# Range

8th February 2020The Range of a distribution gives a measure of the width (or the spread) of the data values of the corresponding random variable. For example, if there are two random variables X and Y such that X corresponds to the age of human beings and Y corresponds to the age of turtles, we know from our general knowledge that the variable corresponding to the age of turtles should be larger.

Since the average age of humans is 50-60 years, while that of turtles is about 150-200 years; the values taken by the random variable Y are indeed spread out from 0 to at least 250 and above; while those of X will have a smaller range. Thus, qualitatively you’ve already understood what the Range of a distribution means. The mathematical formula for the same is given as:

Range=L–S

where L – the largets/maximum value attained by the random variable under consideration and S – the smallest/minimum value.

**Properties**

- The Range of a given distribution has the same units as the data points.
- If a random variable is transformed into a new random variable by a change of scale and a shift of origin as –

Y = aX + b

where Y – the new random variable, X – the original random variable and a,b – constants. Then the ranges of X and Y can be related as –

R_{Y} = |a|R_{X}

Clearly, the shift in origin doesn’t affect the shape of the distribution, and therefore its spread (or the width) remains unchanged. Only the scaling factor is important.

- For a grouped class distribution, the Range is defined as the difference between the two extreme class boundaries.
- A better measure of the spread of a distribution is the Coefficient of Range, given by:

Coefficient of Range (expressed as a percentage)=L–SL+S×100

Clearly, we need to take the ratio between the Range and the total (combined) extent of the distribution. Besides, since it is a ratio, it is dimensionless, and can, therefore, one can use it to compare the spreads of two or more different distributions as well.

- The range is an
*absolute measure*of Dispersion of a distribution while the Coefficient of Range is a*relative measure*of dispersion.

Due to the consideration of only the end-points of a distribution, the Range never gives us any information about the shape of the distribution curve between the extreme points. Thus, we must move on to better measures of dispersion. One such quantity is Mean Deviation which is we are going to discuss now.