cuts off lowest 25% of data = 25th percentile
• second quartile = median = cuts data set in half = 50th percentile
• third quartile = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile

The difference between the upper and lower quartiles is called the interquartile range.

Often it is necessary to interpolate between data values to accomplish this, as in the following example.

```  i    x[i]
1    102
2    105
------------- first quartile, Q1 = (105+106)/2 = 105.5
3    106
4    109
------------- second quartile, Q2 = (109+110)/2 = 109.5
5    110
6    112
------------- third quartile, Q3 = (112+115)/2 = 113.5
7    115
8    118
```
Taking the mean of the values either side of the quartiles is an arbitrary decision: in the example above, the quartiles could be any value in the ranges [105,106], [109,110] and [112, 115].

If the sample size is not a multiple of four, some of the quartiles may be numbers in the original data set, as in this example:

```  i    x[i]
1    102
2    105 -- Q = 105
3    106
------------- Q = 107.5
4    109
5    110 -- Q = 110
6    112
```
In both of the above cases, the first and third quartiles can be taken to be the median values of the lower and upper halves of the data, respectively. However, there is more than one school of thought on how to apply this definition when the overall median is one of the original data values. The next two examples are illustrations of some of the rules of thumb which have been adopted; neither always produces correct results, and it would be better to use a precise formulation as shown later.

One may include the median in both "halves" of the data:

```  i    x[i]
1    102
2    105
3    106 -- Q1 = 106
4    109
5    110
)- Q2 = 110 (note line 5 has been duplicated
5    110              to illustrate the point)
6    112
7    115 -- Q3 = 115
8    118
9    120
```
Or not include the median in either "half":

```  i    x[i]
1    102
2    105
------------- Q1 = 105.5
3    106
4    109
5    110 -- Q2 = 110
6    112
7    115
------------- Q3 = 116.5
8    118
9    120
```
More precise mathematical formulations are possible: the quartiles of the distribution of a random variable X can be defined as the values x such that:

With these definitions the quartiles in the last example are 106, 110 and 115:
```P(X ≤ 106) = 1/3 and P(X ≥ 106) = 7/9;
P(X ≤ 110) = 5/9 and P(X ≥ 110) = 5/9; and
P(X ≤ 115) = 7/9 and P(X ≥ 110) = 1/3.
```