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# Average or Central Tendency: Arithmetic Mean, Median, and Mode

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- Let's say I've got a set of numbers.
- 2, say I've got three 3's, I've got a couple of 4's, and
- I've got a 10 there.
- And what we want to do is find the middle of these numbers.
- We want to represent these numbers with the center of the
- numbers, or the middle of the numbers, just so we have a
- sense of where these numbers roughly are.
- And this central tendency that we're going to try to get out
- of these numbers, we're going to call the average.
- The average of this set of numbers.
- And you've, I'm sure, heard the word average before, but
- we're going to get a little bit more detailed on the
- different types of averages in this video.
- The one you're probably most familiar with, although you
- might have not seen it referred to in this way, is
- the arithmetic mean, which literally says, look, I, the
- arithmetic mean of this set of numbers, is literally the sum
- of all of these numbers divided by the number of
- numbers there are.
- So the arithmetic mean for this set right here is going
- to be 2 plus 3 plus 3 plus 3 plus 4 plus 4 plus 10, all of
- that over, how many numbers do I have?
- 1, 2, 3, 4, 5, 6, 7.
- All of that over 7.
- And what is this equal to?
- This is 2 plus 9, which is 11, plus 8, which is 19, plus 10,
- which is 29.
- So this is going to be equal to 29/7, or you could say it's
- equal to 4 and 1/7.
- If I got my calculator out, we could figure out
- the decimal of this.
- But this is a representation of the central tendency, or
- the middle of these numbers.
- And it kind of makes sense.
- 4 and 1/7, it's a little bit higher than 4.
- We're kind of close to the middle of our number range
- right there.
- And you might say, well, it's a little skewed to the right
- and what caused that?
- And well, gee, 10 is a little bit larger than all of the
- other numbers.
- It's kind of an outlier.
- Maybe that skewed this average up, the arithmetic mean.
- So there are other types of averages, although this is the
- one that, if people just say, hey, let's take the average of
- these numbers, and they don't really tell you more, they're
- probably talking about the arithmetic mean.
- The other forms of average, though, are the median, and
- this literally is the middle number.
- If there are two middle numbers, you actually take the
- arithmetic mean of those two middle numbers.
- You actually find the number halfway in between those two
- middle numbers.
- So the median of this set right here-- let me just
- rewrite them.
- So I have a 2, a 3, 3, 3, 4, 4, 10.
- So, let's see, we have seven numbers right here.
- The middle number, if I go 1, 2, 3, to the
- right, we're there.
- If we go 1, 2, 3 to the left, we're there.
- The middle number is that 3 right there.
- I just listed them in order, and I said, well, look, 3, you
- could think of it as the fourth number from the right,
- and it's also the fourth number from the left.
- 3 is the middle number.
- And this case, it is the median.
- So in this case, 3, if you use the median, is our average.
- And that also makes sense.
- I mean, it's literally the middle number, and if you look
- at this set of numbers, it kind of does represent the
- central tendency of this set.
- Now just to be clear, it was very clear what the middle
- number was, because I had an odd number of numbers.
- I had three on each side of the three, so it was very easy
- to figure out the median, the middle number.
- But if I had a situation-- let's say I have the situation
- where I have 2, 3, 4, and 5.
- Let's say that's my set of numbers.
- Well, here, there is no one middle number.
- The 3 is closer to the left than it is to the right.
- The 4 is closer to the right than it is to the left.
- There's actually two middle numbers here.
- The two middle numbers here are the 3 and the 4.
- And here, when you have two middle numbers, which occurs
- when you have an even number in your data set, there the
- median is halfway in between these two numbers.
- So in this situation, the median is going to be 3 plus 4
- over 2, which is equal to 3.5.
- And if you look at this data set, that's not what our
- original problem was, but if you look at this data set
- right there, you're actually going to find that the
- arithmetic mean and the median here is the exact same thing.
- Let's calculate it.
- What 's the arithmetic mean over here?
- It's going to be 2 plus 3 plus 4 plus 5, which is what?
- 5 plus 9, which is equal to 14, over 4.
- And what's this equal to?
- 14/4 is 3 and 2/4, or 3 and 1/2, the exact same thing.
- So for this data set, they were the same thing.
- For this data set, our median is a little bit lower.
- It's 3, while our arithmetic mean is 4 and 1/7.
- And I really want you to think about why that is.
- And it has a lot to do with this 10 that sits out there.
- All of these other numbers are pretty close to whichever
- average you want to pick, whether it's the arithmetic
- mean or it's the median.
- But this 10 is kind of an outlier, or it
- skews the data set.
- Maybe it's so much larger than the other numbers, that it
- makes the arithmetic mean seem larger than maybe is
- representative of this data set.
- And that's something important to think about.
- When you're finding the average for something, most
- people will immediately go to the arithmetic mean.
- But in a lot of cases, median will make a lot more sense, if
- you have these really large or really small numbers that
- could skew the data set.
- I mean, you can imagine, if this wasn't a 10-- or let's
- imagine adding another number here.
- If I added the number 1 million, if I added 1 million
- to this data set, if that was the eighth number, the
- arithmetic mean is going to be this huge number.
- It's going to be much larger than what is representative of
- most of the numbers in this data set.
- But the median is still going to work.
- The median is still going to be about 3 and a half, right?
- If you had 1 million here, it would be 1, 2, 3, 4.
- The middle two numbers would be that.
- It would be 3 and 1/2.
- So the median is less sensitive to one or two
- numbers at the extremes that otherwise would skew the mean.
- Now, the last form of average I want to talk
- about is the mode.
- It has nothing to do with ice cream.
- The mode is literally the most frequent number.
- And in this data set, it's pretty clear what the most
- frequent number is.
- I only have one 2, I have three 3's, I have two 4's, I
- have one 10, and even if want to include the million, I only
- have one million there.
- So here, the number that occurs most
- frequently is the 3.
- So, once again, the mode seems like a pretty good measure of
- central tendency or a pretty good average
- for this data set.
- Now the mode, it's a little tricky to deal with, and you
- won't see it used that often, because it becomes a little
- ambiguous when-- you know, look at this data set:
- 2, 3, 4, and 5.
- What is the mode there?
- All of these numbers are equally frequent.
- So if you have a situation like this, then you might
- just-- the mode really loses its meaning.
- It might force you anyway to take the median or the
- mean in some form.
- But if you really do have numbers that one shows up a
- lot more than the other, then the mode starts to make sense.
- So, hopefully, this has given you a pretty good overview of
- how to represent the central tendency of a data set.
- Very fancy word.
- But it's just saying, look, we're trying to represent with
- one number all of this data.
- And you might say, hey, why do we even worry about that?
- It only has seven numbers here or eight numbers here.
- But you can imagine if you had 7 million numbers or 7 billion
- numbers, and you don't want to show someone all of that data.
- You just want to give someone a sense of what those numbers
- are on average.
- And as we said, the arithmetic mean is what I see being used
- the most. But in situations where you might have numbers
- that would skew the arithmetic mean, because they're so large
- or they're so small, the median might
- make a lot of sense.

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