Level G Lesson 27
Box and Whisker Plots
In Lesson 27 the objective is the student will create and interpret box-and-whisker
plots. We have three essential questions that will be guiding the lesson. Number 1:
How do find the first quartile of a set of data? Number 2: How do you find the third
quartile of a set of data? Number 3: What is the interquartile range and how do you
find it.
The solve problem for our lesson is: Jeremiah conducted a survey of some of his
friends. He asked them how much money they received each week for allowance.
The list of data below shows his friends’ answers. Make a box-and-whisker plot of
the data, and then tell what percent of his friends received more than 15 dollars a
week for allowance.
We’re going to start with our S step by underlining the question. In this particular
word problem although we do not have a question mark, we have something we’re
looking for which is what percent of his friends received more than 15 dollars a week
for allowance.
We’re going to complete this statement: this problem is asking me to find the percent
of Jeremiah’s friends that got more than 15 dollars a week in allowance.
We’re going to be working with interpreting and making box-and-whisker plots. If you
look at the box-and-whisker plot that we have created underneath our number line,
we have four distinct parts. We have this part, this part, this part and this part. Four
equal parts of a dollar are called quarters. When we have four parts in our coordinate
planes we call those quadrants. The four parts of the box-and-whisker plot are called
quartiles. The lines on the side of the box and whisker plot are called whiskers. Each
whisker ends with a point. The two points are called the lower extreme and the upper
extreme. Why do you think they are called this? When we talk about the word
extreme it means “the most” or “the least” and in this case it’s far away from another
number. 22 and 39 are our lower and our upper extremes. What is the upper
extreme for this box-and-whisker plot? 39. What is the lower extreme? 22. We have
a box in the middle of the box-and-whisker plot. There are three points on that box.
This point on the left is called the first quartile or lower quartile. In this set of data
we’ve used 25 is the going to be our lower quartile. The line inside our box always
indicates the median or the middle point of our data. That means that half the data
from our data set was greater than the median and half of the data in our data set
was less than the median, in this case 34. The last point on our box in this data set is
36. 25 was the first or the lower quartile. 36 is the third or the upper quartile. The
upper quartile for this set of data is 36. Remember that each quartile of the box-andwhisker plot represents the same percent of data. Since there are four parts, what
percent of the data does each part represent? And that would be 25%. You know that
the range of a set of date is the greatest value minus the least value. Based on that,
what do you think the interquartile range would be? Inter means between. It means
the range of the box portion. Our upper quartile was 36. Our lower quartile was 25.
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The range of our interquartile is 11. The range between our upper and lower quartile
is 11.
Now we’re going to learn how to make or create a box-and-whisker plot. We’re going
to be working with the set of data shown in the box below. We have the values of 20,
16, 24, 14, 18, 29, 17, 28, 18, 22, 12, 26, and 14. When we’re constructing a boxand-whisker plot, the very first thing we want to do it put our data in order form least
to greatest and I have done that here. We have 12, 14, 14, 16, 17, 18, 18, 20, 22,
24, 26, 28, and 29. The second step is to find the median of our data. Remember,
the median is the middle number. One way we can do that is to begin by crossing out
the low, the high, the low, the high in pairs until we find our middle number. In this
case we have 13 pieces of data or an odd amount, so we’ll have one number for our
median. If you have an even amount of data then you’ll have two pieces of data for
the median. To find the median when you have an even number of values, you find
the mean of the two data values in the middle once you’ve listed them from least to
greatest. Now, let’s look at the upper half of our data. That is half that is greater than
18. We’re going to find the median of the upper half. We have 20, 29, 22, 28 and
we’re left with two pieces of data. This gives us a chance to practice finding the
median using mean. We take our two values, 24, and 26, we add them and then we
divide by 2. My median for the upper quartile is 25. Alright, we found the median of
our upper quartile, which is 25. Now we’re going to focus on our lower quartile.
Looking for the median there again and I end up with two values, 14, 16. I’ve added
those two values in the middle together for a sum of 30 and I divided by 2 and I find
that my median of my lower set of data or my lower quartile is going to be 15. The
next step is to find our lower extreme and our upper extreme. Our lower extreme is
our lowest or least piece of data and that’s going to be 12. Our greatest value in our
data set is 29. That will be the upper extreme. When we are drawing our number line
that we’re going to use for our box-and-whisker, we need to draw a number line that
will include both the lower and upper extremes. Now that we found our median, our
lower quartile, our upper quartile, and our lower and upper extreme, what we need to
do it start by drawing a number line that will include both our lower and upper
extreme. The number line we’ve drawn starts at 10 and goes to 30. The first step in
our box-and-whisker plot, once we have the number line drawn, is to put a point for
our lower extreme, our first quartile, our median, our third quartile and our upper
extreme. After we have put a point on our number line for those values, we’re going
to make box from our first quartile to our third. We’re going to place a vertical line in
our box to mark our median. Our next step is to draw a line from our upper quartile to
the upper extreme and from our lower quartile to our lower extreme. Those create our
whiskers for our box-and-whisker plot. Each part of my box-and-whisker plot
represents 25 percent of the data. 25 percent of the data values are represented by
each of the four sections of our box-and-whisker plot. Each of those is a quarter of
the data. The last question we have is: what is the interquartile range? Remember
our interquartile range is upper quartile minus the lower quartile and our interquartile
range is 10.
