Finding Sample Variance Standard Deviation Using The Shortcut

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

The Formula - Knowing Its Parts F The calculation of a sample statistic requires the use of a formula. In this case, use: 2 ( x) x x 2 - n Sample variance: s 2 = n -1 • 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? ) 2

Finding Summations x and 2 x F 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: Sample = { 6, 10, 13, 11, 12, 8 } x = 6 + 10 + 13 + 11 + 12 + 8 = 60 x 2 = (6)2 + (10)2 + (13)2 + (11)2 + (12)2 + (8)2 = 36 + 100 + 169 + 121 + 144 + 64 = 634 3

Finding the Numerator F First, find the numerator: 2 ( x) x 2 - n s 2 = = n -1 Previously determined values: x 2=634, x=60, n=6 2 2 ( ) 60 ( x) (634) x 2 - n 6 34 2 = s = = n -1 4

Finding the Answer (a) 2 ( x) x 2 - n s 2 = n -1 F Lastly, find the denominator and divide. You have the answer! 2 ( x) x 2 - n 34 34 2 s = = 6. 8 n -1 6 -1 5 The sample variance is 6. 8 Note: Variance has NO unit of measure, it’s a number only 5

Finding the Standard Deviation (b) F The standard deviation is the square root of variance: s = s 2 F Therefore, the standard deviation is: s= s 2 = 6. 8 = 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 6