Have you ever been on a track team? Take a look at this dilemma.
One afternoon, Mr. Watson had all of the boys on the distance running team line up on the field.
“Hey coach, what’s up?” Manuel asked taking his place on the team.
“I am checking on heights,” Mr. Watson explained taking out his tape measure. He began to measure each boy in centimeters.
“Why?” Carl asked curiously.
“Well, I have the heights of Markwell’s team and I want to compare our heights with theirs. I am wondering if there is a correlation between speed and height. So I am going to start with the heights,” Mr. Watson explained.
Mr. Watson wrote the following heights from smallest to largest.
Markwell Cougars: 170, 172, 175, 176, 176, 176, 178, 181, 182, 183, 183, 183, 185, 185, 187, 188, 188, 189, 190, 195
Hawks: 169, 175, 176, 176, 178, 179, 180, 183, 183, 186, 186, 186, 187, 187, 187, 187, 187, 188, 190, 191, 192
There are many different ways to display this data. In this Concept, you will learn about histograms and by the end of this it, you will be able to create a display of the heights of both teams.
Guidance
Measures of central tendency are an important method for interpreting a set of data. However, humans tend to be very visual. That is, many people understand things best when they can see them. For that reason, we have a variety of tools which allow us to see a set of data. These tools include plots and graphs. Each type of visual tool has advantages and the best type of plot or graph depends on the situation. Indeed, sometimes it is a matter of preference as many different graphs could be used to illustrate the same data.
Let's take a look at frequency tables and histograms.
What does the word frequency mean? Frequency is used to measure how often something occurs. When we think about the word frequency, we think about “how often”.
We can use a frequency table to measure and visually show how often a data value occurs.
Take a look at this situation.
A teacher is preparing for parent conferences. In order to provide parents with the most information possible about their children, he wants to organize the grades of the class so that they can compare the grades to the rest of the class.
The math percents have been calculated and his students earned the following grades: 88, 86, 92, 65, 72, 75, 81, 84, 85, 93, 99, 50, 78, 80, 86, 76, 74, 95, 81, 87, 90, 72, 76, 61, 85, 84, 78, 83.
Grades are determined by percent where 0-59% is an F, 60-69% is a D, 70-79% is a C, 80-89% is a B, and 90-100% is an A, so that makes the most logical intervals.
Intervals are always chosen depending on the range of the data. He will make a frequency table to illustrate the information.
For each student who scored in the given range, he puts an X.
Interval | Tally | Frequency |
---|---|---|
90-100 | XXXXX | 5 |
80-89 | XXXXXXXXXXXX | 12 |
70-79 | XXXXXXXX | 8 |
60-69 | XX | 2 |
0-59 | X | 1 |
This tally is useful in the sense that it communicates to parents how many students in the class scored in the A range, B range, etc. It would not be as important for the parents to see the individual scores of each student as opposed to seeing the total number of each grade. That way, if their child earned a B, then they would know that the child falls in a category that most other students scored in. If a child earned a D, for example, it would indicate that they are below the general level of the other students and might need additional help.
Notice that a frequency table showed you how often a particular score was earned. We could see it in a visual way. Sometimes, frequency tables use X’s and other times, they can use lines for tally marks.
That is a great question. Yes, we could create a histogram.
A histogram is similar to a bar graph in that it uses columns to illustrate data on \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-axes. In a histogram, we can use the same intervals as we did for the frequency table. The bars in the histogram will have no space between them.
The histogram shows the same information as the frequency table does. However, the histogram is a type of graph, meaning that it is visual representation. Of course, we look at all of the data with our eyes, all data is visual. But the bars on the histogram are interpreted more easily by size than numerical data.
Answer each question about histograms and frequency tables.
Example A
True or false. An interval is the frequency that an event happens.
Solution: False.
Example B
True or false. To create a histogram, you first need a frequency table.
Solution: True
Example C
True or false. If I wanted to create a histogram on the number of people who went to the town movie theater on the weekend, I would first need to figure out how many people went to the movies on each weekend day and night.
Solution: True. This data would create the frequency table.
Now let's go back to the dilemma from the beginning of the Concept.
The first thing to do is to create a frequency table using the data.
Markwell Cougars | Hawks | |||
---|---|---|---|---|
Interval | Tally | Frequency | Tally | Frequency |
160-169 | X | 1 | ||
170-179 | XXXXXXX | 7 | XXXXX | 5 |
180-189 | XXXXXXXXXXX | 11 | XXXXXXXXXXXX | 12 |
190-199 | XX | 2 | XXX | 3 |
Now, you can use this data to create a histogram that compares the data.
You can see from the histogram that both teams have more players in the 180-189 interval. However, while the Cougars have more players in the 170-179 interval, the Hawks have slightly more in the taller interval. The Hawks have a slight height advantage.
Vocabulary
- Histograms
- A visual display of data that uses bars, \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes with no spaces between the intervals.
- Frequency Table
- a table that shows how often different values occur in a data set. They are arranged using tally marks or X’s.
- Double Histograms
- a visual display of data using bars and intervals to compare data sets that contain two different sets of values.
Guided Practice
Here is one for you to try on your own.
Create a histogram of the mass of geodes found at a volcanic site. Scientists measured 24 geodes in kilograms and got the following data: .8, .9, 1.1, 1.1, 1.2, 1.5, 1.5, 1.6, 1.7, 1.7, 1.7, 1.9, 2.0, 2.3, 5.3, 6.8, 7.5, 9.6, 10.5, 11.2, 12.0, 17.6, 23.9, 26.8.
Solution
First, let's think about intervals.
The minimum item is .8 kg and the maximum is 26.8. To get a good idea of the data, we could use intervals that encompass perhaps 4 kg intervals, 5 kg intervals, or 6 kg intervals. Let’s try intervals of 5.
Begin with a frequency table.
Interval | Tally | Frequency |
---|---|---|
0-5 | XXXXXXXXXXXXXX | 14 |
5.1-10 | XXXX | 4 |
10.1-15 | XXX | 3 |
15.1-20 | X | 1 |
20.0-25 | X | 1 |
25.1-30 | X | 1 |
Now we can create a histogram for this data.
Video Review
Practice
Directions: Use what you have learned about histograms to answer each question.
Number of Hours Slept | Tally | Frequency |
---|---|---|
5 | I | 1 |
6 | I I | 2 |
7 | I I I I | 4 |
8 | I I I | 3 |
9 | I I I | 3 |
10 | I I I | 3 |
11 | I I | 2 |
12 | I I | 2 |
- Create a histogram that illustrates this data.
- Explain why you chose the intervals that you chose.
- What can you interpret from your histogram?
Compare the stem-and-leaf plot to the histogram of Melanie’s Christmas gift expenses. She told her husband, “Most of the gifts were about $60.”
- Is she telling the truth?
- Which tool is more useful in making a decision about her truthfulness?
\begin{align*}\begin{array}{c|c c c c c c} 4 & 1, & 6 \\ 5 & 3, & 2 \\ 6 & 8, & 8, & 9, & 9 \\ 7 & 7, & 8, & 9, & 9, & 9, & 9 \\ \end{array}\end{align*}
- Looking at this histogram, can you conclude that most people exercise between 6 - 11 hours per week?
- What is the fewest number of hours?
- What is the range of hours?
- Why do you think they chose the interval that they did?
10 – 15 Conduct your own survey and collect data. Choose attendance rates in your class or vacation days per year for example. Then create a frequency table, histogram and analyze your data. Explain why you chose the interval that you did and which data set had the greatest and least results.