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All right.

This video we're going to look at measures of central tendency beginning with some terms.

I'm sure you're already familiar with that is mean median and mode.

One thing to note here I want to show you the symbols for mean we actually have two different symbols

depending on if we're talking about the population or a sample.

So if we're talking about us and by the way throughout these videos I mean introduce terms for us our

symbols that represent a population on the top.

I'm going to sort of do these all this old branching thing you'll see on the top a symbol for for population

which is the Greek symbol mew and below look for for sampling on the line below.

It's an X with a bar over it.

So this is population

and sample mean.

Let's quickly go through how to calculate each of these which again I'm sure you're familiar with.

I mean you just take the sum of all the values that you have and divide that sum by the number of values

you have.

So it's just some of your values divided by n and representing the number of data points Median is a

little bit different to calculate your median you line up all your values from smallest to largest and

the median is the value that is directly in the middle.

If you have haven't that that works very well if you have an odd number of values but if you have an

even number of values you take the two values in the middle and you average them and that's your median

mode is just the value that comes up most often in your data.

Now one question I get a fair amount is well how do I know which is the better representation of central

tendency.

People don't usually say measure of central tendency but the better sort of middle value from my data

mean or median.

Nobody really uses modal all that often but you know when is it more appropriate to use mean or median.

Well up there both they represent slightly different takes on the data.

So you know if you've got both.

Sometimes you look at both of them and that kind of helps guide your thinking.

But there is certainly some situations where one or the other might be more appropriate to you.

So oftentimes when you're when you're talking about money especially when it comes to things like investments

mean might be more appropriate for instance if you're looking at the returns of an investing company

or you know what or whether the average returns for a venture venture capitalist I mean is a good representation

you know because a venture capitalist they're going to sell some home runs and get a big payback and

they're going to take a lot of losses and that's OK as long as they're there they're mean return.

Ends up being high.

The average return ends up being high median.

You might use median if say you were trying to price a product to a certain population like let's say

you're entering a new market and you want to understand sort of what the average person can afford.

You might not use mean because for instance let's say there's there's a there's there's there's a discrepancy

in terms of incomes in that that new market.

You know some people are making a huge amount of money.

A lot of people are just scraping by.

If you're trying to sort of market your product to the masses looking at mean might not be so good because

the average might have been skewed by those those really high earners.

I'm even that even though they might represent a small part of the population.

So in that case using the median household income might be a lot more appropriate in terms of thinking

about what the right price to gain market share would be.

So that's that those are just some some some quick thoughts on I mean otherwise known as average or

at or at arithmetic mean versus median all.

Oh get into the different scene arithmetic and geometric mean in a moment.

So I want to.

But before that I want introduce a couple other concepts related to sort of related to the median.

The first is percentiles.

This is gonna come up a huge amount in statistic.

This idea the statistics this idea of percentiles when we say the percentile what we mean is the percentage

of results in your data that falls below a certain number.

So for instance if I said this value is at the fifth percentile it means 5 percent of all of your values

fall below that number if I said if you tell me you scored in the ninety ninth percentile for the S.A.T.

first of all congratulations that's very good.

That means ninety nine percent of all the other test takers everyone else who took the S.A.T. that year

had a score that was below your value.

So again a percentile is the value at which whatever percent falls below that number eighth percentile

means 68 percent of the results are below that value.

Another concept I want to introduce is quartile is very very much related

trials are typically represented by the 25th percentile the fiftieth percentile and the seventy fifth

percentile.

This quarter's your data so everything below you is basically saying everything below the twenty fifth

percentile.

That's one quartile.

Everything between the 25th percentile and the fiftieth percentile.

That represents a quarter of your data.

Everything between the fiftieth and 70 percentile.

That's a quarter of your data and everything at the seventy fifth percentile and above that represents

a quarter a quarter of your data.

It's often represented in in one when people report but report their core tiles as everything between

the 70 the 25th and the seventy fifth percentile.

