Statistics: The Average

Introduction to descriptive statistics and central tendency. Ways to measure the average of a set: median, mean, mode

welcome to the playlist on statistics, something I've been needing to do for some time.

So I just want to get right into it, and I want to do as many examples as possible

and hopefully give you the feel for what statistics is all about

So I'm just gonna start off in case you're not familiar with it

I think a lot of people have an intuitive feel for what statistics is about

Well, in very general terms, it's kinf of getting your head around data

and it can oddly be classified in maby 3 categories

You have: Descripitive

So, say you have a lot of data, and you want to tell someone about it whitout giving him all the data

Maybe you can find indicative numbers that somehow represent the data

without having to go all over the data

So that would be "Descriptive statistics"

Well, I'll kind of group it together, there's "Inferential statistics"

So that's when you use data to make conclusions about things

So let's say you've sampled some data from a population,

and we'll talk a lot about sample versus population,

but I think you have just a basic sense of what that is,

right: if I survey 3 people who are going to vote for president,

I clearly haven't survey the entire population, I surveyed the sample.

But what inferential statistics are all about are:

"if we can make some maths on the sample, maybe we can make "inferences", or conclusions,

about the population as a whole.

Anyway, that's just a big picture of what statistics is all about,

so let's just get into the middle of it, and we'll start with the Descriptive

So the first thing that I would want to do, and I think most people would want to do

when they are given a whole set of numbers and they are told to describe it

they would think: "well, maybe I could come up with some number

that is most indicative of all of the numbers in that set

or some number that represents kind of the central tendency

this is a word you see a lot in statistics books

the "central tendency" of a set of numbers

And this is also called the average.

And I'll be a little more exact here that I normally am with the word average

when I talk about it in this context, it just means

that the average is one number, that somehow

is giving us a sense of the central tendency

or maybe a number that is most representative of a set

and I know that sounds all very abstract,

but let's do a couple of examples:

So there's a bunch of ways you can actually measure

the central tendency, or the average of a set of numbers

and you've probably seen this before,

they are the "mean"... Well actually there is types of means,

but let's stay with the arithmetic mean

And later on, we'll do geometric means,

and maybe we'll cover the harmonic mean one day

There is the "mean", the "median"

and the mode.

And in statistics speak, these are all

kind of representative of data set or a population central tendancy

or sample central tendency

And they are all, collectively, they can all be forms of an average

and I think when we'll see examples it will make a litlle more sense

In everyday speak, when people talk about an average,

And I think you've already computed averages in your life,

They are usually talking about the arithmetic mean

So normally some is asking you to calculate

the average of these numbers, and they are expecting you to do something,

they want you to figure out the arithmetic mean.

They don't want you to figure out the median or the mode.

But before we go any further, let's figure out what those things are.

So let me figure out a set of numbers.

So let's take...1...another 1, a 2, a 3, let's say I have a 4...

That's good enough. We just want a simple example.

So, the mean, or the arithmetic mean,

and what's probably what you're most familiar with

when people talk about average,

and that's essentially: you add up all the numbers,

and you divide by the number that they are.

So that's: 1+1+1+3+4

and you're going to divide by: 1,2,3,4..5 numbers

1+1 is 2, 2+2 is 4, 4+3 is 7, 7+4 is 11

So that's equal to 11 over 5

That's what? 2 and 1/5, so that's equal to 2.2

And some could say: "hey you know that's a pretty representative number

of this set, that's the number that all of this numbers,

you can kind of say, are closest to.

Or 2.2 is the central tendancy of this set"

And in common speak, that would be "the average".

But if we want to be a little bit more particular,

this would be the arithmetic mean of this set of number that you see

it kind of represents them.

If I didn't want to give you the list of 5 numbers,

I could say: "Well you know I have a set of 5 numbers,

and their mean is 2.2"

and it can of tells you a little bit of, at least, where the numbers are

Right, we'll tell a bit more about how far the numbers are from that mean

in probably the next video.

So that's one measure.

Another measure, instead of averaging it this way,

you can average it by: putting the numbers in order,

which I actually already did,

so let's just right them down in order again.

1,1,2,3,4

And you just take the middle number!

So let's see: there's 1,2,3,4...5 numbers,

so the middle number is gonna be right here, right?

The middle number is 2

There is two numbers greater than 2,

and there is two numbers less than 2.

And this is called the median,

which is actually very simple computation:

you just have to sort the numbers,

and then you find whatever number

where you have an equal number of "greater than", or "less than that number"

So the median of this set is 2.

And you see, that's fairly close to the mean.

And, there is no right answer.

One of these isn't a better answer for the average,

they are just different ways of measuring the average.

So here it's the median.

And I what you might be thinking:

"well, that was easy enough when we have 5 numbers,

but what if we have 6 numbers, what if it was like this:

what if this was our set of numbers?

1,1,2,3,4...let's add another 4 there

So now there no midle number right?

