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Objective 3-3 Describe data using the measures of variation, such as the range, variance, and standard deviation. 3

Distribution Shapes 3-49 Frequency distributions can assume many shapes. The three most important shapes are positively skewed, symmetrical, and negatively skewed. 49

Positively Skewed 3-50 Y X M o d e < M e d i a n < M e a n P o s

Symmetrical 3-51 Y X M e a n = M e d i a n = M o d e S y m m e t r i c

Negatively Skewed 3-52 Y X M n < M e d i a n < M o d e N e g a t

3-3 Measures of Variation - Range 3-53 The range is defined to be the highest value minus the lowest value. The symbol R is used for the range. R = highest value – lowest value. Extremely large or extremely small data values can drastically affect the range. 53

3-3 Measures of Variation - Range Now you try: Consider the following data to constitute the population: 10, 60, 50, 30, 40, 20. Find the range.

3-3 Measures of Variation - Population Variance 3-54 54

3-3 Measures of Variation - Population Standard Deviation 3-55 55

3-3 Measures of Variation - Example 3-56 Consider the following data to constitute the population: 10, 60, 50, 30, 40, 20. Find the mean and variance. The mean  = (10 + 60 + 50 + 30 + 40 + 20)/6 = 210/6 = 35. 56

3-3 Measures of Variation - Example 3-57 The variance  2 = 1750/6 = 291.67. 57

3-3 Population Variance - You Try You have found the following ages (in years) of 4 turtles. The turtles are the entire population of turtles at your local zoo. 46, 67, 105, 57 Based on your sample, what is the average age of the turtles? What is the estimated variance of the ages?

3-3 Measures of Variation - Sample Variance 3-58 The unbias ed estimat or of the population variance o r the samp le varianc e is a statistic whose valu e approxim ates the expected v alue of a variance. It is deno ted by s 2 , ( ) where s X n and sample mean size = 1 - å 58

3-3 Measures of Variation - Sample Standard Deviation 3-59 The sample standard deviation is the squ are root of t he sample variance. å ( X - X ) 2 s = s = 2 . n - 1 59

3-3 Sample Variance – You Try 3-61 Find the variance and standard deviation for the following sample: 16, 19, 15, 15, 14. 61