Hi, this is Rob. Welcome to Math Antics! In this lesson, we’re gonna learn about three important math concepts called the Mean, the Median and the Mode. Math often deals with data sets, and data sets are often just collections (or groups) of numbers. These numbers may be the results of scientific measurements or surveys or other data collection methods. For example, you might record the ages of each member of you family into a data set. Or you might measure the weight of each of your pets and list them in a data set. Those data sets are fairly small and easy to understand. But you could have much bigger data sets. A really big data set might contain the cost of every item in a store, or the top speed of every land mammal, or the brightness of all the stars in our galaxy! Those data sets would contain a lot of different numbers! And if you had to look at a big data set all at one time… it would be pretty hard to make sense of it or say much about it besides “well that’s a lot of numbers”! But that’s where Mean, Median and Mode can really help us out. They’re three different properties of data sets that can give us useful, easy to understand information about a data set so that we can see the big picture and understand what the data means about the world we live in. That sounds pretty useful, huh? So let’s learn what each property really is and find out how to calculate them for any particular data set. Let’s start with the Mean. You may not have ever heard of something called “the mean” before, but I’ll bet you’ve heard of “the average”. If so, then I’ve got good news! Mean means average! “Mean” and “average” are just two different terms for the exact same property of a data set. The mean (or average) is an extremely useful property. To understand what it is, let’s look at a simple data set that contains 5 numbers. As a visual aid, let’s also represent those numbers with stacks of blocks who’s heights correspond to their values: one, eight, three, two, six Right now, since each of the 5 numbers is different, the stacks of blocks are all different heights. But what if we rearrange the blocks with the goal of making the stacks the same height? In other words, if each stack could have the exact same amount, what would that amount be? Well, with a bit of trial and error, you’ll see that we have enough blocks for each stack to have a total of 4. That means that the Mean (or average) for our original data set would be 4. Some of the numbers are greater than 4 and some are less, but if the amounts could all be made the same, they would all become 4. So that’s the concept of Mean; it’s the value you’d get if you could smooth out or flatten all of the different data values into one consistent value. But, is there a way we can use math to calculate the mean of a data set? After all, it would be very inconvenient if we always had to use stacks of blocks to do it! There’s got to be an easier way!! [crash] To learn the mathematical procedure for calculating the Mean, lets start with blocks again. But this time, instead of using trial and error, let’s use a more systematic way to make the stacks all the same height. This way involves a clever combination of addition and division. We know that we want to end up with 5 stacks that all have the same number of blocks, right? So first, let’s add up all of the numbers, which is like putting all of the blocks we have into one big stack. Adding up all of the numbers (or counting all the blocks) shows us that we have a total of 20. Next, we divide that number (or stack) into 5 equal parts. Since the stack has a total of 20 blocks, dividing it into 5 equal stacks means that we’ll have 4 in each, since 20 divided by 5 equals 4. So that’s the math procedure you use to find the mean of a data set. It’s just two simple steps. First, you add up all the numbers in the set. And then you divide the total you get by how many numbers you added up. The answer you get is the Mean of the data set. Let’s use that procedure to find the mean age of the members of this fine looking family here. If we add them all up using a calculator (or by hand if you’d like) the total of the ages is 222 years. But then, we need to divide that total by the number of ages we added which is 6. 222 divided by 6 is 37. So that’s the mean age of all the members in this family. Alright, that’s the Mean. Now what about the Median? The Median is the middle of a data set. It’s the number that splits the data set into two equally sized group or halves. One half contains members that are greater than or equal to the Median, and the other half contains members that are less than or equal to the Median. Sometimes finding the Median of a data set is easy, and sometimes it’s hard. That’s because finding the middle value of a data set requires that its members be in order from the least to the greatest (or vice versa). And if the data set has a lot of numbers, it might take a lot of work to put them in the right order if they aren’t already that way. So to make things easier, let’s start with a really basic data set that isn’t in order. It’s pretty easy to see that we can put this data set in order from the least to the greatest value just by switching the 2 and the 1. There, now we have the data set {1, 2, 3} and finding the Median (or middle) of this data set is easy! It’s just 2 because the 2 is located exactly in the middle. That almost seems too easy, doesn’t it? But don’t worry… it gets harder! But before we try a harder problem, I want to point out that sometimes the Mean and the Median of a data set are the same number, and sometimes they’re not. In the case of our simple data set {1, 