Median: Videos & Practice Problems

The median is a measure of center that represents the middle value of a sorted data set. To find it, arrange the numbers from smallest to largest. For an odd number of values, the median is the middle number. For an even number, it is the mean of the two middle numbers. Unlike the mean, the median is resistant to outliers, making it a better measure of center in skewed distributions. Understanding when to use mean versus median is crucial for accurately summarizing data, especially in the presence of extreme values.

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. So 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 five, ten, 14, 12, and three. 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 three. What I 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 larger and larger datasets.

So I've got three over here, then I've got five, which is the next biggest number. Next one is going to be 10, then I've got 12, and then I've got 14. Alright? So that's my data set 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 equals five over 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. All right. 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 kind of cross off starting from the outside in, or you can just mark them off, one, two, three, four, 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, and now we added a sixth number.

That's 76. Right, we've seen that number before. So in this case, what happens is now we have n equals six over 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 three, five, 10, 12, 14, and then 76, which goes at the end over 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 two on each side.

And then if I cross off these two, I'll be left with nothing. So I basically have two 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.

So what we're going to do here, we're going to take ten and twelve and calculate the mean by adding them together. To calculate the mean, we're just going to take ten and twelve, add them together, and divide by two. Most of the time, you can kind of 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.

A real estate analyst is studying house prices in a neighborhood. Find the median of the data below.

• A. $305,000
• B. $300
• C. $300,000
• D. $299,444

Answer: (C)

Calculating the Median Example 1

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

Remember that you can always take these histograms and convert them back into original data values. What this histogram is telling me is that there is one student who is taking 11 credits, then there are four students taking 12 credits, and so forth. Basically, what I can do here is take the frequency per class or category and then just recreate the data values.

If there is a frequency of one student taking 11 credits, that means I have one number that's an 11, and then four with 12 credits. This tells me there are four data values at twelve. That's how you can go from a histogram back into original data values. There are four twelves.

If we continue to the right, we see that there are three thirteens, two fourteens, and one fifteen. This is my data set that I used to build this histogram. Now, I can find the middle number by chopping off numbers from the left and the right. The numbers start with an 11 and go up to a 15. The numbers are already in ascending order from smallest to largest, making it best to start from the leftmost bar and work your way to the right when you recreate the data values.

The dataset is already sorted, so I have 12 going to fourteen, twelve with thirteen, another twelve with a thirteen, and then I'm left with one number in the middle, which is 13. I don't need to calculate the mean because this is the middle number, which is the median. Therefore, the median number of credits is 13. Although the most common number of credits is 12, the median is 13. Cool? That's how you handle these types of problems.

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 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 what we've seen is 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 five 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 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, 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 plus 12 over two, which is 11. So in other words, we threw in an outlier, the median only increased by one. Alright.

So one disadvantage of using the median is that you're only really looking at maybe one or two values in the datasets, so that's kind of a disadvantage. You don't get a clear picture of all the data. However, we can see here is one clear advantage is that this outlier only changed the median by one. 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 resistant because significant outliers can drastically change it, whereas the median is resistant because even an outlier like 76 only changed the median by one. 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 gonna 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 gonna calculate anything. So we've got this histogram, which tells us the salary of graduate students in the thousands. We've got most of the data that's clustered between thirty and sixty thousand, 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 gonna call that a capital M. The problem is that when these outliers get introduced, they're gonna 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 going to be the best here. By the way, this is why in the real world, when we talk about salaries, oftentimes we 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 or 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 2

Welcome back, everyone. So in this example, hopefully, you had a chance to look over it. It shows the prices in thousands of dollars of eight 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 two 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 eight.

Right? There's eight homes here. So what I'm just going to do is I'm going to go ahead and add up everything: $2.75 + $2.29 + $8.50 + $240 + $305 + $287 + $310 + $342.

Alright? Once you add all these things up and plug them into your calculator, add them and divide by eight. What you should end up with for a mean price is, you should get something that is, I believe it is let's see. We've got $354.75.

And, again, that's in thousands of dollars. Right? So this is going to be 354.8 in thousands. Alright? 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're going to have to sort these things into smallest to largest and then figure out what the middle number is. Okay?

So if I try to do that so remember, the way to do this is you look for your lowest number. I see $2.29. So in order to calculate the median, what I'm going to do is I'm going to reorganize the data. I've got $2.29, and I've got the next highest number is $2.40. The next highest number would be $2.75.

Next highest will be $287. We've got that number over here. Then the next one is going to be $305, then $310. 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, when we could do this is just sort of chop off and cross off from outward inwards. Any way 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 seven. 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 + $305 ÷ 2, and you should end up with $296. Okay?

So in other words, the median price is actually $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 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, like, 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 up in two different places.

So we've got $229 over here, and we've got, $240, $275, $287, 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 segments. Alright. So my median will actually fall somewhere in here.

