**Finding the “sample” variance and “sample” standard deviation **

Dataset: 16, 9, 8, 13, 19, 12, 10, 15, 17, 20

Here, n = 10 because “n” is the number of data values in our dataset.

**The formula for variance for a “sample” is**

Variance s^2 = Σ ( x – mean )^{2} / ( n – 1 )

The “Σ” stand for “sum”

“mean” is the sample mean of your dataset.

“x” is each value in your dataset

Remember that the variance looks at the average of the differences of each value in the dataset compared to the mean. In other words, it looks at how far each data value is from the mean on average. Variance is a measure of “variation”.

**This formula requires a few steps**

Sample Variance s^2 = Σ ( x – mean )^{2} / ( n – 1 )

**Step 1**: Find the mean of your data

mean = (16+ 9+ 8+ 13+ 19+ 12+ 10+ 15+ 17+ 20)/10 = 139/10 = 13.9

**Step 2**: Subtract the mean from each x value in your dataset

16 -13.9 =2.1

9 -13.9 =-4.9

8 -13.9 =-5.9

13 -13.9 =-0.9

19 -13.9 =5.1

12 -13.9 =-1.9

10 -13.9 =-3.9

15 -13.9 =1.1

17 -13.9 =3.1

20 -13.9 =6.1

**Step 3**: Square each of the resulting differences

2.1^2 = 4.41

-4.9^2 = 24.01

-5.9^2 = 34.81

-0.9^2 = 0.81

5.1^2 = 26.01

-1.9^2 = 3.61

-3.9^2 = 15.21

1.1^2 = 1.21

3.1^2 = 9.61

6.1^2 = 37.21

**Step 4**: Sum up all of the square values

4.41

24.01

34.81

0.81

26.01

3.61

15.21

1.21

9.61

37.21 +

—————

156.9

**Step 5**: Divide by n- 1 because this is a sample variance. The formula for population variance divides by n, not n-1.

156.9/(10-1) = 156.9/9 = 17.43

**So the sample variance is 17.43**

**The sample standard deviation is the sqrt(17.43) = 4.18**