2, 3}, the Median is 2 and the Mean is also 2, as you can see if we rearrange the amounts or follow the procedure we learned to calculate the Mean. But what about the first data set that we found the mean of? We determined than the Mean of this data set is 4. But what about the Median? Well, the Median is the middle, and since this data set is already in order from least to greatest, it’s easy to see that the 3 is located in the middle since it splits the other members into two equal groups. So for this data set, the Mean is 4 but the Median is 3. So to find the Median of a set of numbers, first you need to make sure that all the numbers are in order and then you can identify the member that’s exactly in the middle by making sure there’s an equal number of members on either side of it. Okay, ...so far so good. But some of you may be wondering, “What if a data set doesn’t have an obvious middle member?” All of the sets we’ve found the Median of so far have an odd number of members. But, what if it has an even number of members? …like the data set {1, 2, 3, 4} There isn’t a member in the middle that splits the set into two equally sized groups. If that’s the case, we can actually use what we learned about the Mean to help us out. If the data set has an even number of members, then to find the Median, we need to take the middle TWO numbers and calculate the Mean (or average) of those two. By doing that, we’re basically figuring out what number WOULD be exactly half way between the two middle numbers, and that number will be our Median. For example, in the set {1, 2, 3, 4} we need to take the middle TWO numbers (2 and 3) and find the Mean of those numbers. We can do that by adding 2 and 3 and then dividing by 2. 2 plus 3 equals 5 and 5 divided by 2 is 2.5 So the Median of the data set is 2.5 Even though the number 2.5 isn’t actually a member of the data set, it’s the Median because it represents the middle of the data set and it splits the members into two equally sized groups. Okay, so now you know the difference between Mean and Median. But what about the Mode of a data set? What in the world does that mean? Well, “Mode” is just a technical word for the value in a data set that occurs most often. In the data sets we’ve seen so far, there hasn’t even been a Mode because none of the data values were ever repeated. But what if you had this data set? This set has 6 members, but some of the value are repeated. If we rearrange them, you can see that there’s one ‘1’, two ‘2’s and three ‘3’s The Mode of this data set is the value that occurs most often (or most frequently) so that would be 3 since there’s three ‘3’s. Now don’t get confused just because the number 3 was repeater 3 times. The mode is the number that’s repeated most often, NOT how many times it was repeated. As I mentioned, if each member in a data set occurs only once, it had no mode, but it’s also possible for a data set to have more than one mode. Here’s an example of a data set like that: In this set, the number 7 is repeated twice but so is the number 15. That means they tie for the title of Mode. This set has two modes: 7 and 15. Okay, so now that you know what the Mean, Median and Mode of a data set are. Let’s put all that new information to use on one final real-world example. Suppose there’s this guy who makes and sells custom electric guitars. Here’s a table showing how many guitars he sold during each month of the year. Let’s find the Mean, Median and Mode of this data set. First, to find the Mean we need to add up the number of guitars sold in each month. You can do the addition by hand or you can use a calculator if you want to. Either way, be careful since that’s a lot of numbers to add up and we don’t want to make a mistake. The answer I get is 108. So that’s the total he sold for the whole year, but to get the Mean sold each month, we need to divide that total by the number of months which is 12. 108 divided by 12 is 9, so the Mean (or average) is 9. Next, to find the Median of the data set, we’re going to have to rearrange the 12 data points in order from smallest to largest so we can figure out what the middle value is. There, that’s better. Since there’s an even number of members in this set, we can’t just choose the middle number, so we’re going to have to pick the middle two numbers and then find the Mean of them. 9 and 10 are in the middle since there’s an equal number of data values on either side of them. So we need to take the Mean of 9 and 10. That’s easy, 9 plus 10 equals 19 and then 19 divided by 2 is 9.5 So, the Median number of guitars sold is 9.5. That means that in half of the months, he sold more than 9.5, and in half of the months, he sold less than 9.5. Last of all, let’s identify the Mode of this data set (if there is one). We let’s see… there’s two ‘8’s in the data set… Oh… but there’s three ’10’s. That looks like the most frequent number, so 10 is the Mode of this data set. It’s the result that occurred most often. Alright, so that’s the basics of Mean, Median, and Mode. They are three really useful properties of data sets and now you know how to find them. But sometimes, the hardest part about Mean, Median and Mode is just remembering which is which. So remember that “Mean means average”, Median is in the middle, and Mode starts with ‘M’ ‘O’ which can remind you that it’s the number that occurs “Most Often”. Remember, to get good at math, you need to do more than just watch videos about it. You need to Practice! So be sure to try finding the Mean, Median and Mode on your own. As always, thank for watching Math Antics, and I’ll see ya next time. Learn more at www.mathantics.com