We’re now going to practice creating a box-and-whisker plot. We have the terms that
describe the different parts of our box-and-whisker plot: Lower extreme, first quartile
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or lower quartile, median or the middle quartile, third quartile which is also known as
the upper quartile, and the upper extreme. The first thing we’ve done is write a
definition for each of those terms. The lower extreme, remember, is the lowest value
in a data set. The lower quartile or first quartile is the middle or our median of the
lower half of the data. The median is the middle value of our total data set. The third
quartile is the middle of the upper half of the data. And the upper extreme is the
largest value in a set of data.
The data set we’re going to use to create our box-and-whisker plot is listed below: 22,
35, 42, 28, 32, 29, 39, 21, 27, 32, 18, and 23. The first thing we need to do is to list
our data in order from least to greatest. What we’ve done here is after listing our
data; we found the median by doing low, high. We have an even set of data. Our
median is going to be between the two numbers in the middle. When we have an
even number in our data set, we have to find the mean of those two values. So, 28
plus 29 is going to be 57 and when we divide that by 2 to find the mean we’ll have
28.5 as our median. What we’ve done here, after we find the median, is we have
written each half of the data set and then we have also found the median of each of
those. The lower half, again we have two pieces of data in the middle and we find
the mean of 22 and 23 is going to be 22.5. In our upper half of the data, we have 32
and 35. Our mean will be 33.5 of those two pieces of data. The next thing we’re
going to do is in each column we’re going to list our value from our data set. Our
lower extreme, our lowest value, is 18. Our first quartile, we know that we took our
two median numbers and we added them and divided by 2 and our median or the
middle of our lower half of the data is 22.5. We found our median which was the
middle value by adding 28 and 29 and dividing that by 2. Our third quartile,
remember, we had the two values of 32 and 35. We add them together and we
divide by 2 for a value of 33.5 for our third quartile. Our upper extreme is the greatest
value in our data set and it’s 42. We now have created a number line that we can use
with our data set and remember when you’re making the number line you want it to
include both your lower and upper extreme. Our lower extreme is 18. Our upper
extreme is 42. We’ve created a scale of 2 on this number line. The first thing we did
was put our five points for the lower extreme, the first quartile, the median, the third
quartile and our upper extreme. We then drew a box to connect our third quartile and
our first quartile. Within the box we’ve drawn a vertical line to mark our median. The
last step is to find the interquartile range. Remember our interquartile range is the
difference between our upper and our lower quartile. 33.5 minus 22.5 gives us an
interquartile range of 11.
We’re now going back to the solve problem from the beginning of our lesson.
Jeremiah conducted a survey of some of his friends. He asked them how much
money they received each week for allowance. The list of data below shows his
friends’ answers. Make a box-and-whisker plot of the data, and then tell what
percent of his friends received more than 15 dollars a week for allowance. At the
beginning of the lesson we underlined the question and we completed the statement:
this problem is asking me to find the percent of Jeremiah’s friends that got more
than 15 dollars a week in allowance.
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Now we’re going to organize the facts. We’re going to start by identifying the facts.
Jeremiah conducted a survey of some of his friends. That’s a fact. He asked them
how much money they received each week for allowance. That’s another fact. The
list of data below shows his friends’ answers. That’s another fact. Make a box-andwhisker plot of the data, and then tell what percent of his friends received more than
15 dollars a week for allowance.
We’re going to eliminate the unnecessary facts. We don’t need to know that he
conducted a survey of some of his friends. We also don’t need to know that he asked
them how much money they received for allowance. We do need to know the list of
data to use and we do need to know to make a box-and-whisker plot. So, we’re going
to list the necessary facts of what his friends received for an allowance.
We’re going to line up our plan and we’re going to start by choosing an operation or
operations. In this case we wrote an N/A because there’s not an applicable operation
that we’ll use. We’re going to write in words our plan of action. Put all the amounts in
order and create a box-and-whisker plot.
Now we’re moving to the V step. Our estimate is going to be 25 percent because we
wanted to know what percent of his friends. Once the amounts are put in order from
least to greatest, 15 is on one of the higher amounts. So there are not many
numbers above it. Then we carry out our plan. We list our values in order from least
to greatest. We found the median the first quartile, and the third quartile. 25 percent
of the data was between the third quartile, which is 15, and the upper extreme.
For our E step we’re going to ask the question: Does your answer make sense? We’re
going to compare your answer to the question? Yes, the whisker is 25 percent of the
data. Is your answer reasonable? And we’re going to compare it to the estimate.
Yes, in fact our estimate was about 25 percent and that was our answer. Is your
answer accurate? We can go back and take a look and make sure we completed the
computations correctly to find the median, first and third quartile. Write your answer
in a complete sentence. The percent of his friends that get more than 15 dollar a
week for allowance is 25%.
Now we’re going to go back and answer the essential questions from the beginning
of our lesson. How do you find the first quartile of a set of data? You find the median
of the lower half of the data. How do you find the third quartile of a set of data? You
find the median of the upper half of the data. What is the interquartile range and how
do you find it? The interquartile range is the range of the two inner quartiles. You find
it by subtracting the lower quartile from the upper quartile.
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