42.5: Using Histograms to Answer Statistical Questions

Lesson

Let's draw histograms and use them to answer questions.

Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Questions

Here are four questions about the population of Alaska. Which question does not belong? Be prepared to explain your reasoning.

1. In general, at what age do Alaska residents retire?
2. At what age can Alaskans vote?
3. What is the age difference between the youngest and oldest Alaska residents with a full-time job?
4. 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?

Exercise \(\PageIndex{2}\): Measuring Earthworms

An earthworm farmer set up several containers of a certain species of earthworms so that he could learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measured the lengths of 25 earthworms in one of the containers. Each length was measured in millimeters.

Figure \(\PageIndex{1}\): By JKpics. Public Domain. Pixabay. Source.

1. Using a ruler, draw a line segment for each length:
   • 20 millimeters
   • 40 millimeters
   • 60 millimeters
   • 80 millimeters
   • 100 millimeters
2. Here are the lengths, in millimeters, of the 25 earthworms.

\(6\quad 11\quad 18\quad 19\quad 20\quad 23\quad 23\quad 25\quad 25\quad 26\quad 27\quad 27\quad 28\quad 29\quad 32\quad 33\quad 41\quad 42\quad 48\quad 52\quad 54\quad 59\quad 60\quad 77\quad 93\)

Complete the table for the lengths of the 25 earthworms.

Table \(\PageIndex{1}\)

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

3. 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.

Figure \(\PageIndex{2}\)

4. Based on the histogram, what is a typical length for these 25 earthworms? Explain how you know.
5. 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?

Are you ready for more?

Here is another histogram for the earthworm measurement data. In this histogram, the measurements are in different groupings.

Figure \(\PageIndex{3}\)

1. Based on this histogram, what is your estimate of a typical length for the 25 earthworms?
2. Compare this histogram with the one you drew. How are the distributions of data summarized in the two histograms the same? How are they different?
3. Compare your estimates of a typical earthworm length for the two histograms. Did you reach different conclusions about a typical earthworm length from the two histograms?

Exercise \(\PageIndex{3}\): 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 male professional baseball players and 50 male professional basketball players.

1. Decide which histogram shows the heights of baseball players and which shows the heights of basketball players. Be prepared to explain your reasoning.

Figure \(\PageIndex{4}\): Two histograms, height in inches, 66 through 90 by twos. Histogram A, 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. Histogram B, beginning with 66 up to but not including 68, height of bar for each interval is 2, 3, 14, 19, 7, 5, 1.

2. Write 2–3 sentences that describe the distribution of the heights of the basketball players. Comment on the center and spread of the data.
3. Write 2–3 sentences that describe the distribution of the heights of the baseball players. Comment on the center and spread of the data.

Summary

Here are the weights, in kilograms, of 30 dogs.

\(10\quad 11\quad 12\quad 12\quad 13\quad 15\quad 16\quad 16\quad 17\quad 18\quad 18\quad 19\quad 20\quad 20\quad 20\quad 21\quad 22\quad 22\quad 22\quad 23\quad 24\quad 24\quad 26\quad 26\quad 28\quad 30\quad 32\quad 32\quad 34\quad 34\)

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. The boundaries of our bins will be: 10, 15, 20, 25, 30, 35. We stop at 35 because it is greater than the maximum.

Next, we find the frequency for the values in each group. It is helpful to organize the values in a table.

Table \(\PageIndex{2}\)

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.

Figure \(\PageIndex{5}\): 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.

Glossary Entries

Definition: Center

The center of a set of numerical data is a value in the middle of the distribution. It represents a typical value for the data set.

For example, the center of this distribution of cat weights is between 4.5 and 5 kilograms.

Figure \(\PageIndex{6}\)

Definition: Distribution

The distribution tells how many times each value occurs in a data set. For example, in the data set blue, blue, green, blue, orange, the distribution is 3 blues, 1 green, and 1 orange.

Here is a dot plot that shows the distribution for the data set 6, 10, 7, 35, 7, 36, 32, 10, 7, 35.

Figure \(\PageIndex{7}\)

Definition: Frequency

The frequency of a data value is how many times it occurs in the data set.

For example, there were 20 dogs in a park. The table shows the frequency of each color.

Table \(\PageIndex{3}\)

color	frequency
white	\(4\)
brown	\(7\)
black	\(3\)
multi-color	\(6\)

Definition: Histogram

A histogram is a way to represent data on a number line. Data values are grouped by ranges. The height of the bar shows how many data values are in that group.

This histogram shows there were 10 people who earned 2 or 3 tickets. We can't tell how many of them earned 2 tickets or how many earned 3. Each bar includes the left-end value but not the right-end value. (There were 5 people who earned 0 or 1 tickets and 13 people who earned 6 or 7 tickets.)

Figure \(\PageIndex{8}\)

Definition: Spread

The spread of a set of numerical data tells how far apart the values are.

For example, the dot plots show that the travel times for students in South Africa are more spread out than for New Zealand.

Figure \(\PageIndex{9}\)

Practice

Exercise \(\PageIndex{4}\)

These two histograms show the number of text messages sent in one week by two groups of 100 students. The first histogram summarizes data from sixth-grade students. The second histogram summarizes data from seventh-grade students.

Figure \(\PageIndex{10}\): Two histograms, text messages sent per week by sixth grade students and seventh grade students, 45, to 155 by tens. First graph, sixth grade students, beginning at 45 up to but not including 44, height of bar at each interval is 2, 0 11, 21, 27, 24, 12, 3, 3, 0, 1. Second graph, seventh grade students, beginning at 75 up to but not including 85, height of bar at each interval is 7, 30, 31, 24, 3.

1. Do the two data sets have approximately the same center? If so, explain where the center is located. If not, which one has the greater center?
2. Which data set has greater spread? Explain your reasoning.
3. Overall, which group of students—sixth- or seventh-grade—sent more text messages?

Exercise \(\PageIndex{5}\)

Forty sixth-grade students ran 1 mile. Here is a histogram that summarizes their times, in minutes. The center of the distribution is approximately 10 minutes.

On the blank axes, draw a second histogram that has:

• a distribution of times for a different group of 40 sixth-grade students.
• a center at 10 minutes.
• less variability than the distribution shown in the first histogram.

Figure \(\PageIndex{11}\)

Figure \(\PageIndex{12}\)

Exercise \(\PageIndex{6}\)

Jada has \(d\) dimes. She has more than 30 cents but less than a dollar.

1. Write two inequalities that represent how many dimes Jada has.
2. Can \(d\) be 10?
3. How many possible solutions make both inequalities true? If possible, describe or list the solutions.

(From Unit 7.2.2)

Exercise \(\PageIndex{7}\)

Order these numbers from greatest to least: \(-4, \frac{1}{4}, 0, 4, -3\frac{1}{2}, \frac{7}{4}, -\frac{5}{4}\)

(From Unit 7.1.4)