QBA 260 Business Statistics Chapter 5

Deviating from the Average Compare two datasets (reported in inches) Dataset 1: 48, 46, 46, 49, 49, 47, 47, 46, 49 Dataset 2: 47, 51, 49, 53, 47, 45, 37, 38, 60 What is the average of each dataset? How are they different?

Comparing Datasets Mean (Average) for both datasets is 47.4 inches Variances differ: Dataset 1 = 1.78 sq. inches, Dataset 2 = sq. inches Mean

Variance and Deviations Variance is a measure of how much the data points in a sample deviate from the sample mean For data: x 1, x 2, x 3,…, x n A deviation is the difference between a data value and a central value such as the average

Deviations Dataset 1: 48, 46, 46, 49, 49, 47, 47, 46, 49 Average = 47.4 Deviations The average deviation is (always) zero XiXi X-BarDeviation (Average) (Sum)

Deviations Dataset 2: 47, 51, 49, 53, 47, 45, 37, 38, 60 Average = 47.4 Deviations The average deviation is (always) zero XiXi X-BarDeviation (Average) (Sum)

Populations and Samples Population variance Sample variance

Sample Variance – Dataset 1 Sample variance XiXi X-BarDeviationSq Dev (Average) (Sum) Sq Inches

Sample Variance – Dataset 2 Sample variance XiXi X-BarDeviationSq Dev (Average) (Sum) Sq Inches

Variance and Standard Deviation The calculated variance is in a different unit of measure than the original dataset Original Dataset measured in inches Variance measured in inches squared (due to the use of squared deviations before we average them) To accommodate this, we calculate the standard deviation which is the square root of the Variance

Standard Deviation Square root of the variance

Excel Functions VARP(…) Population Variance VARPA(…) Population Variance (includes cells that contain true/false data) VAR(…) Sample Variance VARA(…) Sample Variance (includes cells that contain true/false data)

Excel Functions STDEVP(…) Population Standard Deviation STDEVPA(…) Population Standard Deviation (includes cells that contain true/false data) STDEV(…) Sample Standard Deviation STDEVA(…) Sample Standard Deviation (includes cells that contain true/false data)

Average Absolute Deviation XiXi X-BarDeviationABS (Average) (Sum) Absolute Deviation = 10.44/9 = 1.16 EXCEL: AVEDEV