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