Median - Online Tutor, Practice Problems & Exam Prep

Calculating the Median

So in recent videos, I showed you how to find the mean of a data set, which we said was a measure of center. This is a number that tries to summarize a data set in one central value that's somewhere around the middle. But why didn't we actually just take the data and find the middle number? That's exactly what I want to talk about in this video because that's exactly what the median is. The median is just another measure of center.

And, basically, to find it, all you're going to do is you're going to sort the data from smallest to largest, and then you're just going to find the middle number. Again, this is probably something that you've seen before in grade school, but we're going to go over some examples in practice so we can get the hang of it. Let's get started here. We're going to jump right into our examples. We're going to find the median of this set of numbers here.

So I got five numbers. I've got 5, 10, 14, 12, and 3. Now before you just go haphazardly trying to pick out the middle number like 14, remember that the first thing that you have to do is you have to sort the data from smallest to largest. Alright? That's the first thing you have to do here.

What I'm going to do is I'm going to take this number. I'm going to take these sets of numbers, and I'm going to rewrite them from smallest to largest. So we've got 3. What I'd like to do is kind of just mark off each one of them so I don't double count them or miss them. This is how I keep track of bigger datasets.

So I got 3 over here, then I've got 5, which is the next biggest number. The next one is going to be 10, then I've got 12, and then I've got 14. Alright? So that's my dataset sorted from smallest to largest.

And now what we can do is just look at the number of values that we have. In this case, we have n=5 here, which is just an odd number. And basically, the rule is whenever you have an odd number, the median is just going to be literally the middle value. Alright? So, there's no trouble here when you have an odd number because all you can do is you may have just seen this method where you can kinda cross off starting from the outside in, or you can just mark them off 1, 2, 3, 4, and then you end up with this middle number over here, which is your median.

So in this case, the median value is the middle number, which is just 10. Alright. That's the median of that data set. Let's move on to the second example over here because now things are a little bit more complicated. So now we see we have the same five numbers.

Now we added a 6th number. That's 76. Right. We've seen that number before. So in this case, what happens is now we have n=6 here, which is an even number.

But let's see what happens first. First, we're going to sort the data from smallest to largest, which is going to be 3, 5, 10, 12, 14, and then 76, which goes at the end here. Alright. And so now we can go ahead and use our method. Alright.

If you try to sort of cross them off from the outside in or mark them off, I got 1, 1, 2, 2. And then if I cross off these 2, I'll be left with nothing. So I basically have 2 middle values here. So what's the median? Well, basically, what we do here when we have two middle numbers is we just take the mean of those two middle numbers.

So in other words, the median of an even, numbered amount of data is actually going to be the mean of the two middle numbers. Okay? So what we're going to do here, we're going to take 10 and 12 and calculate the mean by adding them together. So we'll get this 10 and 12. To calculate the mean, we're just going to take 10 and 12, add them together, and divide by 2.

Most of the time, you can just visually inspect what the middle number is going to be. But we can see here clearly, if we calculate this, that it's going to be 11. So in other words, the median of this data set is 11, even though 11 doesn't actually show up in this data. And that's perfectly fine. Alright?

So that's how you find the median of a set of numbers. Now that we've understood how to do that, let's go ahead and take a look at some practice. Thanks for watching.

Find the median of the sample data below.

• A. 24
• B. 18
• C. 22
• D. 33

Calculating the Median

Welcome back, everyone. Let's work out this example together. So, in this problem, we have a histogram that shows us the number of college credits that a sample of students is taking in a certain semester, and we're going to find the median number of credits. One of the things that you may be noticing here right off the bat is that we don't have a list of numbers. So, the question is, if we're not given a list and we're just given a histogram, can we still find the median number of credits?

Let's see how we might be able to do that. Remember that you can always take these histograms and turn them back into original data values. What this histogram is telling me is that if I'm looking at college credits per semester, there's one student who is taking 11 credits, then there are 4 students taking 12 credits, and so on and so forth. So, basically, I can take the frequency for each class or category and recreate the data values.

If I have a frequency of 1 student who's taking 11 credits, that means I have one number that's 11. Then there are 4 students taking 12 credits, which tells me that there are actually 4 data values of twelve. That's how you can go from a histogram back into original data values. There are 4 twelves.

