LESSON 2: Measures of Dispersion

Measures of Dispersion

The Range 

To measure the spread or dispersion of data, we must introduce statistical values known as the range and the standard deviation.

Solution: 

The greatest number of ounces dispensed is 10.07 and the least is 5.85. The range of the numbers of ounces dispensed is 10.07 – 5.85 = 4.22 oz.

The Standard Deviation 

        The range of a set of data is easy to compute, but it can be deceiving. The range is a measure that depends only on the two most extreme values, and as such it is very sensitive. A measure of dispersion that is less sensitive to extreme values is the standard deviation.

        Because the sum of all the deviations of the data values from the mean is always 0, we cannot use the sum of the deviations as a measure of dispersion for a set of data. Instead, the standard deviation uses the sum of the squares of the deviations.

        Most statistical applications involve a sample rather than a population, which is the complete set of data values. Sample standard deviations are designated by the lowercase letter s. In those cases in which we do work with a population, we designate the standard deviation of the population by s, which is the lowercase Greek letter sigma. 

We can use the following procedure to calculate the standard deviation of n numbers.

        The batteries from Dependable have the smallest standard deviation. According to these results, the Dependable company produces the most consistent batteries with regard to life expectancy under constant use. 

The Variance 

        A statistic known as the variance is also used as a measure of dispersion. The variance for a given set of data is the square of the standard deviation of the data. The following chart shows the mathematical notations that are used to denote standard deviations and variances.