Two Formulas a Population Variance b Sample Variance s 2 4 The Standard

Two formulas a population variance b sample variance

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The Variance -- A common measurement of the variability between numbers in a data set. Two Formulas a. Population Variance (   b. Sample Variance (s 2 ) 4. The Standard Deviation – Probably the most common and widely used measure of variability. Two Formulas a. Population Standard Deviation b. Sample Standard Deviation Note: The standard deviation is simply the square root of the variance.
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Module 3: Measures of Variability – Ungrouped Data 5. The Coefficient of Variation (CV) – Another measure of variability that is less common than the variance or standard deviation. The coefficient of variation is always expressed as a percent. a. Population CV  x 100 =  b. Sample CV = x 100 = 
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Module 3: Measures of Variability – Ungrouped Data Example 1: For the following small data set, calculate the following: (5, 9, 16, 17, 18) a. The Population Variance b. The Population Standard Deviation c. The Population Coefficient of Variation d. The Sample Variance e. The Sample Standard Deviation f. The Sample Coefficient of Variation
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Module 3: Measures of Variability – Ungrouped Data Solution for Example 1: For the following small data set, calculate the following: (5, 9, 16, 17, 18) a. 2 = = 26 b. = = 5.0990 c. Population CV = (100) = 39.2231% d. s 2 = = 32.5 e. s = = 5.7009 f. Sample CV = (100) = 43.8531% Note: The sample statistics are larger then the population parameters, reflecting the added uncertainty associated with sample vs. population data. x x 2 5 25 9 81 16 256 17 289 18 324 ∑x = 65 ∑x 2 = 975 mean = 65/5 = 13
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Module 3: Measures of Variability – Ungrouped Data Solution for Example 1 using Excel Note: The data was in rows 12 through 16 in Column A. There is no CV function in Excel. You must divide the standard deviation by the mean and express as a %. x 5 9 16 17 18 5 = Sample Size =COUNT(A12:A16) 13 = Mean =AVERAGE(A12:A16) 26 = 2 =VARP(A12:A16) 5.0990 = =STDEVP(A12:A16) 39.2232% = Population CV =(A21/A19) 32.5 = s 2 =VAR(A12:A16) 5.7009 = s =STDEV(A12:A16) 43.85% = Sample CV =(A24/A19)
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Module 3: Measures of Variability – Ungrouped Data Example 2: Assume the following data is a sample randomly obtained from a population. Calculate: a. Sample Size (n) b. The Sample Mean ( ) c. The Sample Variance (s 2 ) d. The Sample Standard Deviation (s) e. The Sample Coefficient of Variation 66 89 79 62 85 72 88 73 72 83 79 86 82 82 60 81 88 80 82 69 65 69 85 70 51 51 83 82 64 59 77
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Module 3: Measures of Variability – Ungrouped Data Example 2 Solution 51 a. n = 31 51 b. = 74.6452 59 c. s 2 = 116.5032 60 d. s = 10.79367 62 e. Sample CV = 14.4600% 64 65 66 69 69 70 72 72 73 77 79 79 80 81 82 82
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Module 3: Measures of Variability – Ungrouped Data Example 3: Assume the following data is the population of bags of cheese curds, measured in grams, a bags of cheese curds at Middlefield Cheese on a particular day. Calculate: a. The Population Size (N) b. The Population Mean ( ) c. The Population Variance ( 2 ) d. The Population Standard Deviation( ) e. The Population Coefficient of Variation
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Module 3: Measures of Variability – Ungrouped Data Example 3 Solution: 150 a. N = 24 89 b. = 135.0417 112 c. 2 = 542.1233 156 d.
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