Finding Sample Variance & Standard Deviation

Finding Sample Variance & Standard Deviation Using the Definition Formula • Given: The times, in seconds, required for a sample of students to perform a required task were: 6, 10, 13, 11, 12, 8 • Find: a) The sample variance, s2 b) The sample standard deviation, s

The Formula - Knowing Its Parts  (x-x)2 Sample variance: s2 = x (x-x) n -1 s2 • x is “x-bar”, the sample’s mean • (x-x) is the “deviation from mean” • The calculation of a sample statistic requires the use of a formula. In this case, use: • s2 is “s-squared”, the sample variance

The Formula - Knowing Its Parts (Cont’d)  (x-x)2 Sample variance: s2 = (x-x)2  (x-x)2 n -1 n -1 • (x-x)2 is the “squared deviation from the mean” •  (x-x)2 is the “sum of all squared deviations” • n -1 is the “sample size less 1” (Do you have your sample data ready to use?)

Finding the Numerator  (x-x)2 s2 = = n -1 Sample = { 6, 10, 13, 11, 12, 8 } and mean x = 10.0  (x-x)2 + + + + + s2 = = n -1 + + + + + + + + + + = = = 16 0 9 1 4 4 • First, find the numerator: ( - )2 6 ( - )2 10 ( - )2 13 6-10 10 10-10 13-10 10 (11-10)2 11-10 (12-10)2 12-10 (8-10)2 8-10 10 (-4)2 (0)2 (3)2 (-4)2 (0)2 (3)2 (1)2 (1)2 (2)2 (2)2 (-2)2 (-2)2 16 0 9 1 4 4 34

Finding the Denominator  (x-x)2 34 Sample variance: s2 = = n -1 n -1 = n = 6 1 2 3 4 5 6  (x-x)2 34 s2 = = n -1 • Next, find the denominator: Sample = { 6, 10, 13, 11, 12, 8 } 5 6 6 6 - 1 = 5 1 2 3 4 5 6 5

Finding the Answer (a)  (x-x)2  (x-x)2 34 34 = = s2 = n -1 n -1 5 5 = s2 = • 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

Finding the Standard Deviation (b) s =  s2 s =  s2 =  6.8 • The standard deviation is the square root of variance: • Therefore, the standard deviation is: = 2.60768 = 2.6 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

Finding Sample Variance & Standard Deviation