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1 Finding Sample Variance & Standard Deviation Given: The times, in seconds, required for a sample of students to perform a required task were: Find: a) The sample variance, s 2 6,10,13,11,12,8 b) The sample standard deviation, s Using the Definition Formula

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2 The calculation of a sample statistic requires the use of a formula. In this case, use: (x-x) 2 n -1 Sample variance: s 2 = The Formula - Knowing Its Parts s 2 is “s-squared”, the sample variance (x-x) is the “deviation from mean” x is “x-bar”, the sample’s mean x s2s2 (x-x)

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3 Sample variance: s 2 = (x-x) 2 n -1 (x-x) 2 The Formula - Knowing Its Parts (Cont’d) (Do you have your sample data ready to use?) n -1 is the “sample size less 1” (x-x) 2 is the “sum of all squared deviations” (x-x) 2 is the “squared deviation from the mean” (x-x) 2 n -1

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4 (11-10) 2 (12-10) 2 (8-10) 2 (-4) 2 (0) 2 (3) 2 (-4) 2 (0) 2 (3) 2 10 13( - ) 2 13-10 10 ( - ) 2 10-10 ( - ) 2 8-1012-1011-10 6 106-10 0914416(1) 2 (2) 2 (-2) 2 First, find the numerator: Finding the Numerator = (x-x) 2 n -1 s 2 = +++++ = (x-x) 2 n -1 s 2 = (1) 2 (2) 2 (-2) 2 = +++++ = +++++ = 34 Sample = { 6, 10, 13, 11, 12, 8 } and mean x = 10.0 0914416

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5 - 1 Finding the Denominator Next, find the denominator: Sample = { 6, 10, 13, 11, 12, 8 } 123456 5 n = 6 123456 (x-x) 2 n -1 Sample variance: s 2 == 34 n -1 =666= 5 5 (x-x) 2 n -1 = 34 s 2 =

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6 Finding the Answer (a) Lastly, divide and you have the answer! 6.8 The sample variance is 6.8 Note: Variance has NO unit of measure, it’s a number only 5 (x-x) 2 n -1 = 34 s 2 = = 5 (x-x) 2 n -1 = 34 s 2 =

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7 Finding the Standard Deviation (b) The standard deviation is the square root of variance: s = s 2 Therefore, the standard deviation is: s = s 2 = 6.8 = 2.60768 = The standard deviation of the times is 2.6 seconds Note: The unit of measure for the standard deviation is the unit of the data 2.6

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1 Principles of Cycle Time Reduction Copyright 2000.

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