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 Shortcut Formula

2  x 2 - n -1 Sample variance: s 2 = (x)2n(x)2n x2x2 The Formula - Knowing Its Parts  The calculation of a sample statistic requires the use of a formula. In this case, use: s 2 is “s-squared”, the sample variance  x 2 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?) xx s2s2 n n

3  x = (6) Finding Summations  x and  x 2 Sample = { 6, 10, 13, 11, 12, 8 }  The “shortcut” formula calculates the variance without the value of the mean. The first step is to find the two summations,  x and  x 2 :  x 2 = (10) 2 (13) 2 (11) 2 (12) 2 (8) 2 = = (6) 2 (10) 2 (13) 2 (11) 2 (12) 2 (8) 2 = 60

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

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

6 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 = = 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