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Population & Sample Variance: Definition, Formula & Examples

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  • 0:02 Variance in Real Life
  • 2:30 Steps to Finding Variance
  • 4:29 Finding Population Variance
  • 7:39 Finding Sample Variance
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Population and sample variance can help you describe and analyze data beyond the mean of the data set. In this lesson, learn the differences between population and sample variance.

Variance in Real Life

Ruby is a third-grade teacher. She has been working with her students on reading. The students can read out loud and to themselves. Ruby has been giving the students timed reading comprehension tests. Each student is timed to read six pages in the classroom reading books. She can average the reading speeds among her students, but she doesn't feel like this information paints a clear picture with her students. Some of her students can read very fast, and some of her students read about average speed. The numbers are the number of minutes it takes for the students to read six pages in the classroom reading books.

Ruby will need to know how to find the population and sample variance of her data. Variance is how far a set of numbers are spread out. This is very different from finding the average, or the mean, of a set of numbers.

For example, take a look at the following set of numbers: 12, 8, 10, 10, 8, 12. If you add these numbers together and divide by the total numbers in the data set, which in this case is 6, you will get an average of 10. Notice that these numbers are all pretty close to the number 10.

Now take a look at this set of data: 28, 4, 6, 4, 2, 16. You'll notice that there is a greater difference between the numbers in the second set of data versus the first set of data. However, both sets of data have an average of ten. We show these differences in data by using variance.

There are two main types of variance: population and sample. Population is all members of a specified group. If we were to collect data on just the members of your household, then everyone living in your house would be considered the population. Sample is a part of a population used to describe the whole group.

If we were collecting data on the members of your household and only collected data about two members out of five members, then this would be considered a sample. Other examples of population and samples would be the total members of a school (the population) versus only the members of a class in the school (the sample), or a random selection of 50 members of a school, which would also be a sample.

Steps to Finding Variance

The symbol for variance is represented by the Greek symbol sigma squared, which looks like this.


Sigma, the symbol for variance
Grek symbol sigma


The formula of population variance is sigma squared equals the sum of x minus the mean squared divided by n.


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I don't know about you, but that sounds and looks like Greek to me. So let's break this down into some more manageable steps.

  1. Find the mean of the set of data.
  2. Subtract each number from the mean.
  3. Square the result.
  4. Add the results together.

Before we get to step five, you need to know that there is a difference between population variance and sample variance. Here is step five for population variance:

5. Divide the result by the total number of numbers in the data set.

The formula for sample variance looks like this:


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You can tell it looks slightly different from population variance.

Here is step five for sample variance:

5. Divide the result by the total number of numbers in the data set minus one.

You're probably asking, 'Why do we subtract one?' Remember that a sample is only part of the population and isn't actually the whole picture. Because of that, statisticians found a way to compensate, by subtracting one from the total number of numbers in the data set. Also, variance will never be a negative number, so if you get one, make sure to double check your work. Variance can only be zero if all of the numbers in the data set are the same. This is because there is zero difference in the numbers.

You're probably feeling a bit confused, so let's look at some examples. Remember, you can pause the video at any time and try to work these problems on your own. Then, play the video to see if you got the answer right.

Finding Population Variance

Let's take the data sets from our previous examples and see how variance can make a difference in how we interpret the data. Ruby's class has a total population of six students. The students have the following reading speeds: 12, 8, 10, 10, 8, 12.

We start with step one. I've listed my data set in the first column. We already know the mean, which is 10, so let's put that underneath step one.

Step two, we need to take each number from the data, set and subtract it from the mean:

  • 10 - 12
  • 10 - 8
  • 10 - 10
  • 10 - 10
  • 10 - 8
  • 10 - 12

Third, square each number. If you look at the first row, you will see that I have 10 - 12 under step two, then the result, -2, is squared. I also have: 2^2, 0^2, 0^2, 2^2 and (-2)^2.

Step four, add all of the numbers together. You'll see that under step four, I've listed the results of squaring the numbers: 4, 4, 0, 0, 4 and 4. Add all of those together for a result of 16.

Step five: divide the result by the total numbers in the data set. I have six total numbers in my data set, so I will divide 16 by 6, and that will give me a result of 2.67. I am dividing by six here because we are looking at the entire population. The variance 2.67 describes the spread of the numbers throughout the data set. This shows us that all of the numbers are relatively close together.

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