Variance and Standard Deviation Information
Variance: the dispersion among the measures in a given population.
Standard deviation: “mean of the means.” Use a normal distribution curve. The
standard deviation is a statistic that tells you how tightly all of the data points are
clustered around the mean in a set of data. When the examples are pretty tightly bunched
together and the bell-shaped curve is steep, the standard deviation is small. When the
examples are spread apart and the bell curve is relatively flat, that tells you that you have
a relatively large standard deviation.
One standard deviation away from the mean in either direction on the horizontal axis (the
red area on the above graph) accounts for somewhere around 68 percent of the people in
this group. Two standard deviations away from the mean (the red and green areas)
account for roughly 95 percent of the people. And three standard deviations (the red,
green and blue areas) account for about 99 percent of the people.
If this curve were flatter and more spread out, the standard deviation would have to be
larger in order to account for those 68 percent or so of the people. So that's why the
standard deviation can tell you how spread out the examples in a set are from the mean.
We define the variance to be
and the standard deviation to be
Variance and Standard Deviation: Step by Step
1.
2.
3.
4.
5.
6.
Calculate the mean, x.
Write a table that subtracts the mean from each observed value.
Square each of the differences.
Add this column.
Divide by n -1 where n is the number of items in the sample This is the variance.
To get the standard deviation we take the square root of the variance.
Example:
The owner of the Ches Tahoe restaurant is interested in how much people spend at the
restaurant. He examines 10 randomly selected receipts for parties of four and writes
down the following data.
44, 50, 38, 96, 42, 47, 40, 39, 46, 50
He calculated the mean by adding and dividing by 10 to get
x = 49.2
Below is the table for getting the standard deviation:
x
x - 49.2
44
-5.2
50
0.8
38
11.2
96
46.8
42
-7.2
47
-2.2
40
-9.2
39
-10.2
46
-3.2
50
0.8
Total
(x - 49.2 )2
27.04
0.64
125.44
2190.24
51.84
4.84
84.64
104.04
10.24
0.64
2600.4
Now
2600.4
= 288.7
10 - 1
Hence the variance is 289 and the standard deviation is the square root of 289 = 17.
What this means is that most of the patrons probably spend between $32.20 and $66.20.
Example:
Find the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12, 14
First work out the mean: 10.222
Now, subtract the mean individually from each of the numbers in the question and square
the result. x refers to the values in the question.
x
(x - x)²
4
9
11
12
17
5
8
12
14
38.7 1.49 0.60 3.16 45.9 27.3 4.94 3.16 14.3
Now add up these results (this is the 'sigma' in the formula): 139.55
Divide by n-1. n is the number of values, so in this case is 8: 17.44
And finally, square root this: 4.18