http://www.ck12.org Chapter 15. Concepts of Statistics
15.5 Variance
Here you will calculate population variance, sample variance and standard deviation from univariate data.
Two groups of students that each have an average test score of 75 might have a score distribution that looks
remarkably different. One class might be made up entirely of grades between 72 and 78 while the other class may
have half the group around 50, with the other half getting near 100. Variance is a way of measuring the variation
in a set of data. What is the mean and variance for the following sample test scores taken from a larger student
population?
75, 73, 78, 90, 60, 51, 87, 79, 80, 77
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/63001http://www.youtube.com/watch?v=6JFzI1DDyyk Khan Academy: Statistics: Variance of a Population
Guidance
The thought process of a person trying to describe the spread of some data for the first time must have been something
like this.
Well, the average is 75. What if I try to just add up how different each number is from 75?
As the person calculates the numbers, they realize pretty quickly that this sum will be zero, essentially by definition.
This is because the numbers that occur below 75 precisely cancel out with the numbers above 75.
Since I cannot add the differences directly, why don’t I just sum the absolute value of the differences?
This is a legitimate method for describing the spread of data. It is called absolute deviation and is simply the sum of
the absolute values of each of the differences.
If I take the average absolute difference, I will be able to judge on average how far away each data point is from the
mean. A larger difference means more spread out.
If you take the average of the absolute deviation, you get the mean absolute deviation. The mean absolute variation
is a legitimate, but limited, way of describing the spread of data. Eventually, a person trying to describe the spread
of data for the first time might consider a method called population variance.
What if instead of using absolute value to solve the issue, I square each difference and then add them together? Of
course I’d have to divide by the number of data points to get the average difference squared