Finding Sample Variance & Standard Deviation

Finding Sample Variance & Standard Deviation Using The Shortcut 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)2 n x x2 - x2 n Sample variance: s2 = s2 n -1 n • The calculation of a sample statistic requires the use of a formula. In this case, use: • s2 is “s-squared”, the sample variance • x2 is the “sum of squared x’s”, the sum of all squared data • x is the “sum of x”, the sum of all data • n is the “sample size”, the number of data (Do you have your sample data ready to use?)

Finding Summations x and x2 6 10 13 11 12 8 x = + + + + + x2 = + + + + + (6)2 (10)2 (13)2 (11)2 (12)2 (8)2 • The “shortcut” formula calculates the variance without the value of the mean. The first step is to find the two summations, x and x2: Sample = { 6, 10, 13, 11, 12, 8 } 6 10 13 11 12 8 = 60 (6)2 (10)2 (13)2 (11)2 (12)2 (8)2 = 36 + 100 + 169 + 121 + 144 + 64 = 634

Finding the Numerator (x)2 n x2 - s2 = = n -1 ( )2 (x)2 n ( ) - x2 - s2= = = n -1 n -1 60 634 6 • First, find the numerator: Previously determined values: x2=634, x=60, n=6 60 634 6 34

Finding the Answer (a) (x)2 n x2 - s2 = n -1 (x)2 n x2 - 34 34 s2 = = = = n -1 6-1 5 • Lastly, find the denominator and divide. 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