1. QBA 260 Business Statistics Chapter 5

2. 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?

3. Comparing Datasets Mean (Average) for both datasets is 47.4 inches Variances differ: Dataset 1 = 1.78 sq. inches, Dataset 2 = 51.03 sq. inches Mean

4. 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

5. 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 4847.40.56 4647.4-1.44 4647.4-1.44 4947.41.56 4947.41.56 4747.4-0.44 4747.4-0.44 4647.4-1.44 4947.41.56 47.40.00 (Average) (Sum)

6. 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 4747.4-0.44 5147.43.56 4947.41.56 5347.45.56 4747.4-0.44 4547.4-2.44 3747.4-10.44 3847.4-9.44 6047.412.56 47.40.00 (Average) (Sum)

7. Populations and Samples Population variance Sample variance

8. Sample Variance – Dataset 1 Sample variance XiXi X-BarDeviationSq Dev 4847.40.560.31 4647.4-1.442.09 4647.4-1.442.09 4947.41.562.42 4947.41.562.42 4747.4-0.440.20 4747.4-0.440.20 4647.4-1.442.09 4947.41.562.42 47.40.0014.22 (Average) (Sum) Sq Inches

9. Sample Variance – Dataset 2 Sample variance XiXi X-BarDeviationSq Dev 4747.4-0.440.20 5147.43.5612.64 4947.41.562.42 5347.45.5630.86 4747.4-0.440.20 4547.4-2.445.98 3747.4-10.44109.09 3847.4-9.4489.20 6047.412.56157.64 47.40.00408.22 (Average) (Sum) Sq Inches

10. 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

11. Standard Deviation Square root of the variance

12. 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)

13. 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)

14. Average Absolute Deviation XiXi X-BarDeviationABS 4847.40.56 4647.4-1.441.44 4647.4-1.441.44 4947.41.56 4947.41.56 4747.4-0.440.44 4747.4-0.440.44 4647.4-1.441.44 4947.41.56 47.40.0010.44 (Average) (Sum) Absolute Deviation = 10.44/9 = 1.16 EXCEL: AVEDEV