Quartiles and Measures of Spread

Two cricketers can have the same average score yet be completely different players: one steady, one wildly up and down. The mean alone hides this. Measures of spread tell us how scattered the data is around the centre. The simplest is the range (highest minus lowest), but a single freak value can blow it up. Quartiles give a fairer picture by looking at where the bulk of the data sits.

Arrange the data in increasing order. The median (Q2) splits it into two halves. Now split each half again: the lower quartile Q1 is the middle of the lower half, and the upper quartile Q3 is the middle of the upper half. So Q1, Q2 and Q3 cut the ordered data into four equal parts, each holding one-quarter of the values. For n values arranged in order, Q1 is the (n+1)/4 th term and Q3 is the 3(n+1)/4 th term.

The inter-quartile range is IQR = Q3 − Q1. It is the spread of the middle 50% of the data and ignores the extreme top and bottom quarters, so unusual high or low values do not distort it. Half of this, (Q3 − Q1)/2, is the semi inter-quartile range or quartile deviation. A small IQR means the data is tightly bunched around the centre; a large IQR means it is widely spread.

For grouped data we cannot list every value, so we use an ogive - the cumulative frequency curve. First build a cumulative frequency table, then plot cumulative frequency (y-axis) against the upper boundary of each class (x-axis) and join with a smooth S-shaped curve. To find the median, go to n/2 on the y-axis, move across to the curve and down to read the value. Q1 is read at n/4 and Q3 at 3n/4 in the same way. This is the standard Selina Class 10 method.