Let's go back to your 12 ordered noodles, arranged from shortest to longest on a new piece of paper or cardboard. As before, two of the noodles will be Min and Max. Now we're going to identify three noodles that divide the 12 (including Min and Max) into four groups of the same size.
First, divide your noodles into four groups with an equal number of noodles in each:
As with the Three-Noodle Summary for an even data set, we need to insert three extra lines, which we'll label Q1, Q2, and Q3, to divide and define the groups:
Note that Q2 is the median (Med) of this data set, since six noodles are to the left of Q2 and six are to the right.
What is the median of the six noodles to the left of Q2? What is the median of the six noodles to the right of Q2?
The median divides the set equally, so the median in a set of six noodles is the value that has three noodles to the left of it and three noodles to the right. Close Tip
Q1, Q2, and Q3 are called quartiles, since they divide the noodles into four groups (i.e., quarters), with an equal number of noodles in each group. The line Q1 is the median of the six noodles to the left of Q2, and Q3 is the median of the six noodles to the right of Q2. Q2 is the median of the entire set of noodles.
The Five-Noodle Summary consists of Min, Q1, Med (Q2), Q3, and Max:
Using the information given in this Five-Noodle Summary, describe what you know about the 12 noodles. For example, what do you know about the ninth noodle, and what information are you still missing?
To convert the Five-Noodle Summary to the Five-Number Summary, use the same procedure you've followed throughout this session. Add a vertical number line so that you can indicate the lengths of the five noodles:
Remove the noodles, and you're left with the Five-Number Summary:
The number Q1 is called the first or lower quartile. The number Q3 is called the third or upper quartile.
If N4 is the length of the fourth noodle, what information would you know about N4 from the Five-Number Summary?
Ralph claims that the Five-Number Summary is enough to know that N4 is closer to Q1 than it is to Med. He says, "Since N4, N5, and N6 are all between Q1 and Med, N4 has to be closer to Q1 than it is to Med." Is his reasoning valid? Why or why not?
Try to build a data set that shows whether or not Ralph's claim is valid. Close Tip