11.15: Displaying by Type of Data
Ahmed conducted a survey of the favorite kind of breakfast cereal for 8th graders. He organized his survey results into the table below.
Cereal |
Amount of Students |
Fruit Flakes |
12 |
Chocolate Puffs |
18 |
Oat-O’s |
9 |
Raisin Delight |
3 |
Honey Crunch |
17 |
Bran Loops |
3 |
Oat Squares |
9 |
Fiber Max |
2 |
Fruities |
10 |
Cinnamon Squares |
17 |
How can Ahmed display his survey data? What can Ahmed say about the kind of breakfast cereal that 8th graders like?
In this concept, you will learn how to create a survey, determine the best sample sizes based on populations, and understand the influence of bias.
Displaying by Type of Data
Data is a set of numerical or non-numerical information. Data can be collected and analyzed in many different ways. In this concept you will consider ways of collecting both numeric and non-numeric data.
A survey is one way to collect information about a population, or group of people. Surveys can collect information in a number of different ways, such as asking questions, measuring something, or counting the frequency or occurrence of something.
A population is a group of people that share similar characteristics.
A sample is the group of people that are surveyed. A sample is a subset, or small group, of the total population. Surveys are conducted on samples because populations are normally too big to survey.
A random sample is a group of people who have been randomly selected to survey. This means that every person in the population has an equal chance of being asked to participate in the survey.
Bias refers to a feeling or prejudice towards something. When conducting a survey using questions, it is important that the questions are unbiased. Good survey questions are short and concise and usually begin with “Who," "What," "When," "Where," "Why," or "How.”
Let's look at an example.
Hollywood Middle School is holding elections for the student body president. Create a survey that will predict the winner of the election.
First, determine the question that will be asked on the survey. Good survey questions are short and concise and usually begin with “Who," "What," "When," "Where," "Why," or "How.” For this survey, one question is enough. Since the survey should predict the winner, the survey question should be "Who will you vote for student body president?"
Next, determine the population and sample. The population for this survey is all students at Hollywood Middle School who can vote in the election. The sample should be randomly selected students who are in the population. The larger number of students surveyed, the greater reliability the survey results have.
The answer is the survey should ask the question "Who will you vote for student body president?" The survey should be asked of randomly selected Hollywood Middle School students who are able to vote. The survey should be conducted at school, where there is an equal chance of participating in the survey.
Examples
Example 1
Earlier, you were given a problem about Ahmed, who conducted a survey of 8th graders favorite kind of breakfast cereal.
He organized his data in the table below.
Cereal | Amount |
---|---|
Fruit Flakes | 12 |
Chocolate Puffs | 18 |
Oat-O’s | 9 |
Raisin Delight | 3 |
Honey Crunch | 17 |
Bran Loops | 3 |
Oat Squares | 9 |
Fiber Max | 2 |
Fruities | 10 |
Cinnamon Squares | 17 |
Ahmed wants to display his survey data, but doesn't know how.
How can Ahmed display his survey data? What can Ahmed say about the kind of breakfast cereal that 8th graders like?
First, consider what type of data Ahmed is trying to display. In this case, the data are numerical values that represent an amount for ten different categories.
Next, consider if Ahmed's data is best shown graphically? In Ahmed's case, his data is best represented graphically.
Next, since the data compares the number of people who like a given cereal, a bar graph is best.
To create a bar graph, first draw the horizontal (x) and vertical (y) axes.
Next, label the horizontal axis. The horizontal axis lists the different kinds of cereal. In this case, the category will be "Cereal."
Next, label the vertical axis. The vertical axis lists the amount of people who like each cereal. In this case, the category will be "Amount."
Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Favorite Cereal Survey.”
Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest values in the table. The smallest value is 2 and the largest is 18. Based on these values label the vertical axis from 0-20 by a unit of 2. Therefore, the vertical axis will start at 0 and go to 20 by increments of 2.
Next, draw the vertical columns. To do this, write each cereal along the horizontal axis. Be sure to leave space between each one. Then draw the column vertically until it reaches the quantity for that activity. For example, draw a vertical column to the number “12” for the cereal “Fruit Flakes.”
Next, analyze the bar graph to draw conclusions about the cereal preferences of 8th graders.
The first answer is the graph should look like the one below. The second answer is: the favorite cereal among 8th graders is Chocolate Puffs; the least favorite cereal is Fiber Max; and Honey Crunch and Cinnamon Squares are the second favorite cereals of 8th graders.
Example 2
Miranda creates a survey to determine what sport 7th graders prefer to play in gym class. She asked her friends on the soccer team, "Is your favorite sport to play in gym class soccer?" Explain why Miranda's survey is unreliable and how she can improve its reliability.
First, consider the question Miranda asked on her survey. Is her question a good survey question? Is the question unbiased? Good survey questions are short and concise and usually begin with “Who," "What," "When," "Where," "Why," or "How.”
Next, consider Miranda's population and sample. What is Miranda's population? What is her sample? Are the participants in her sample randomly selected? Is her sample unbiased?
The answer is Miranda's survey is unreliable because: the survey question and sample are biased. Miranda should use a good survey question, such as "What is your favorite sport to play in gym class?" Miranda should ask her survey question to a random sample of students, not only her friends on the soccer team. Miranda could ask every other 7th grade student that leaves gym class her survey question. This would ensure a randomly selected and large sample.
Example 3
Twenty-five households were asked to participate in a survey in which they were asked to approximate the number of hours they watch T.V. each day. The results of the survey are listed on the table below. Create a display of the data, then determine the percentage of people that watched T.V. between 4-5 hours per day.
First, determine what is the best kind of display for this data. Since these values are numerical and show the frequency of how much T.V. people watch, a histogram is the best type of display.