So again going back to the idea of S.A.T. scores oftentimes colleges will report their S.A.T. scores

in court files so they'll say that's the twenty fifth to seventy fifth percentile is say for S.A.T.

scores would be between twelve hundred and fourteen hundred.

I don't even remember if we're still on a sixteen hundred score scale.

I know that's changed back and forth but yeah that would be our entire quartile range would be between

the 25th and the seventy fifth percentile.

So you say you know art are twenty four.

Twelve hundred is at the twenty fifth percentile meaning twenty five percent of our results were below

twelve hundred fourteen hundred is the seventy fifth percentile meaning seventy five percent of the

results are below fourteen hundred in our entire quartile range would be between twelve hundred and

fourteen hundred.

Right the last measure of central tendency that I want to introduce is something called the geometric

mean

and I want to distinguish this from the arithmetic mean which is what we were talking about earlier.

I sort of teased this earlier.

So the arithmetic mean is what we generally think about when we're talking about an average.

That's where you take the sum of all your your your your values and you divide by the number of values

you have.

That's the earth medic mean the geometric mean is a little bit different.

We calculate it by taking the product

of our values

and we take that to the one over in power.

Now the geometric mean so you can see it's a different calculation.

The geometric mean is used to calculate growth rates over time and is often called the time weighted

rate of return.

It's not a huge part of statistics and it's not a big part of this course.

However I wanted to introduce it here because it is it's great for finance.

The arithmetic mean does not take into account compounding.

So in situations where there is compounding where you have growth rates over time you will want to use

the geometric mean it's a more accurate reflection of average the average changes.

So for instance if you've got an investment that is growing over time this compounding associated or

shrinking over time I should say.

But there's this compounding associated with the changes over time.

The geometric mean is a really good measure.

Likewise if you have a population that is changing over time and here I'm using the term population

to mean like people living in an area or a wildlife population again in these situations a geometric

mean will be useful.

So let me illustrate with an example.

Oh and one thing I should note here because it's taking into account compounding the geometric mean

is always going to be lower than the arithmetic mean just little factoid.

So let's say I let's say you running a sales department and you've been given a mandate from your supervisor

to double sales within five years and you want to know how much does your sales have to grow each of

those five years so that you will hit your target of doubling growth within five years.

Some of you may be thinking right now well OK doubling growth that's one hundred but I have to increase

growth by increased sales by 100 percent and I've got five years to do it.

So I'll just take one hundred percent divided by five and that is the number I need to hit every year

in terms of growth to get to that one hundred percent increase.

And you would be wrong.

The reason is again because of compounding and in fact you don't need to get all the way to 20 percent

growth in order to hit your target of one hundred percent increase in your sales.

That's because let's say you start off at one hundred thousand dollars in sales.

I'm just making up numbers here.

If you increase your sales by 20 percent in year one now you're at one hundred twenty thousand dollars.

And then if you take 20 percent of that it's a larger than 20 thousand dollars increase the next year

and you're just going to keep growing and you'll actually end up well ahead of your target of doubling

sales and of course you're not that person you you only want to do the minimum.

So let's let's actually figure out using the geometric mean what your target growth needs to be year

over year.

So thinking about this year the year end numbers here your is your your increase year over year.

Let's just.

We don't need to to solve for what that number is going to be we just need to realize that we need to

hit to the product of our year over year growth over those five years needs to be two because you're

looking to double your your sales.

So what we've got here is it's red to actually perform my calculation the product of our end numbers

meaning our growth year over year is 2.

And we've got five years to do it.

So up here I'm going to take one over five and that equals I think something like one point one for

nine that's rounded on meaning we need to be at about 15 percent growth per year

one point one for nine is it would be like one hundred and fifteen percent.

And of course you know that would be an increase so you really need to be just growing by 15 percent

per year.

So remember I guess a geometric mean not a huge part of statistics we're not gonna be using it a whole

lot but it is something you're going to want to keep in mind when you get to subjects like finance.

Remember it is for growth rates over time good for things like investments.

And it takes into account compounding.