If you keep going to the right here, there are 3 students taking 13 credits. So I'm going to write down 3 thirteens. Then, there are 2 students taking 14 credits, so I note down 2 fourteens. Finally, there's 1 student taking 15 credits, so I have one 15. This data set is what I used to build this histogram. I can now simply find the middle number.

I can just chop off numbers from the left and the right. I've got 11 which goes with 15, the 12s which go with the 14s. By starting from the leftmost bar and working my way to the right, my numbers are already in ascending order from smallest to largest. It's always best, when working with a histogram, to start with the leftmost bar and then try to recreate the data values.

It's already sorted. So I got 12 that matches with 14, and another 12 with 13, and then another 12 with 13, leaving me with one number in the middle here. I don't have to take a mean, and that single middle number is 13.

So, this is the median of my dataset. The median number of credits here is 13. Interestingly, the most common number of credits is actually 12, but the median is 13. Cool?

That's how you handle these types of problems, folks. Let me know if you have any questions, and let's move on.

Comparing Mean vs. Median

In recent videos, we've talked independently about mean and median, which are measures of center. These are numbers that summarize the data in one's value that's roughly around the center or the middle. Now I want to talk about these things a little bit more together in this video because clearly, we've seen that mean and median have their own distinct pros and cons. So we're just going to jump right into this example, and I'm going to show you the advantages and disadvantages of using mean versus median in certain datasets. Alright?

Let's go ahead and get started, and we'll do a couple of examples together. So let's just go ahead and take a look right at our example. We have all these numbers here that we've seen before. These are sample data. And, most of the calculations you've already done in other videos.

So let's just go ahead and take a look. So when we had the sample of 5 numbers and calculated the mean, add up all the values and divide by the total number. We would get a number that was 8.8. And then all of a sudden, we introduced an outlier, an extreme value relative to everything else. What that made us do is that we had to add the 76 in and divide by the new number.

Then we got a mean that was equal to 20. Alright. So we used all of the values in the dataset to get these numbers, which is actually an advantage when it comes to the mean. It's an advantage because you're using all of the values and it helps you get a better picture of the whole entire dataset. However, what we can see here is that one clear disadvantage is that one extreme value, like this huge outlier of 76 over here, can change the mean by a lot.

In this case, it changed it by 11.2, which is huge relative to the other numbers there. Alright? So one extreme value can change the mean by a lot. Therefore, whenever you have these types of problems where they're asking you what measure of center is the best, the best situation to use the mean is where the data is symmetric and without outliers. Alright.

Because those outliers are significantly going to change that mean. Let's move on to the median. Alright? So the median was pretty straightforward. You just organize the data from smallest to largest, and you look at the middle numbers.

In one case, when we found that the sample of numbers had a median of 11, then when we threw in this outlier of 76, it's still just the two middle numbers. We found that it was 10 + 2, which is 12. So in other words, when we threw in an outlier, the median only increased by 1. Alright? So one disadvantage of using the median is that you're only really looking at maybe one or two values in the dataset, so that's kind of a disadvantage.

You don't get a clear picture of all the data. However, what we can see here is one clear advantage is that this outlier only changed the median by 1. Alright? It didn't have such a drastic effect like the mean did. Alright?

And so what happens here is that this median is going to be the best measure of center whenever the data is roughly symmetric with outliers because it's basically just going to ignore some of those outliers. This idea that the mean and median can significantly change or not relative to outliers is called resistance. So sometimes what you may see in your books is the mean is not resistance because significant outliers can drastically change it, whereas the median is resistant because even an outlier like 76 only changes the median by 1. Alright? So that's the idea here.

Those are some advantages and disadvantages to mean versus median. But let's take a look at our last problem over here, which is a little bit more of a problem that you may actually see, which is without calculating, we're going to determine if the mean or median best represents the center of the graph data. In other words, which one of these things is the best measure of the center? Is it the mean or the median? So let's take a look here.

And, again, we're not going to calculate anything. So we've got this histogram, which tells us the salary of graduate students in the 1,000. We've got most of the data that's clustered around 36,000, but we have a couple of drastic outliers out here. So what's going on? Well, if I just imagine that these outliers just don't exist, then basically what we can say here is that the mean and the median would be roughly around the same point.

Right? So the mean would probably be somewhere over here, that's x̄, and the median would also be somewhere around here, that's I'm just going to call that a capital M. The problem is that when these outliers get introduced, they're going to significantly pull that mean all the way to the right, such that the mean actually may end up somewhere over here. I don't actually know because I can't calculate this. Right?

