Using the computational formula, compute SS then calculate the sample variance and standard deviation for the following scores: 5, 11, 9, 6, 8, 14.

Answer 1

The total is 53

the mean is 53/6 = 8 + 5/6

Note that 5/6 = 0.833333333... and 1/6 = 0.166666....

Here's the table courtesy of MS Excel

x dev dev^2 x^2

5 -3.833333333 14.69444444 25

11 2.166666667 4.694444444 121

9 0.166666667 0.027777778 81

6 -2.833333333 8.027777778 36

8 -0.833333333 0.694444444 64

14 5.166666667 26.69444444 196

53 0 54.83333333 523

The denominator of the variance is n-1 = 5

The numerator of the variance by definition

is the sum of the squares of the deviations

which is 54.83333333..... = 54 +54/6

The shortcut formula says the numerator is

sum of x^2 - n * mean^2 =

523 - 6* (8 + 5/6)^2 =

523 - 6 * (8+ 5/6)(8+5/6) =

523 - 6 * (64 + 80/6 + 25/36) =

523 - 6 * (384/6 + 80/6 + 25/36) =

523 - (384 + 80 + 25/6) =

523 - (464 + 25/6) =

523 - 464 - 25/6 =

59 - (4 + 1/6)

55 - 1/6 =

54 + 5/6

so the variance is (54 + 5/6) divided by 5

= 329/6 * 1/5 = 329/30 = 10 + 29/30

the standard deviation is the square root of the variance, which

is approximately