The quartiles are three values which divide the sorted data set into four (approximately) equal parts. We cannot divide a set of 17 (say) data values into four equal parts since that would require splitting a datum. What we can do is interpolate between data items which bracket the desired position.
i x[i] 1 102 2 105 ----------- the first quartile, Q[1] = (105+106)/2 = 105.5 3 106 4 109 ------------ the second quartile or median or Q[2] = 109.5 5 110 6 112 ------------ the third quartile, Q[3] = (112+115)/2 = 113.5 7 115 8 118