So all we know here is that the median is much more resistant to extreme values, whereas the mean gets pulled a lot. So clearly, we can see here that because this data is roughly symmetric around most of the data points, but there are a couple of outliers, the median is going to be the best measure of center. Alright? So the mean is definitely not going to be the best here. By the way, this is why in the real world, when we talk about salaries, we often talk about the median salary.

Most people in the US make somewhere in this range of income. However, the one or few people who make billions of dollars significantly affect that mean. Whereas the median is much more representative of what the typical citizen of the US actually makes. Alright? So in this case, it would be the median.

Alright, folks. That's it for this one. Let me know if you have any questions.

Comparing Mean vs. Median Example 1

Welcome back, everyone. So in this example, hopefully, you had a chance to look over it, shows the prices in 1,000 of dollars of 8 homes in a city. Alright? And we're given these numbers over here. We're going to determine or find the mean and median price.

And once we find those mean and median prices, we're going to use that to determine which of the 2 is more representative of the sample. So this is a situation where we're going to have to calculate some stuff and then interpret. Alright? Let's get started. So in order to calculate the mean and median price, it doesn’t really matter which one you start with.

I'm just going to go ahead and calculate the mean. So the mean price is just going to be x̄. And remember, how do I calculate the mean? You add up everything and divide by the total number. So in this case, we have n is equal to 8.

Right? There are 8 homes here. So what I'm just going to do is I’m going to go ahead and add up everything: 275 plus 229 plus 850 plus 240, plus 305, then I've got 287, and then I've got 310, and then 342. Alright?

Once you add all these things up and plug them into your calculator, add them and divide by 8, what you should end up with for a mean price is, I believe, it is let's see. We've got 354.75. So 354.75. And, again, that's in 1,000 of dollars. Right?

So this is going to be 354,800. Let's just actually round this up, and we could just call this 354.8. Alright? So that's going to be our mean of the sample data. Okay.

So now that's the mean price, we're going to have to calculate now the median price. Now, the median price is just going to be we are going to have to sort these things into the smallest to largest and then figure out what the middle number is. Okay? So if I try to do that remember, the way to do this is look for your lowest number. I see 229.

So in order to calculate the median, what I'm going to do is I’m going to reorganize the data. I've got 229, and I've got the next highest number is 240. The next highest number would be 275. Next highest will be 287.

We've got that number over here. Then the next one is going to be 305, then 310. That's those two numbers here. Then I've got 342. And then finally, the 850.

Alright. So those are my eight numbers sorted from smallest to largest. Now, again, one way to do this is just sort of chop off and cross off from outward inwards. Anyway, I do this, I'm going to work my way inwards and find that I end up with actually two middle numbers. Alright?

So remember, this is 8. So I end up with two middle numbers over here. So what I have to do is I have to take the mean of those two middle numbers. So it's going to be 287+3052, and you should end up with 296. Okay?

So in other words, the median price is actually 296. And, again, that's in 1,000. Right? So in other words, it's $296,000. Okay.

So what happens here is now that we have both of our answers, right, that's the mean price and that's the median price, which one of these is more representative of the sample? So if you take a look at your mean, the mean is $354,000. But if you take a look at that number, that actually tends to be higher than almost every other number in this dataset except for the 850. So if you were to sort of plot these things on a graph, that's one way I sort of or on a line, that's kind of one way I like to think about this. You'll see that the mean and the median kind of lineup in two different places.

So I've got 229 over here, and I've got 240, 275, which are going to be closer together. I'm kind of just assuming that all of this stuff is, like, on a line somewhere like this. Let's see. 287.

I've got 305, 310, which are pretty close, 342, and then I've got 850 all the way over here. Alright? So what happens here is that my two numbers, the mean and the median, are going to fall somewhere in this sort of line segment. Alright. So my median will actually fall somewhere in here.

So this is going to be where my median is. So this is the median. And then way out here, 354 is going to be where x bar sort of lines up. Alright? So which one of these is sort of representative of where the center or the typical sort of data value would line up?

And it would actually be something like the median. And it's again because of this outlier. This outlier tends to push that mean all the way out towards the right here, such that it's not really representative of the rest of the data that you have, whereas the median is much better. Okay? So the answer to this problem here is that the median is a much better measure of or representative of this dataset.

Hopefully, that makes sense. Let me know if you have any questions, and I'll see you in the next.