So this is going to be where my median is. So this is median. And then way out here, $354 is going to be where x x̄ 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 a representative of this data set.

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

Comparing Mean vs. Median Example 2

Welcome back, everyone. So, in this example, we have a histogram that shows us the annual salaries of several CEOs. Presumably, this number is in billions or something like that. We're going to go ahead and first estimate the mean and the median of the dataset. It just can be an estimate.

It doesn't have to be super rigorous. And then we're ultimately going to find the answer to this problem. We're going to use this to answer which is the most appropriate measure of the center of the data for this histogram. So let's just go ahead and get started here. We have all the data for the annual CEO salaries.

Again, presumably this is in billions. How do we go about finding the mean and the median? Well, remember that the median is basically going to separate the lower 50% and upper 50% of the data. The median is going to be somewhere in the middle. So, if we look at this data over here, we notice that we have a ton of data points that are going to be in the low tens, twenties, and thirties.

And then as we go towards the right, the data tends to skew and get a little bit lower and lower. So what's going to happen is the median is definitely not going to be straight in the middle of these numbers. It's not going to be over here somewhere. It's definitely going to be somewhere over here where you're going to get a lot of low numbers and there's going to be a bunch of higher numbers, but there's going to be fewer of them. So, therefore, the median in this case is going to be somewhere over here.

In this case, I'm just going to go ahead and estimate that the median is roughly somewhere around 30. Now what about the mean? In this case, what happens here is the mean is a little bit different because we have these extreme values that are over here on the right that are basically going to pull the mean closer to the right side of the graph. So, in this case, what happens is that the mean sort of gets pulled because, again, these eighties, nineties, and hundreds will vastly shift and skew that data off to the right over here.

So, my guess is just by looking at this, just estimating, is that the mean is going to be somewhere over here. So this mean is probably going to be somewhere, again, towards the middle or to the right of the graph at around 50. Now if you didn't get exactly what the numbers that I got, that's okay because it's just an estimate and that's what the point of the problem was. The point of the problem is actually to use this to answer which is the most appropriate or the more appropriate measure of the center of the data. So here's the median and the mean for this.

But really, the answer is between these two, which one is more representative or appropriate to measure the center of the data. And in this case, what you should know about these types of graphs is that whenever the graph is skewed to the right, this is always going to happen. So when it's skewed to the right like this, what you should know is that the median will always be less than the mean. And what happens is that the median is always just going to be the more appropriate measure when you have data skewed to the right because it's better at representing the more typical CEO salary. Obviously, you're going to have some of them that are way off to the right, but the typical CEO salary is going to be somewhere in the middle of this cluster over here.

And so, therefore, the median is a more appropriate measure of the center of the data. Alright? So, a little bit of a strange problem, but you may actually get some problems like this, so you'll need to know how to do them. Alright? So that's it for this one.

Let me know if you have any questions. Thanks for watching.

Here’s what students ask on this topic:

What is the median and how is it calculated in a data set?

The median is a measure of center that represents the middle value of a sorted data set. To calculate the median, first arrange the numbers in the data set from smallest to largest. If the number of values (n) is odd, the median is the middle number. If n is even, the median is the mean of the two middle numbers. This involves adding the two middle numbers and dividing by two. The median is particularly useful in skewed distributions as it is resistant to outliers, unlike the mean, which can be significantly affected by extreme values.

How does the median differ from the mean in terms of resistance to outliers?

The median and mean are both measures of center, but they differ in their resistance to outliers. The median is resistant to outliers because it only considers the middle value(s) of a sorted data set, ignoring extreme values. This makes it a reliable measure of center in skewed distributions. In contrast, the mean is sensitive to outliers because it involves summing all values and dividing by the total number of values. An extreme value can significantly alter the mean, making it less representative of the data set's central tendency when outliers are present.

When is it more appropriate to use the median instead of the mean?

The median is more appropriate to use instead of the mean when the data set contains outliers or is skewed. In such cases, the mean can be significantly affected by extreme values, leading to a misleading representation of the data's central tendency. The median, being resistant to outliers, provides a more accurate measure of center in these situations. For example, in income data where a few individuals earn significantly more than the rest, the median salary is a better indicator of the typical income than the mean.

How do you find the median in a data set with an even number of values?

To find the median in a data set with an even number of values, first arrange the numbers from smallest to largest. Since there is no single middle number, identify the two middle numbers. The median is then calculated as the mean of these two middle numbers. This involves adding the two middle numbers together and dividing the sum by two. This method ensures that the median accurately represents the center of the data set, even when there is an even number of values.

Why is the median often used to report income data?

The median is often used to report income data because it provides a more accurate representation of the typical income level in the presence of outliers. Income data can be skewed by a small number of individuals earning significantly more than the majority. These outliers can drastically increase the mean, making it unrepresentative of the general population's income. The median, being resistant to such extreme values, reflects the central tendency of the majority, offering a clearer picture of what most people earn.