Min, Max and the Five-Number Summary Part C: Quartiles and the Five-Number Summary (35 minutes)

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.

Problem C1
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.

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:

Problem C2
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.

Problem C3
If N4 is the length of the fourth noodle, what information would you know about N4 from the Five-Number Summary?

Problem C4
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.

In This Part: More Five-Number Summaries

In the previous example, there were 12 noodles. Twelve is a convenient number of data values for introducing quartiles, because it is an even number and it is divisible by four. In this case, the quartiles separate the data into groups that each contain three values.

Quartiles always produce four groups of data with an equal number of data values in each group. But when the total number of data values is not divisible by four, it’s trickier to determine exactly how many values will be in each of the four groups.

Determining quartiles is a two-step process:

• First, find the median, or Med. Med divides the ordered data into two groups of equal size. One group contains data values to the left of Med, and the other contains data values to the right of Med.

•  Next, find the median of the data values to the left of Med, which is the first quartile (Q1). Similarly, the third quartile (Q3) is the median of the data values to the right of Med.

Let’s illustrate how this works for 13 data values (i.e., noodles). Since the total number of noodles is now odd, the median will be one of the original 13 noodles. Note that there is the same number of noodles to the left of Med as there is to the right of Med. Since you cannot divide the 13 noodles into two equal groups without splitting a noodle, take one noodle in the middle as the median and divide the other 12 noodles into two equal groups. This will occur whenever there is an odd number of noodles. The two equal groups will have exactly half of the noodles, with one noodle left in the middle as the median.

Now we find Q1, the median of the six noodles to the left of Med, and Q3, the median of the six noodles to the right of Med. Because there is an even number of noodles to the left and right of Med, Q1 and Q3 will be represented by lines between a pair of noodles.

Note that there are three noodles to the left of Q1, three noodles between Q1 and Med, three noodles between Med and Q3, and three noodles to the right of Q3. Also note that each group of three noodles is approximately one-fourth of the total of 13 noodles. As with the calculation of the median, the quartiles split each half of the noodles into two equal groups; if there is an odd number of noodles in a half, one will be left in the middle as the quartile.

In This Part: Review

Review how you would find the Five-Noodle Summary using a set of 12 noodles and a set of 13 noodles.

Problem C6
Explain how you would create a Five-Noodle Summary for 14 noodles. How many noodles are in each of the four groups?

Remember that the median of a group may be represented by a noodle or by a line drawn halfway between two noodles. Since a quartile is the median of half the data, it may also be represented by a noodle or by a line drawn halfway between two noodles.

Problem C7
Explain how you would create a Five-Number Summary for 15 noodles. How many numbers are in each of the four groups?

TAKE IT FURTHER

Problem C8
How many numbers are in each of the four groups if you started with 57 noodles? With 112 noodles? Can you find a rule that would allow you to determine the number of values in each group without creating a Five-Number Summary?

In general, the Five-Number Summary divides ordered numeric data into four groups, with each group having the same number of data values. If you know only the Five-Number Summary (Min, Q1, Med, Q3, and Max), these five values still give you a lot of information:

• All the data values are between Min and Max.
• Med divides the ordered data into two groups, with an equal number of values (approximately half) in each group:
• One group contains data values to the left of Med.
• One group contains data values to the right of Med.
• The quartiles divide the ordered data into four groups, with an equal number of values (approximately one-fourth) in each group:
• One group contains values to the left of Q1 (and includesMin).
• One group contains values between Q1 and Med.
• One group contains values between Med and Q3.
• One group contains values to the right of Q3 (and includes Max).

Problem C9
What information is learned from the interquartile range, the length of the interval between Q1 and Q3? Think about why this might be useful in describing the variation in your data.