To create a histogram, first a frequency table must be made. To create a frequency table, first identify the bins, or intervals, which will be the left hand column of the table. The bins in this case will be the amount of time people watch T.V. Next, calculate the frequency, or occurrences, of the bins. This data will be in the right hand column of the table.
Bins: Hours of Television |
Frequency |
0 – 1 |
5 |
2 – 3 |
8 |
4 – 5 |
9 |
6 – 7 |
3 |
Next, create the histogram. To do this, first draw the horizontal (x) and vertical (y) axes.
Next, label the horizontal axis. The horizontal axis lists the different categories of data. In this case, the category will be "Hours."
Next, label the vertical axis. The vertical axis lists the quantity or amount of the data. In this case, the category will be "Frequency."
Next, title the graph. The title of the graph should be short and clear. It should explain what data is presented in the graph. In this case, the title will be “Hours of Television Watched.”
Then, determine the units on the vertical axis. To do this, start by reviewing the smallest and largest frequencies in the table. The smallest value is 3 and the largest is 9. Based on these values label the vertical axis from 0-10. Since the range is small the vertical axis should use a unit of 2.
Next, draw the vertical columns. To do this, write each bin along the horizontal axis. Then draw each column vertically until it reaches the frequency for that time. For example, draw a vertical column to the number 1 for the bin 0-2. Continue to draw all the columns and do not leave space between them.
Next, determine the percent of people that watched T.V. between 4-5 hours per day. To do this, divide the number of people who watch between 4-5 hours per day by the total number of people surveyed. Then multiple by 100 to convert the decimal to a percentage.
\begin{align*}9÷25=0.36\end{align*}
\begin{align*}0.36×100=36\%\end{align*}
The answer is the graph should look like the one below and the percent of people that watch between 4-5 hours per day is 36%.
Example 4
A survey will be conducted to determine what activities 6th grade students at Judson Middle School like to do on the weekends. Of the three options below, explain which option is the best way to conduct the survey?
- Go to the mall and survey every tenth person that walks by.
- Survey 6th graders at a local movie theater.
- Survey every third person leaving each 6th grade class.
First, analyze the options to conduct the survey and determine which one will be the least biased. Then determine why the other options are more biased.
The answer is the best way to conduct the survey is C. This is the most unbiased approach and ensures that everyone in the population is of equal chance to participate in the survey. A and B are not good options because not every 6th grade student has an equal chance of participating in the survey. Also, those students how are asked at the mall or movie theater are a biased group and may not accurately represent all 5th grade students at Judson.
Example 5
Oscar created a survey to find out if 8th grade students liked the lunch served in the cafeteria. He asked the students who brought their lunch, "Is the cafeteria pizza good?" Explain why Oscar's survey is unreliable and how he can improve its reliability.
First, consider the question Oscar asked on his survey. Is his question a good survey question? Is the question unbiased? Good survey questions are short and concise and usually begin with “Who, What, When, Where, Why, or How.”
Next, consider Oscar's population and sample. What is Oscar's population? What is his sample? Are the participants in his sample randomly selected? Is his sample unbiased?
The answer is Oscar's survey is unreliable because: the survey question and sample are biased. Oscar should use a good survey question, such as "Do you eat the lunch served in the cafeteria? Why or why not?" or "What do you think of the lunch served in the cafeteria?" Oscar should ask his survey question to a random sample of students, not only the students who bring their own lunch. Oscar could ask every other 8th grader that leaves the cafeteria his survey question. This would ensure a randomly selected and large sample.
Review
Use the information provided to answer the following questions.
One hundred people were asked to participate in a survey about travel. Participants were asked to state whether they had visited each of the cities listed below.
City | Number of People Who Visited: |
---|---|
Waikiki | 85 |
New York City | 80 |
San Francisco | 87 |
Chicago | 54 |
Dallas | 35 |
Orlando | 38 |
Atlanta | 50 |
Seattle | 44 |
Denver | 32 |
- Create a bar graph of the data.
- What percent of the people visited Waikiki?
- What percent of the people visited Denver?
- What percent of the people did not visit Dallas?
- What percent of the people did not visit Chicago?
- Which city had more people visited than any other city?
- Which city had less people visited than any other city?
Thirty students were selected at random at Montgomery High School. Each participant was asked to state the number of textbooks they were carrying at that moment. The results of the survey are depicted below. Choose the best display to depict the data. Then use the graph you created to answer the questions below.
\begin{align*}& 0 \qquad 2 \qquad 4 \qquad 3 \qquad 1\\
& 2 \qquad 5 \qquad 0 \qquad 4 \qquad 3\\
& 2 \qquad 5 \qquad 4 \qquad 3 \qquad 2\\
& 1 \qquad 0 \qquad 0 \qquad 2 \qquad 3\\
& 5 \qquad 6 \qquad 1 \qquad 2 \qquad 4\\
& 5 \qquad 6 \qquad 3 \qquad 2 \qquad 3\end{align*}
- Which graph is the best display of the data?
- Create that graph here.
- What percent of the students had the most common amount of books in their backpack?
- What percent of the students had between four and seven books in their backpack?
- How many students had 1 book in their backpacks?
- How many students had 0 books in their backpacks?
- What percent had 1 book?
- What percent had 0 books?
Create a survey question. Decide who will be the sample population. Write a few sentences to describe who will take part in the survey and how you will administer the survey. Administer the survey, record the results on a table. Choose the most appropriate display for the data. Use these questions for guidance.
- What is your sample population?
- How many people are in the sample?
- Does your question use who, what, when, where or how?
- Is your sample biased, why or why not?
- Are your results what you expected?
Review (Answers)
To see the Review answers, open this PDF file and look for section 11.15.
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In this concept, you will learn how to create a survey, determine the best sample sizes based on populations, and understand the influence of bias.
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