Measures of Variability for Ungrouped Data

Measures of Variability For Ungrouped Data

A measure of central location, like the mean, allows us to identify the point where most of the data tend to cluster. From this central point, another measure is needed to provide an adequate description on how the data scatters or deviates away from the mean. A measure of variability, a numerical value computed from the given observations measures how the data spread from the central location. This is often used in comparing two sets of data. The lesser the measure is, the closer the values of the observations from the central value. To illustrate this idea, consider the two sets of data given below:

Data Set A | 78 79 80 81 82 |

Data SetB | 70 75 80 85 90 |

Verify that the two sets of data have equal means which is 80. However, a closer look at the data set will show that there is a difference in the variability between the two data sets. Data Set A has values that are more clustered around the mean. Data Set B has values that are more scattered from the mean. Thus, it can be said that the “measure of variability” for data Data Set B is larger than the “measure of variability for Data Set A. Hence, “Data Set B is more scattered than Data Set A”.

Range

The range of a set of observations is the difference between the largest and the smallest values in the set. It is denoted by R.

R = Highest value – Lowest value

Consider the two data sets given above. For Data Set A, the range is 82 – 78 = 4, while the range of Data Set B is 90 – 70 = 20. This means that Data Set B is more scattered or more dispersed or more variable than Data Set A because Data Set B has a larger numerical value of the range than Data Set A.

Variance

Given the finite population observations x1, x2, …, xn, the population variance is:

(population variance)

For ease...