# Mean, Median, and Mode

Hey guys! Welcome to this video over mean, median, and mode.

Mean, median, and mode all represent averages. Now, typically when people think of finding the average, they have the mean in mind. The mean is the one where you add up all the numbers, and divide by how many numbers there are. And it’s true, that is the most common kind of average, but really, all three are a type of average.

Let’s take a look at how to solve for each average.

## Mean

First, the mean.

Let’s say we have a list of nine numbers:

23, 16, 54, 27, 31, 16, 33, 24, 19

So, to find the mean, we need to add up all of the numbers in the list, and then divide by the amount of numbers that there are, which we know is 9 because we have 9 numbers.

When we add up all the numbers, we get 243. So now we take this number and divide it by 9 to get 27.

And that is all there is to finding the mean! You add up all the numbers on the list, then divide by how many numbers there are.

## Median

Finding the median is a little easier because you don’t have to do any addition or division, which can get pretty crazy depending on how many numbers you have in your collection of data.

The median is literally the number in the “middle” of a list of numbers. However, before we just look at the number in the center of our data, we must first arrange the set of numbers in ascending order, which means smallest to largest.

So, using our same list, if we arrange it in ascending order we would get:

16, 16, 19, 23, 24, 27, 31, 33, 54

So we can see that the number in the middle now is 24, and we know this because we have an equal amount of numbers on each side of 24. So, we have four numbers here (to the left), and four numbers here (to the right), putting our 24 in the middle.

Now, we can easily see the middle number in this list, because we only have nine numbers; but what happens when we have a list of one thousand numbers? I mean you can use the same method, but it may take you a little longer. So, mathematicians have graciously worked to give us a formula to make this process quicker.

### Formula for Median

$$\frac{(n+1)}{2}$$

Now in this formula, $$n$$ is the amount of numbers on our list. Now, let’s see if this works for us. Our $$n$$ is equal to 9 because we have 9 numbers in our list, and we’ll just use the same list. So with that we have, $$\frac{(9+1)}{2}$$, which is equal to $$\frac{10}{2}$$, which is equal to 5. So 5 is clearly not 24, but what 5 is telling us is that the 5th number in the list is our median, which we can see by looking at this list.

So, as you can see, finding the median is relatively simple, but it’s especially simple when we are looking for the median in a list with an odd amount of numbers. Like, in our case, we have been working with a list of nine numbers. But what about when we have a list with an even amount of numbers? Well, you still would use the same general method. You would set them up in ascending order, except now you have to take the mean of the two numbers in the middle.

So, let’s just add a number to our list of 9 numbers. Let’s say 60.

16, 16, 19, 23, 24, 27, 31, 33, 54, 60

So now we have 10 numbers, which is an even amount of numbers. So we need to take the mean of the fifth and sixth numbers. So we take the sum of 24 and 27, then divide by the amount of numbers that we are summing, which in this case is just 2. So we have, $$\frac{24+27}{2}=25.5$$. So this makes 25.5 our median.

So remember, when you have an odd amount of numbers, like in our example here, you just take the number in the middle and that’s your median. But when you have an even amount of numbers, you have to take the two middlemost numbers and then take their mean to get your median.

## Mode

Okay, now onto mode.

Good news: the mode is definitely the simplest of the three to find. The mode is the number that appears the most amount of times.

Taking a look at our list that we have been using, we can see that 16 is the only number that is being repeated, and that is our mode. Simple enough.

Now, if there are no numbers that are repeated, then there is no mode. Also, let’s pay close attention to our definition. The mode is the number that appears the most amount of times. So, it may be that you have a number repeated, but there is another number that is repeated more times. The number that is repeated more is the mode, but you can have multiple modes. If two numbers are repeated the same amount of times, they are both modes.

I hope that this video was helpful. For further help be sure to check out our other videos by subscribing to our channel below.

See you guys next time!

## Practice Questions

Question #1:

Find the mean of the following data:

4, 5, 7, 5, 7, 10, 7, 12, 10, 15, 17

8

9

11

7

The correct answer is 9. To calculate the mean, add up all of the numbers in the list and then divide that by the total amount of numbers. In this case, $$99÷11=9$$.

Question #2:

Find the mean of the following data:

70, 72, 75, 76, 80, 113

51

61

71

81

The correct answer is 81. The sum of all the numbers in the list is 486, and there are 6 numbers in the list. Dividing 486 by 6 gives the answer 81.

Question #3:

Find the mean of the following data:

150, 150, 156, 156, 161, 163

180

256

156

194

The correct answer is 156. There are 6 numbers in the list, and the sum total of these numbers is 936. In this case, $$936 ÷ 6 = 156$$.

Question #4:

Find the median of the following data:

5, 4, 6, 7, 6, 8, 3, 5, 2

5

2

9

7

The correct answer is 5. To find the median, locate the number that is in the middle of the list. First, the list should be arranged so the numbers are in ascending order (2, 3, 4, 5, 5, 6, 6, 7, 8). Now, simply locate the number that is in the middle of the list. This would be 5.

OR

Use the formula $$\frac{(n+1)}{2}$$, where n represents the amount of numbers in the list. In this case, $$n=9$$. So, $$9+1=10$$, which would then be divided by 2 to get 5. This number is saying that the fifth number in the list (once the list is arranged in ascending order) is the median, which in this case is 5.

Question #5:

Find the median of the following data:

16, 18, 13, 15, 19, 20, 24

13

16

18

24

The correct answer is 18. First, arrange the list in ascending order (13, 15, 16, 18, 19, 20, 24). The number in the middle of the list, 18, is the median.

OR

Use the formula $$\frac{(n+1)}{2}$$, where n represents the amount of numbers in the list. In this case, n represents 7. So, $$7+1=8$$. Divide 8 by 2 to get 4. This means that the median is the fourth number in the list in its ascending order, which in this case is 18.

Question #6:

Find the median of the following data:

1, 3, 9, 8, 7, 6, 7, 3

6.5

7.5

6

7

The correct answer is 6.5. First, arrange the list in ascending order (1, 3, 3, 6, 7, 7, 8, 9). Next, find the number in the middle of the list. In this case, there are an even number of numbers in the list, so the median falls between the two middle numbers, which is the average of the two middle numbers. The two middle numbers in this list are 6 and 7, so 6.5 is the median.

Question #7:

Find the mode of the following data:

3, 4, 4, 4, 5, 5, 8, 9

9

4

8

3

The correct answer is 4. The mode is the number that appears most frequently in the list of numbers. In this case, 4 appears most frequently, so 4 is the mode.

Question #8:

Find the mode of the following data:

23, 34, 23, 34, 44, 24, 44, 11, 23, 43

43

44

34

23

The correct answer is 23. The mode is the number that appears most frequently in the list of numbers. In this case, 23 appears most frequently, so 23 is the mode.

Question #9:

Find the mode of the following data:

5.5, 6.3, 7.2, 4.4, 7.2, 5.6, 5.1

7.2

4.4

5.5

5.6