Here are four questions about the population of Alaska.
Describe the questions as precisely as you can.
In general, at what age do Alaska residents retire?
At what age can Alaskans vote?
What is the age difference between the youngest and oldest Alaska residents with a full-time job?
Which age group is the largest part of the population: 18 years or younger, 19–25 years, 25–34 years, 35–44 years, 45–54 years, 55–64 years, or 65 years or older?
7.2
Activity
Measuring Earthworms
An earthworm farmer sets up several containers of a certain species of earthworms so that he can learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measures the lengths, in millimeters, of 25 earthworms in one of the containers.
Using a ruler, draw a line segment for each length:
20 millimeters
40 millimeters
60 millimeters
80 millimeters
100 millimeters
Here are the lengths, in millimeters, of the 25 earthworms.
6
11
18
19
20
23
23
25
25
26
27
27
28
29
32
33
41
42
48
52
54
59
60
77
93
Complete the table for the lengths of the 25 earthworms.
length
frequency
0 millimeters to less than 20 millimeters
20 millimeters to less than 40 millimeters
40 millimeters to less than 60 millimeters
60 millimeters to less than 80 millimeters
80 millimeters to less than 100 millimeters
Use the grid and the information in the table to draw a histogram for the worm length data. Be sure to label the axes of your histogram.
Based on the histogram, what value could be given as the center as a typical length for these 25 earthworms? Explain how you know.
Write 1–2 sentences to describe the spread of the data. Do most of the worms have a length that is close to your estimate of a typical length, or are they very different in length?
7.3
Activity
Tall and Taller Players
Professional basketball players tend to be taller than professional baseball players.
Here are two histograms that show height distributions of 50 professional baseball players and 50 professional basketball players.
Describe the distribution of each histogram. Comment on the center and spread in your description.
Decide which histogram shows the heights of baseball players and which shows the heights of basketball players. Be prepared to explain your reasoning
A
A histogram, height in inches, 66 through 90 by twos. Beginning with 70 up to but not including 72, height of bar for each interval is 1, 0, 4, 12, 10, 12, 4, 2, 0, 1.
B
A histogram, height in inches, 66 through 90 by twos. Beginning with 66 up to but not including 68, height of bar for each interval is 2, 3, 14, 19, 7, 5, 1.
Student Lesson Summary
Here are the weights, in kilograms, of 30 dogs.
10
11
12
12
13
15
16
16
17
18
18
19
20
20
20
21
22
22
22
23
24
24
26
26
28
30
32
32
34
34
Before we draw a histogram, let’s consider a couple of questions.
What are the smallest and largest values in our data set? This gives us an idea of the distance on the number line that our histogram will cover. In this case, the minimum is 10 and the maximum is 34, so our number line needs to extend from 10 to 35 at the very least.
(Remember the convention we use to mark off the number line for a histogram: We include the left boundary of a bar but exclude the right boundary. If 34 is the right boundary of the last bar, it won't be included in that bar, so the number line needs to go a little greater than the maximum value.)
What group size or bin size seems reasonable here? We could organize the weights into bins of 2 kilograms (10, 12, 14, . . .), 5 kilograms, (10, 15, 20, 25, . . .), 10 kilograms (10, 20, 30, . . .), or any other size. The smaller the bins, the more bars we will have, and vice versa.
Let’s use bins of 5 kilograms for the dog weights. A bin size of 2 would show more precision, but would have a lot of bars to consider. A bin size of 10 might be too big and lose the shape of the distribution with only 3 bars. The boundaries of our bins will be: 10, 15, 20, 25, 30, 35. We stop at 35 because it is greater than the maximum value.
Next, we find the frequency for the values in each group. It can be helpful to organize the values in a table.
weights in kilograms
frequency
10 to less than 15
5
15 to less than 20
7
20 to less than 25
10
25 to less than 30
3
30 to less than 35
5
Now we can draw the histogram.
A histogram, the horizontal axis is labeled “dog weights in kilograms” and the numbers 10 through 35, in increments of 5, are indicated. On the vertical axis the numbers 0 through 10, in increments of 2, are indicated. The data represented by the bars are as follows: Weight from 10 up to 15, 5. Weight from 15 up to 20, 7. Weight from 20 up to 25, 10. Weight from 25 up to 30, 3. Weight from 30 up to 35, 5.
The histogram allows us to learn more about the dog weight distribution and describe its center and spread.
