Mean - Online Tutor, Practice Problems & Exam Prep

Calculating the Mean

Data using different charts and graphs. But a lot of times when we interpret data, we'll have to do it a bit more numerically. So we're going to shift our focus in this video. I'm going to start talking about how we can calculate certain important variables from the datasets. The first one we'll talk about is called the mean, and, luckily, that's a word you've probably heard and seen before.

So in this video, I'm going to show you how to calculate the mean from a dataset. We'll discuss some different notations that we need to know, some important conceptual information, and then we'll just do some examples. Let's get started here. The mean, again, that's probably something you've heard at some point in math class, maybe even grade school, is essentially just an average of a dataset. Alright?

And when you take an average of a set of numbers, all you're going to do is you're just going to add up all of those values, and then you're going to divide by the total number of values. Alright? So let's just go ahead and take a look at this example here. Let's say I have a sample of numbers: 5, 10, 12, 14, and 3.

I've got 5 numbers in this dataset. And just to calculate a mean or an average, we're going to use the word "mean" here, I'm just going to take all those numbers and add them together first. So 5 plus 10 plus 12 plus 14 plus 3. Then I'm going to divide it by the total number of values that I have, which in this case is 5. Alright?

So what happens here is when you plug this into your calculator, make sure you do that top part in parentheses because of order of operations, but you should get a number that's 44 divided by 5. Then when you calculate that, you're just going to get 8.8. Alright? So in other words, this dataset here, whatever it represents, has a mean of 8.8. Alright?

That's how you calculate that. Now, a lot of times in math, we're going to take these complicated sort of list of instructions, and we'll turn them into shorter, equations with symbols. So I want to talk to you about what the mean is. The mean, when you see it, you're going to see this sort of symbol here, x⎯, and we just call it x-bar. And, basically, the equation here is this funny looking E that I'll talk about in a second, ∑xn.

So, basically, what this equation is saying is this little, this big E symbol here is the Greek capital letter sigma. And, basically, that just means that you're going to add up a bunch of values. It's always going to mean that. We're going to add up all the values in the datasets, which is your x, and then you're just going to divide by the total number of values, which in this case we know is n.

We've seen that before. Alright? So that's what that equation means. So this x-bar here, when we calculated this, was just a mean of 8.8. Alright?

So what does that mean? Well, the mean is what we call a measure of center or measure of central tendency. And, basically, it's a fancy word for saying it summarizes a dataset in one central value. We have numbers in the sample that range from 3 all the way to 14. That's where my min and my max.

And I calculated a number of 8.8, which is somewhere more or less in the center or in the middle. That's what a mean, a measure of center means. Alright? So that's really it for this first example. Let's go ahead and move on to our second one over here.

So what I want you to do is imagine that this sample of data over here is actually part of a larger population. So now we actually have an extra number in the mix here. We've added this 76. But ultimately, whether you're dealing with a population or a sample, the mean is always the same. You're just going to add up everything and divide by the total number.

So we got 5 plus 10 plus 12 plus 14 plus 3 plus 76. Again, we're going to put that all in parentheses. And now when you divide by the total number, what do you think you're dividing by? Is it 5? Well, be careful here because we've added another number into the mix.

There's actually 6 data values here. So one of the things that you might see is you might see little n's with samples and you might see big N's with populations. But, ultimately, you calculate the mean the exact same way. So you're just going to divide by 6 over here. This ends up being a 120 over 6, which when you calculate mean is going to be 20.

Alright? So just some notation here, whenever you see a population, you may see some different symbols attached to this. You may see mu instead of x-bar, and then you may see big N instead of little n. So basically, if you see this equation here where mu is equal to ∑xN, don't freak out because all that's happening here is you're just calculating the mean. You don't really need to know when to use one versus the other.

And if you're ever unsure, x-bar is probably your safest bet here. Alright? So just so just wanted to let you know that. Okay. Cool.

So let's talk about these means here for a second. When we calculated the sample, we got 8.8. Then when we threw in this extra number of 76, we got a mean of 20. So what's going on there? Basically, what happens is while the mean uses all the data values, the values in the datasets, any extreme values that you have, any outliers like the 76 over here, are going to significantly change your mean.

Alright? And we threw in the 76. That's a number that's so big relative to the other numbers that it kind of shifts the mean, and you still end up with a number that's sort of in between 3 and 76, but that 76 has shifted the mean by a lot. And now you have a mean that's 20 instead of 8.8. So that's something you'll have to be aware of.

Alright? So that's it. That's it for the introduction as to calculate the mean. Let's go ahead and take a look at some practice problems.

Find the mean of the sample data below.

• A. 24
• B. 15
• C. 23
• D. 18

Calculating the Mean Example 1

Back, everyone. Let's go ahead and work this out together here. So the data below shows heart rates in terms of beats per minute from a sample of adult males and females. So, the question we're trying to ask is, does there appear to be a difference in mean heart rates between these two samples? Alright?

So a lot of times in these sorts of problems, we're going to have to first figure out what to calculate and then sort of interpret that. So we're going to calculate the mean heart rates for both of these samples, then we're going to have to compare the two. Alright? That's what's going on here. Again, to calculate sample means, we're going to use this equation.

So, add up all the values and divide by the total number. In this case, what we have here is that n equals 10 for both of these samples. What happens here is that in order to calculate the average for males or the mean for males, I want to use all of the data in this row here which is ten numbers. And then for females, I want to use all the data in this row, which is, again, ten numbers. So don't confuse the two and just try to do 20 because the sample mean is going to be. You're going to use all of these numbers over here just for males, which is 10, and so on and so forth.

Alright? So what I'm going to do here is I'm going to put a little column over here where we're going to calculate the x-bars for both of these things. So let's go ahead and start with the first one, the xbar for males. We're going to use all of the top row information here. Remember, I have to add up everything and divide by the total number.

So this is going to be 84 plus 70. By the way, if you want to pause and just work on this on your own, you're free to do that. So you've got 88, sorry, 68. Then I've got 59 plus 61 plus 77 plus 90 plus 65 plus 56 plus 72. Alright.

So, remember, when you do all this in one step, you should really enclose these things in parentheses. That way, you'll get the correct answer. And you just divide it by 10. So, in other words, the sample mean over here, the heart rate for males, is going to turn out to be 70.2. Alright?

So that's 70.2 over here. I'm going to put that over here. Okay. Let's do the same exact thing, but for females now. So, we're going to have xbar for females.

We're going to use all the blue information and this is going to be I have 80 plus 73 plus 88 plus 91 plus 69 plus 85 plus 91 plus 81 plus 79 plus 77. Alright? Again, all the all the parentheses divide by 10. What you should get over here is 81.4. So this is going to be the data for, the sample mean for heart rates or mean heart rates for females.

So, we have 70.2 versus 81.4. And by the way, this is pretty accurate when you actually compare these mean heart rates. And basically, what happens here is we can clearly tell that the average or mean heart rate for females is higher than it is for males. This is just a biological difference, but it's actually true.

So, in other words, what we can say here is, and our question is, does there appear to be a difference in mean heart rates? And yes, there is. And the way we would write this is that xbar for females is greater than the xbar for males. Alright? So, again, all these things involve a calculation and then you have to interpret that data.

And that's basically what it's telling us over here. So this would be the answer to your question. Alright? Thanks for watching. Let me know if that makes sense.

And let's move on.