In this video,
we will determine the five-number summary, range,
interquartile range, as well as the outliers,
given a set of data.
We will also make a box plot or box and whisker plot
from the data.
Notice how we are given data on the number
of text messages per day,
and we have nine data values, which is odd.
The first step is to order the data from least to greatest,
which I've already done here on the right to save time.
Before we determine the five-number summary, though,
it's important to recognize there are different methods
for determining quartile one, quartile three,
as well as the outliers.
In this video, we will be using the Moore and McCabe Method,
where to find quartile one,
we find the median of the lower half of the data,
not including the median,
and to find quartile three,
we find the median of the upper half of the data,
again, not including the median.
And then to find the outliers,
we'll use Tukey's Method,
where 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 are considered outliers,
where the interquartile range is quartile three
minus quartile one.
So going back to our problem,
let's determine the median,
as well as quartile one and quartile three.
Because we have an odd number of data values,
the median will be a data value in the list.
Remember, the median is the data value in the middle
when the data is ordered from least to greatest.
And therefore, 28 is the median
because there are four values to the left
and four values to the right.
28 is in the middle.
The median can also be referred to as quartile two.
And now let's find quartile one,
which is the median of the four values to the left of 28.
Because we have an even number of data values,
quartile one is going to be the mean or average
of 20 and 24.
Again, because we have two middle values on the left,
we have to take the average of 20 and 24
to find quartile one.
So quartile one is equal to the quantity
20 plus 24 divided by two,
which is equal to 22.
So 22 is quartile one.
And quartile three is going to be the median
of the four values on the right of the median,
which is the mean or average of 34 and 41.
So the quantity 34 plus 41 divided by two
is equal to 75 divided by two, which is 37.5.
So 37.5 is quartile three.
And of course, the minimum is 18,
and the maximum is 65.
And therefore, the five-number summary is 18,
comma 22, comma 28,
comma 37.5, comma 65.
And now let's determine the range and interquartile range.
The range is the maximum minus the minimum.
The interquartile range is quartile three
minus quartile one.
So the range is 65 minus 18,
which is equal to 47.
The interquartile range is Q3 minus Q1,
which is 37.5 minus 22,
which is equal to 15.5.
And now see if we have any outliers.
To begin, we need to compute quartile one
minus 1.5 times the interquartile range,
as well as quartile three or Q3 plus
1.5 times the interquartile range.
Q1 minus 1.5 times the interquartile range
is 22 minus 1.5 times 15.5.
And quartile three plus 1.5 times the interquartile range
is 37.5 plus 1.5 times 15.5.
This gives us the values of negative 1.25 and 60.75.
If we have any data values less than negative 1.25
or greater than 60.75,
then we do have outliers.
Well, we don't have any data values
less than negative 1.25,
but we do have one data value more than 60.75,
which is the data value of 65.
So 65 is an outlier.
Let's record the outlier.
And now let's create the box plot.
The first step is to create a number line
that includes all the data values from 18 to 65.
Notice the number line goes from 10 to 70.
Next, we'll plot the five points
from the five-number summary,
which are 18,
22, 28,
37.5, and 65.
Next, we form a box from quartile one to quartile three,
which is here.
And then we sketch a vertical line segment
through the median.
And now because we have an outlier,
we need to be careful about forming the whiskers.
On the right, the whisker does not go all the way out to 65
because 65 is an outlier.
65 needs to stay as a point.
The next data value less than 65 is 41,
and therefore, we plot 41,
which will be the end of the whisker on the right.
And then again, we leave 65 as a point
because it's an outlier.
On the left, 18 is not an outlier.
The left whisker goes from quartile one to the minimum.
And now we have our box and whisker plot
showing the outlier.
Sometimes, you will see small vertical segments
at the end of the whiskers, shown like this.
I hope you found this helpful.