And now we'll look at several examples
of determining the
five-number summary, range,
interquartile range, and outliers,
as well as creating a box plot
when we have an even number
and odd number of data.
Notice in this example
we have 10 data values
that give the number of
text messages per day.
The first step is to order the
data from least to greatest,
which I've already done here
on the right to save time.
Let's begin by determining
the five-number summary,
which is the minimum, quartile one,
the median, quartile
three, and the maximum.
It's important to recognize, though,
there are different methods
for determining quartile one,
quartile three, as well as outliers.
In this video, to find quartile
one and quartile three,
we'll be using the Moore and McCabe method
where the quartile one is the median
of the lower half of the data,
not including the median,
and quartile three is the median
of the upper half of the data,
again, not including the median.
And then for the outliers,
we'll use Tukey's method
where the outliers will be values
that are less than quartile one minus 1.5
times the interquartile range
or greater than quartile three plus 1.5
times the interquartile range
where the interquartile
range is Q three minus Q one.
Let's begin by determining the median
as well as quartile
one and quartile three.
Well, the median is
the value in the middle
when the data is ordered
from least to greatest.
And because we have 10 data values,
there are actually two values
in the middle, 26 and 28.
Because we have two values in the middle,
the median is going to be the
mean or average of 26 and 28.
So the median is 26
plus 28 divided by two,
which is equal to 27.
So the median is 27.
And now let's find quartile one,
which is the median of
the lower half of the data
or the median of these five data values.
Because we have an odd
number of data values,
the median of the lower half is going
to be a data value in the list.
This is 24.
Notice there are two values
to the left and right of 24
on the lower half of the data.
So this is quartile one.
And then quartile three is
the median of the upper half,
which would be 34.
Notice there are two values
to the left and right of 34
in the upper half of the data.
And of course the minimum is 18,
and the maximum is 65.
And therefore, the
five-number summary is 18
comma 24 comma 27
comma 34 comma and 65.
Next, we're asked to find the
range and interquartile range
where the range is the
maximum minus the minimum,
and the interquartile range
is Q three minus Q one.
So the range is 65 minus 18.
65 minus 18 is equal to 47.
The interquartile range
is Q three minus Q one,
which is 34 minus 24,
which is equal to 10.
And now we need to find the outliers.
To help us determine the outliers,
we first need to calculate Q one
minus 1.5 times the interquartile range,
as well as Q three plus 1.5
times the interquartile range.
Well, Q one, we already know, is 24.
So we have 24 minus 1.5 times
the interquartile range,
which is 10,
which is equal to 24 minus
15, which is equal to nine,
which means any data values less than nine
would be an outlier.
Notice how we don't have any
data values less than nine.
And now let's find Q three plus 1.5
times the interquartile range,
which is 34 plus 1.5 times 10,
which is equal to 34 plus 15,
which is equal to 49.
Any data values more than 49 are outliers.
Notice how this indicates
65 is an outlier.
So we'll go ahead and
record 65 as the outlier.
And now we'll make the box
plot or box and whisker plot.
We begin with a number line
that includes all the data values.
Notice how the data
values go from 18 to 65.
The number line goes from 10 to 70.
Let's first plot the five-number summary.
We make a point at 18, 24, 27,
34, and 65.
Next step is to create the box
from quartile one to quartile
three, which is here.
Draw a line segment through the median.
And you need to be careful
here because 65 is an outlier.
So on the right, we are not going
to draw a whisker all the way out to 65.
The next value less than 65 is 41,
so we plot 41, which is here.
41 will be the end of
the whisker on the right.
And then we leave the point at 65,
indicating 65 is an outlier.
On the left, 18 is not an outlier,
and therefore we sketch the whisker
from Q one to the minimum.
Sometimes you will see
small vertical segments
at the end of the whiskers like this.
And now we have the box
plot or box and whisker plot
for the given data including the outlier.
I hope you found this helpful.
