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

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

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[1] = 105
  3    106
------------- Q[2] = 107.5
  4    109
  5    110 -- Q[3] = 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 are two schools of thought on how to apply this definition when the overall median is one of the original data values.

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