Measures of Variation

Range, Variance, & Standard Deviation

Use of Variance and Standard Deviation Useful when comparing data sets to determine which is more variable. Used to determine the consistency of a variable. Example: If you’re producing a product and the length from the first to the 1,000 th has a variation of 2mm production is going well. If larger may have a problem with production. Used to determine the number of data values that fall within a specified interval in a distribution. Used quite often in inferential statistics.

Steps to find Variance of Raw Data 1: Find the squares of each value, in the data set, and add them together. (To be used later) 2: Find the sum of your original values and square that answer. 3: Divide Step 2 by sample size. 4: Answer of Step 1 – Answer of Step 3 5: Divide answer of Step 4 by the (sample size – 1) s² = ∑X² - {(∑X)² ÷ n} n – 1

Steps to find the Standard Deviation of Data 1: Take the variance of the data and find it’s square root. s = √ [ ∑X² - ∑(X)² ÷ n] ÷ (n – 1)

Example Find the variance and standard deviation of the following set of data: 16, 17, 11, 22, 20, 15, 13, 19, 9 s² = ∑X² - {(∑X)² ÷ n} n – 1

Finding Variance of Ungrouped Data 1: Square each value of the set and multiply it by the frequency. (Column D) 2: Find the sum of Step 1. (To be used later) 3: Multiply each value of the set by the frequency. (Column C) 4: Find the sum of Step 3 5: Take answer from Step 4 and square it. 6: Take answer from Step 5 and divide it by total frequency. 7: Step 2 answer – Step 6 answer 8: Step 7 divided by (sample size -1)

Formula for Variance and Standard Deviation of Ungrouped Data s² = ∑ f ∙ X² - [ (∑f ∙ X)² ÷ n] n – 1 s = √ Answer above

Example Find the variance and standard deviation for the following data set. s² = ∑ f ∙ X² - [ (∑f ∙ X)² ÷ n] n – 1

Steps to find Variance of Grouped Data 1: Find the midpoint of each class. (Column C) 2: Square the midpoint and multiply it by the frequency of the class. (Column E) 3: Find the sum of Column E. (To be used later) 4: Multiply the midpoints by the frequency of the class. (Column D) 5: Find the sum of Column D. 6: Square the sum of Column D and divide it by the total frequency. 7: Step 3 answer – Step 6 answer 8: Step 7 answer divided by (sample size – 1)

Variance and Standard Deviation formulas for Grouped Data s² = ∑ f ● X² m - [ (∑ f ● X m )² ÷ n] n – 1 s = √ Answer above