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Identifying & Calculating Averages

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  • 0:02 Averages
  • 1:18 Weighted Averages
  • 1:58 The Formula
  • 2:24 Example
  • 5:01 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to calculate any kind of average, even the average of certain events when some events repeat. Learn the general formula to use to calculate any average.

Averages

Everyone has a statistic. It is from these statistics that we get averages, or typical values. We can say that the average pay of an office worker is $12 per hour. Even though the average pay of an office worker is $12 per hour, you may not get paid exactly that much if you found an office job. Because the average is found by calculating the typical value of a large group of numbers, your pay could be more or less than $12. The way averages work is that if you found an office job that pays you $13 an hour, then you know that someone else has an office job that pays less than $12, $11 an hour for example. Why? Because after making the calculations, you have to get an answer of $12. If everyone got a pay higher than $12, then the average must be also be higher than $12. If we asked just a few office workers what their pay is, and we got answers of $10, $11, $12, $13, and $14, we would find an average of $12.

We will talk about the formula in just a bit. These numbers are spread out evenly. However, most often, our numbers are not spread out this evenly. You will find that some numbers repeat more often than others.

Weighted Averages

When some of your numbers repeat more often than others, we call their average a weighted average because the numbers that repeat more often than others will bring the average closer to those numbers. In the real world, we find weighted averages more common than averages where the numbers are spread out evenly. For example, in the real world, when you ask office workers about their pay, you might find that you get these answers: $9, $11, $12, $12, $12, $13, and $15. Do you see how the $12 is repeated? Even though we have these two kinds of averages, the formula we use to find our averages is the same.

The Formula

What is this formula? It is this:

average formula

The w stands for the weight, or how many times x happens, and x stands for our data. The symbol before these letters is the summation symbol that tells you to add everything up. What this formula is telling you to do is to multiply each different data by the number of times it appears, sum it all up, then divide by the total number of data points that you have.

Example

Let's use this formula to calculate the averages of the pay of office workers. The first set of data we have is $10, $11, $12, $13, and $14. We have five different data numbers, five different x's: $10, $11, $12, $13, and $14. Each of these numbers happens just once, so w is 1 for each of these. Summing it all up we have (1 * $10) + (1 * $11) + (1 * $12) + (1 * $13) + (1 * $14) = 60. We then divide it by the total of our w's. We have 1 + 1 + 1 + 1 + 1 = 5. So, 60 / 5 = 12. Our average pay is $12, which is what we expect. This is the average where our data is spread out evenly.

Let's look at finding the weighted average now. Our data is now $9, $11, $12, $12, $12, $13, and $15. We see that our $12 repeats. So, how many different data values do we have? We have 5. We can create a table to help us out with our calculations. In the first column, we will write our w values, the number of times each value repeats. In the second column, we will write our x values. Our $9 repeats once, our $11 repeats once, our $12 repeats three times, our $13 repeats once, and our $15 repeats once. Our table looks like this all filled in now:

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