>> In this video we're going to do the quartiles
and the five-number summary for small sample sizes.
Okay. So the mean and median only tells us part of the story
in describing the distribution.
A more powerful numerical description
of a distribution requires both a measure of center
and a measure of spread.
One way to describe spread is by using the five-number summary.
Definition: The five-number summary consists
of a minimum value; first quartile, Q1; the median,
or the second quartile; the third quartile, Q3;
and the maximum value.
Not only does the five-number summary tell you the spread
of the data in quarters, but also,
these five numbers are very portable
and it gives you an idea of how the data is distributed.
So before we start there is a disclaimer.
It turns out that different methods arise from the lack
of universal agreement [inaudible] among statisticians
on how to calculate quantiles Q1 and Q3.
So I do have two videos
that describe how to calculate Q1, Q3.
And maybe one of these two methods are the ones
that you are going to use in your class.
Okay. Here's an example.
Find the five-number summary from a sample
of 15 students' travel time in minutes from home to school.
So here are the data.
We're going to want 5 numbers summary:
Min, Q1, median, Q3, max.
Now the data should be in order from smallest to largest
so make sure you do that in your dataset.
This one's ready to go.
Then I'm going to label each data point.
Five is the first data point.
Sixty is the last data point.
So I have 15 data points all together.
So min is easy which is 5.
Max is 60.
Next thing I want to do is to locate the median.
I do have a video on how to create
or how to locate the median.
If you want, you can watch that for more examples.
But in this case to find the median it's best
to use the n plus 1 over 2 to find the location of the median.
So what's n?
Well, that's 15 data points plus 1 divided by 2,
means it's the eighth position.
That means the median is 20.
Again, if that was a little bit too fast for you, I got a video
that shows you how to find the median.
And I actually have two ways to find the median.
Now let's find Q1.
So I'm going to clear the median calculation.
I'm going to show where the median is and I'm going
to find and locate Q1 now.
First thing I want to do is label
and locate my data starting
from the min all the way up to the median.
You'll note that I don't count the median at all
or I don't say this is my eighth position.
Okay. And that's where the controversy comes around.
Do we include the median or not?
In this video we're not going to include the median to locate Q1.
So to figure out the location
of Q1 this time our formula changes just a bit.
The variable instead of n,
n means how many data points you have, you know, in your dataset.
Well, finding Q1 in this case we're only going
to find a subset of it.
We're only going to look at 7 data points.
So that's, Q will be 7.
We're going to have 1 divided by 2 and we're going to get 4.
This 4 is the fourth position of our dataset and that means
that Q1 is 10, the data is 10.
So now our Q1 is 10.
So it's very similar to the median, Q1,
because the median is, you know, half of the data,
and Q1 is half of the half.
All right.
Now let's look for Q3.
Okay. So in Q3 again we're going to write the location, 1, 2, 3,
4, 5, 6, and 7, right.
We're not going to include the median.
So there are 7 data points available.
We're going to write our formula, Q plus 1 divided by 2.
The Q stands for how many data points you have right
of the median in this case and that's going
to by 7 plus 1 divided by 2, and then that's going to be 8 over 2
and that's the fourth position.
Where is that fourth position?
Right there.
And that Q3 now is 30.
So now we have the five-number summary that we can use
for perhaps in the [inaudible] which I do have a video
for you to watch on that.
So the next dataset is find the five-number from a sample
of 20 students' travel times in minutes, from home to work.
So I'm going to write down the 20 times from the students.
First I make sure that these are in order
and then I find their location like the previous dataset.
So I need five numbers for my five-number summary.
The min and max is going to be easy.
Min is 5, max is 85.
And now we're going to find and locate where the median is.
And that's going to be n plus 1 divided by 2.
n is 20 because I have 20 data points.
Okay. So 20 plus 1 is going to be 21 divided by 2 is 10.5.
And then 10.5 place so the median is between 20 and 25.
So the median, I'm going to have to take 20 plus 25 divided by 2
to find the average and that's going to be 22.5.
So that's going to be the median.
Okay. I'm going to rewrite the dataset here.
And now the median is going to be there.
I'm going to find the location.
So in this case the location is going
to be just the first half of the data.
That means 1 through 10.
The median in this case is between 20 and 25
so I can count 20 because I'm not technically counting
the median.
And so this time to locate Q1 it's going
to be Q plus 1 divided by 2.
Q is the location values from 1 through 10 so it's going to be,
Q is going to be 10 plus 1 divided
by 2 is going to be 11 over 2.
That's going to 5.5 place.
That's between 5 and 6.
So the data that I'm looking for are 15 and 15.
So Q1 is going to 15 plus 15
over 2 equals 30 over 2 which is 15.
Looks a little confusing but remember we want
to first find the location which are the orange numbers
and then we look at the data
which the orange numbers are telling us.
And if there are two values that we need to work with,
then we're going to have to average it.
All right.
Then let's look at the other half, 1 through 10.
Let's locate Q3.
Same thing, Q plus 1 over 2.
So Q in this case is, again, 10 values, not 20.
That would be n. Okay.
So that's 10 plus 1 over 2 is 5.5 place which between 5 and 6.
That means I'm going to look at Q3 with using
and seeing these two pieces of data so I'm going
to take 40 plus 45 divided by 2 which is 42.5.
And that's the data that I need.
Doing this for small samples, what is a small sample?
Well, that's debatable.
Maybe it's less than 30.
Maybe it's just 10 values, 20 values.
That is going to be something that is debatable
and you're going to have to think about.