Question about Variability

Question

Calculate SS (sum of squares, variance, and standard deviation for the following Sample of scores: 2,3,5,5,3,6.

Answer 1

First, find the mean of the sample score.

x-bar = 2 + 3 + 5 + 5 + 3 + 6) = 24/6 = 4

Second, find variance by adding the sum of the difference of the individual score and the average and divide it by 5 because of 6 - 1.

s² = [(4 - 2)² + (4 - 3)² + (4 - 5)² + (4 - 5)² + (4 - 3)² + (4 - 6)²] / 5 = (4 + 1 + 1 + 1 + 1 + 4)/6 = 12/5 = 2.4

Finally, square root the variance to get standard deviation.

s = √2.4 ≈ 1.55

Answer 2

To do this manually, build a table where the first column is the list of numbers. Calculate the mean by adding all the numbers and dividing by the number of numbers (in this case there are 6 numbers). We define the numbers as values of "x" and the mean as x-bar.

x

---

2

3

5

5

3

6

---

sum = 24, x-bar = 24/6 = 4

Now, add a second column that is x-bar subtracted from the x-value (so x - x-bar) like this:

x (x - x-bar)

----------------

2 -2

3 -1

5 1

5 1

3 -1

6 2

Now add a third column that is the square of the second column (so (x - x-bar)² like this:

The sum of the squares is typically the sum of the third column. Adding the numbers in column 3 gives us 12 so SS = 12

The Variance (for a sample, which this is defined as a sample), is SS/(n - 1) so 12/(6 - 1) = 12/5 = 2.4

The standard deviation is the square root of the Variance so √2.4 = 1.549193.

Answer 3

n=6, n-1=5

2+3+5+5+3+6 = 24

24/6=4 = mean

(4-2)^2 = 4

(4-3)^2 = 1

(4-5)^2 = 1

(4-5)^2 = 1

(4-3)^2 = 1

(4-6)^2 = 4

SS = 12

variance = 12/(n-1) = 12/5= 2.4

standard deviation = sqr(12/5)= about 1.55