Measures of spread: Range

In a nutshell

Measures of spread, also called measures of dispersion or variation, describe how spread out the data is.

Range

The range is the difference between the maximum and the minimum values of the data set.

Interquartile range

The interquartile range is the difference between the upper and the lower quartile.

${IQR=Q_3-Q_1}$

$IQR$	Interquartile range
$Q_3$	Upper quartile
$Q_1$	Lower quartile

Interpercentile range

The interpercentile range is the difference between the values of two chosen percentiles.

Note: The range is affected by extreme values, whereas the interquartile and interpercentile ranges aren't. However, the range takes into account all of the data, while the interquartile range only considers the middle $50\%$ .

Example 1

For the following values, find the range, the interquartile range and the $10th$ to $90th$ interpercentile range.

$3,5,5,7,9,10,12,12,15,16,20,20,21$ 

The range will be given by:

$Maximum-minimum=21-3=\underline{18}$ 

For the interquartile range you have to find each quartile:

 $Q_1=\dfrac{13}{4}=3.25=4th$ value (rounding up).

$Q_3=\dfrac{13\times3}{4}=9.75=10th$ value.

So, the lower quartile will be $7$ and the upper quartile is $16$ .

$Q_3-Q_1=16-7=\underline{9}$ 

Now, find the $10th$ and the $90th$ percentiles:

$0.1\times 13=1.3\approx2nd$ value (rounding up), which means $P_{10}=5$ .

$0.9\times13=11.7\approx 12th$ value, which means $P_{90}=20$ .

$P_{90}-P_{10}=20-5=\underline{15}$ 