I mean, 2 is not the middle number,

because ther is two numbers less than 2,

and there three greater than 2

And 3 is not the middle number,

because ther's two larger and three less than 3

So there's no middle number.

So when you have a set with even numbers,

and someone tells you to figure out the median,

what you do is you take the middle two numbers

then you take the arithmetic mean of those two numbers

So in this case, for this case,

the median would be 2.5

Fair enough

Let's put this aside,

because I want to compare

the median and the mean and the mode

for the same set of numbers

But that's a good thing to know,

because sometimes, it can be a little bit confusing.

And you know, these are all definitions,

this are all kind of mathematical tools

for getting you head around numbers,

there's nothing like:

oneday, someone saw one of these formulas

at the face of the sun, and say:

"oh, that's part of the universe,

this is how the average should be calculated"

These are human constructs

to get our heads around large sets of data

This isn't a large set of data,

we have five numbers,

but if we have five billion numbers,

you can imagine that you don't like

to think about every number individually.

But anyway, before I talk more about that,

let me tell you what the mode is.

And the mode, to some degree,

is the one that most people probably forget

or never learn, and when then see it on an exam

it confuses them, because "oh, that's sounds very advanced"

But in some way it is the easiest

of all of the measures of central tendancy

The mode is essentially:

"what number is most common in the set"

So, in this example, there is two 1,

and there is one of every thing else, right?

So the mode here is: 1

So mode, you kind of say is

"the most common number".

And then you can say:

"hey, what if our set was: 1,1,2,3,4,4

So here I have two 1, and I have two 4.

And this is were the mode gets a little bit tricky,

because either of these would have been a decent answer

for the mode.

You could have actually say that

"the mode of this is 1"

or "the mode of this is 4".

And it gets a little bit ambiguous,

and you probably want a little of clarity

from the person asking you.

Most times on a test, when they ask you,

there's not gonna be this situation.

There will be a most common number in the set.

So then I say: "Why wasn't one of these good enough?"

Why we do we use averages?

Why don't we use arithmetic mean all the time?

What are the median and mode good for?"

Well I'll try to do a little one example of that,

and see if it "rings true with you".

And then you can think a little bit more.

Let's have this set of numbers:

3,3,3,3,3, and, I don't know... and 100!

Sa what's the arithmetic mean here?

So 1,2,3,4,5 threes and 1 one hundred.

So it would be 115/6, right?

Because there's 1,2,3,4,5,6 numbers.

And 115 is just the sum of all of these.

So that's equal to:...19 and 1/6

Fair enough, I just added the numbers,

and divided by how many they are,

But my question is: "is this really a representative of this set?"

I mean, I have a ton of threes,

then I have one 100 all of a sudden,

and we are saying that the central tendency

is 19 and 1/6.

Does 19 and 1/6 really seem an indicative of this set?

Maybe is does, depending on the application,

but it just seems a little bit off.

My intuition would be that the central tendency

is something closer to 3.

Because their is a lot of threes here.

So what will the median tell us?

I already put these numbers in order, right?

If I gave to you in another order,

you would put them in this order.

And you would say: "what's the middle number?"

Let's see: the middle two numbers,

since they have an even number, are 3 and 3.

So if I take the arithmetic mean of 3 and 3,

I get 3.

And this is maybe a better measurement

of the central tendency, or the average

of this set of numbers, right?

Essentially what it does is,

by taking the median,

I wasn't affected by this very large number

that's very different from the others.

The statistics call that an outlier.

A number, that, you know,

if talked about average home prices,

maybe every house in the city

is 100,000$, and then

there's one house that cost a trillion $.

And then someone tells you:

"the average price is.. I don't know.. 1 million $"

then you might have a very wrong

perception of that city.

But the median house price, would be 100,000$

and you would get a better sense

of what houses in that city are like.

So, similarly, this median maybe

gives you a better sense of what

the numbers in this set are like.

Because the arithmetic mean

was influenced by this outlier.

And be able to tell what an outlier is,

is kind of those things that

statisticians will say:

"well I know it when I see it"

There isn't really a formal definition for it,

but it tends to be

a number that really kinds of sticks out.

Sometimes it's, you know, a measurement error or whatever.

And then finally the mode.

What is the most common number in the set?

Well there is five 3 and one 100,

so the most common number is...

3

So in this case, when we have this outlier,

the median and the mode tend to be

maybe a little bit better about giving you an indication

of what this numbers represent.

Maybe this was just a measurement error.

But I dont know.

We don't actually know what these numbers represent

if these where house pricies,

I would argue that these are more indicative

measures of what are the prices in the area.

But if this something else,

if these are scores on a test,

maybe there is one kid in the class, out of 6 kids,

who did really really well

and everyone else didn't study.

And this is more indicative of how

students at that level are doing on average.

Anyway, I'm done talking about all of this.

And I encourage to play with a lot of numbers,

and deal with the concepts yourself.

In the next video, we'll explore

more descriptive statistics,

and instead of talking about the central tendency,

we'll talk about how spread apart

things are away from the central tendency.

See you in the next video!