Hello and welcome to my secret math tutor in this video, we're gonna cover how to use Chebyshev's theorem
Now this is a really interesting theorem
But essentially what it says is it gives you the proportion of data that is within K standard deviations of the mean
So you could have have you know
Some sort of distribution doesn't have to be normal or anything like that could be any distribution and you're curious
How much of my data falls within say two standard deviations or three standard deviations and this quick and handy formula
Essentially just tells you that percentage
Now the interesting part is it gives you a lower number.
So it actually could be more than what the formula says
But at least it gives you a ballpark answer and it says it must be more than this percentage.
Let's get into a couple of examples and see how this exactly plays out.
For this first one we're going to compare what Chebyshev's theorem says versus the actual proportion of data
All right, so I have 30 data points over here
And we want to know what proportion of the data is within two standard deviations of the mean
So since this is all about chebyshev's theorem. We'll go ahead and start with that
so we'll simply say 1 divided by 2 squared or 1 minus 1 over 2 squared that's
the same as 1 minus 1/4
That's 3/4 which is the same as 75%
So, you know
I really don't even have to do anything to the data and what it's saying is that
At least 75% of my data is within 2 standard deviations of the mean
Now watch how much time this takes when I'm using the the actual percentage or I'm looking for the actual percentage
First thing is we need to calculate the mean and the standard deviation for our data. Now we have a lot of data
So I'm going to use my calculator
To take care of this and we'll speed up the video so you don't have to sit through all of that
Alright and with that it looks like we have a mean of around thirty six point one and the sample standard deviation of twelve point
Eight. So let's go ahead and write down those values and see what we need to do with them
So meaning that was around thirty six point one
standard deviation
Around a twelve point eight
So we want to know a range of values
Within two standard deviations. So what I'm going to look at is our mean thirty six point one and we'll subtract
two standard deviations
See what that gives us. We'll also do the same thing
Thirty-six point one and we'll add two standard deviations. So twelve point eight
Alright this gives me ten point five
And sixty one point seven
So this is a lower and an upper
Range for where I can expect my data and if we're looking at the data
It's all organized from the largest all the way down to the smallest
We can see that most of our data does fall within ten and a half up to sixty one point seven
There's only two values up here in the corner, which do not fall in that range so I can say that 28 out of 30
fall within two standard deviations
All right reducing that fraction getting a percent that turns out to be ninety three point three percent
So we can really make in comparison between what chebyshev's theorem says versus the actual you know
Just using this quick and nice formula that said we should expect at least seventy five percent and sure enough
That's what we're finding. The actual is, you know a little bit higher
93.3%
But it gives us a ballpark. We know at least 75 percent fell within that range
Alright, let's do this with another example
But something a little bit different to see how you might actually use this
so imagine the average price of a new car is
$36,000 with a standard deviation of
$4,100 we want to know what is minimum percentage of cars. That should sell between twenty-two thousand and fifty thousand
So here we have a range and we're looking for that percentage. All right
Visually, what you want to think of is where everything is in relation to everyone else
So this mean that we're getting for the sale price of a car
it's somewhere in our distribution and
We're looking at this range up here up to fifty thousand
Or small as 22,000
So I'm essentially looking for this K value that I can use in chebyshev's formula
Because we already have the mean and we already have the standard deviation. All right, so let's work with that
I know that my mean of
36,000
Plus some number of standard deviations
There's our standard deviation should give us that
50,000 so I'm really just borrowing that top value
You could also do the same thing with the bottom value using these two. Anyway, we want to solve for K
so the first thing we can do is
subtract
36,000 from both sides
Let's see, it'll give us 14,000
Now we want to divide both sides by the 4,100
And it looks like a little bit more than three standard deviations is what this range is right now that we have the standard deviation
Now we can use chebyshev's to give us that percentage. So remember that's
It's 1 minus 1 over K squared
So we have our seen the the number of standard deviations. So this be 1
minus 1 over 3
Point let's just make it three point four squared
And log it was point nine one three
So as a percentage
We expect that the minimum percentage of cars that are in this range to be about
91.3%
All right. Hopefully that helps out. If you'd like to see some more videos, please visit mysecretmathtutor.com