Here are data that show the numbers of siblings of 10 students in Tyler’s class.
1
0
2
1
7
0
2
0
1
10
Without making any calculations, estimate the center of the data based on your dot plot. What is a typical number of siblings for these sixth-grade students? Mark the location of that number on your dot plot.
Find the mean. How does the mean compare to the value that you marked on the dot plot as a typical number of siblings? (Is it a little larger, a lot larger, exactly the same, a little smaller, or a lot smaller than your estimate?)
6.2
Activity
Your teacher will give you an index card. Write your first and last names on the card. Then record the total number of letters in your name. After that, pause for additional instructions from your teacher.
Here is a data set on numbers of siblings.
1
0
2
1
7
0
2
0
1
10
Sort the data from least to greatest, and then find the median.
In this situation, do you think the median is a good measure of a typical number of siblings for this group? Explain your reasoning.
Here is the dot plot showing the travel time, in minutes, of Elena’s bus rides to school.
Find the median travel time. Be prepared to explain your reasoning.
What does the median tell us in this context?
6.3
Activity
Your teacher will give you six cards. Each has either a dot plot or a histogram. Sort the cards into 2 piles based on the distributions shown. Be prepared to explain your reasoning.
Discuss your sorting decisions with another group. Did you have the same cards in each pile? If so, did you use the same sorting categories? If not, how are your categories different?
Pause here for a class discussion.
Use the information on the cards to answer these questions.
Card A: What is a typical age of the dogs being treated at the animal clinic?
Card B: What is a typical number of people in the Irish households?
Card C: What is a typical travel time for the New Zealand students?
Card D: Would 15 years old be a good description of a typical age of the people who attended the birthday party?
Card E: Is 15 minutes or 24 minutes a better description of a typical time it takes the students in South Africa to get to school?
Card F: Would 21.3 years old be a good description of a typical age of the people who went on a field trip to Washington, D.C.?
How would you decide which measure of center to use for the dot plots on Cards A–C? What about for those on Cards D–F?
Student Lesson Summary
The median is another measure of center for a distribution. It is the middle value in a data set when values are listed in order. Half of the values in a data set are less than or equal to the median, and half of the values are greater than or equal to the median.
To find the median, we order the data values from least to greatest and find the number in the middle.
Suppose we have 5 dogs whose weights, in pounds, are shown in the table. The median weight for this group of dogs is 32 pounds because three dogs weigh less than or equal to 32 pounds and three dogs weigh greater than or equal to 32 pounds.
20
25
32
40
55
Now suppose we have 6 cats whose weights, in pounds, are listed here. Notice that there are 2 values in the middle: 7 and 8.
4
6
7
8
10
10
The median weight must be between 7 and 8 pounds, because half of the cats weigh less than or equal to 7 pounds, and half of the cats weigh greater than or equal to 8 pounds.
When there are even numbers of values, we take the number exactly in between the two middle values. In this case, the median cat weight is 7.5 pounds because .
The dot plot shows the number of stickers on 30 pages. The mean number of stickers is 21 (marked with a triangle). The median number of stickers is 20.5 (marked with a diamond).
<p>A dot plot for stickers on a page. The numbers 8 through 34, in increments of 2, are indicated. A diamond is indicated at 20.5 stickers and a triangle is indicated at 21 stickers. Data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 11 stickers, 2 dots; 12 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 2 dots; 17 stickers, 1 dot; 18 stickers, 2 dots; 19 stickers, 1 dot; 20 stickers, 3 dots; 21 stickers, 1 dot; 22 stickers, 3 dots; 23 stickers, 1 dot; 24 stickers, 2 dots; 26 stickers, 2 dots; 28 stickers, 1 dot; 30 stickers, 1 dot; 32 stickers, 2 dots; 33 stickers, 1 dot; 34 stickers, 1 dot.</p>
In this case, both the mean and the median could describe a typical number of stickers on a page because they are fairly close to each other and to most of the data points.
Here is a different set of 30 pages with stickers. It has the same mean as the first set, but the median is 23 stickers.
<p>A dot plot for “stickers on a page.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 stickers, and a diamond is indicated at 23 stickers. The data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 13 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 1 dot; 17 stickers, 1 dot; 19 stickers, 1 dot; 20 stickers, 2 dots; 21 stickers, 2 dots; 22 stickers, 3 dots; 23 stickers, 6 dots; 24 stickers, 5 dots; 25 stickers, 4 dots; 26 stickers, 1 dot.</p>
In this case, the median is closer to where most of the data points are clustered and is therefore a better measure of center for this distribution. That is, it is a better description of the typical number of stickers on a page. The mean number of stickers is influenced (in this case, pulled down) by a handful of pages with very few stickers, so it is farther away from most data points.
In general, when a distribution is symmetrical or approximately symmetrical, the mean and median values are close. But when a distribution is not roughly symmetrical, the two values tend to be farther apart.
The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order of value.
For the data set 7, 9, 12, 13, 14, the median is 12.
For the data set 3, 5, 6, 8, 11, 12, there are 2 numbers in the middle. The median is the average of these 2 numbers. , and . The median is 7.
