Statistics: Introduction to Surveys & Data Collection
Learning Module
- Creator(s):
In this module on Statistics, you will learn the basic components of creating and analyzing opinion surveys. The limitations of these surveys, such as bias, will be discussed, as well as the importance of randomness. The construction of charts and graphs to analyze data will also be covered. Relative frequency histograms will introduced and will be used to model data and describe trends in the data.
As for standards, the class focus on the following from the Common Core State Standards listed below, which includes content standards and the math practice standards:
The final assessment will be to create and administer a survey that is not biased and interpret the data as directed in the updates. The project will be peer reviewed and you will have the opportunity to revise the survey as you work through the updates.
The 8 updates listed below should be done in sequential order and can be a part of any junior level high school course. Each update would correlate with one class period.
The sequence of information, and many of the examples and discussion points are taken from the College Preparatory Mathematics curriculum that is used in my current school. I chose this topic because I am looking for a way to incorporate technology and peer reviews into the current work in order to make this chapter more meaningful and useful for my students.
Before you can figure out a way to find information, you need to know what information you are looking for. Clear and unbiased research questions are a must for research. Research questions should be statistical and allow you the opportunity to collect various data points.
Watch the video below on research questions:
Click on the following exercise to practice what you learned in the video.
TASK: Post your research question in the comment section of Update 1 and provide feedback to at least 3 of your peers.
GOAL: Students will define and write research questions
STANDARD: S-IC.6. Evaluate reports based on data
MATH PRACTICE STANDARDS: make sense of problems and perserver in solving them
In this lesson, students will write research questions that are statistical in nature. Have students watch the video on creating research questions and complete the practice set of identifying good research questions. As students begin to work on creating research questions, pay attention to the types of questions they have and make sure they are something that will allow for a valid survey. Some research questions may be too narrow in scope and students may have a hard time coming up with multiple survey items. Some research questions may be too broad. These will be hard to answer using a simple survey.
Survey questions are needed for your own investigation and it is important to understand how the questions that you ask can influence your results. Influence on a study that favors a certain outcome is called bias.
Watch the video below on types of bias:
TASK:
GOAL: Students will write research questions and consider issues of bias as they write survey items to investigate those questions.
STANDARD: S-IC.6. Evaluate reports based on data
MATH PRACTICE STANDARDS: use appropriate tools, attend to precision, critique the reasoning of others
In this lesson, students will consider issues of bias as they write survey items to investigate their research questions from update 1. Have students watch the video on understanding bias. Once students have watched the video and come up with some examples for their survey, have them share their ideas in a google doc. Peers will comment and leave feedback via google comments recognizing potential sources of bias in the research questions. Using this feedback, the author can then revise before submitting to the teacher.
In order to decide whether you like a particular type of ice cream, do you need to eat the entire container? Usually, you can get a good sense of the flavor of ice cream by taking a small taste, or a sample.
When conducting a survey, often times is it not possible to survey every person or object in the population in which you are interested. (For example, all the teenagers in the United States.) However, just as you can learn about the flavor of ice cream with a small taste instead of eating the whole container, you can learn about a population by sampling. In this update, you will consider ways that statisticians take samples that represent the whole population well.
TASK: Reflect on your research question and survey questions from update 1 and update 2. Do they take into account random sampling? How do you know? If they don’t, what changes do you have to make to ensure you are getting an accurate sample? Post your reflection below.
GOAL: Students will compare nature of samples selected using intentional choice with those selected randomly. They will learn that random sampling produces samples that represent populations better than samples chosen intentionally.
STANDARD: S-IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. S-IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
MATH PRACTICE STANDARDS: make sense of problems and persevere in solving them
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Students will compare the representative nature of samples selected using intentional choice with those selected randomly. They will learn that random sampling produces samples represent populations better than samples intentionally selected by the surveyor.
Students will need to reflect on their prior work and make adjustments if needed.
Observe student work and discussion to make sure the following question are being addressed:
What if random sampling is not possible? In update 3, you learned that random samples generally represent populations from which they are selected more closely than intentionally-selected samples. For this reason, randomization is an important part of statistics. However, we need to consider an alternate option as random sampling is not always possible or practical.
For example, if you wanted to survey voters in the United States about their choice for a presidential candidate, random sampling might not be practical. Even if you designed a method to select 10,000 names randomly from the population of American voters, how would you find those 10,000 people to survey in a timely and cost effective manner?
Sometimes statisticians are restricted to sampling strategies based on who is available to them, a practice known as convenience sampling. While this method can be cost effective and easier to implement, there is no way to make a generalization to the larger population based on results.
Watch the video below on convenience sampling:
TASK: Respond to the following:
GOAL: Students will consider populations represented by particular convenience samples. They will incorporate some level of random selection into their own plans for sampling.
STANDARD: S-IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
MATH PRACTICE STANDARDS: critique the reasoning of others
This update focuses on when true randomization is not possible. Convenience sampling is introduced as a possible method, but make sure students realize that this is not necessarily a viable option when making a connection to a broader population.
You may wish to have a discussion with students about the following examples to help incorporate randomness into their sample.
For example, one way is to divide the population into smaller clusters and choose some clusters to sample. If someone were trying to randomly sample students in a high school, this would be like sampling every 3rd period English class. A second method is to divide the population into groups that are alike for a particular characteristic and then sample from each group. This might look like randomly choosing a few students from different mathematics courses, some from each of Algebra 1, Geometry, Algebra 2, and Calculus. A third method is to systematically choose every 5th or 10th or whatever, person to sample.
Relative frequency histograms are one way to visually represent data from your survey. Your teacher will walk you through the following example to show you how to create a relative frequency histogram on your calculator.
VISORS FOR RUNNERS, part 1 (taken from cpm.org)
The Style & Comfort Headgear Company makes printed visors as gifts for women running in charity marathons. Marathons in different cities have different numbers of participants. S & C Headgear needs a method for deciding how many visors of each size to print for each event. The company collected the data below for women’s head circumference by measuring 40 randomly selected runners at charity events.
Questions about percentages are easier to answer if you make a table of relative frequencies. Relative frequency is the proportion of hats: the percent of hats in that size out of the total 40 hats, written as a decimal. As such, we convert the data in the first table into a new table, tabulating the frequency each size occurs. Finish the table below:
After completing the table, create a relative frequency histogram and answer the following questions:
GOAL: Students will create a relative frequency histogram.
STANDARDS: S-ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
MATH PRACTICE STANDARDS: use appropriate tools strategically, attend to precision
The data that students receive from their survey will need to be converted into graphical form. Relative frequency histograms are one way to show data. Walk student through the student exercise and show them how to enter data and create the histograms on their calculators. The video below show you through how to do this on a TI-84 Plus :
Read the math notes box below and note the differences between observational studies and experiments.
Think about the following newspaper headlines below. Decide if an observational study was done or an experiment.
TASK: Find an article from a trusted newspaper source that describes a research study. Post below and include whether an observational study or experiment was done to collect data.
GOAL: Students will combine ideas from prior lessons an apply their learning to real world situations.
STANDARDS: S-IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
MATH PRACTICE STANDARDS: critique the reasoning of others, attend to precision
This update ties together some ideas from prior updates in this module. The main goal of update 6 is to increase students’ analytical skills when it comes to articles they encounter via newspapers or social media. They will look for bias within these applications and think critically about studies that have been done and are reported to the public. This activity will lead them to a be better informed citizens and not to always believe the first thing they read/see. The emphasis on this update is not for student to place correct labels on studies, but to active “thinkers” when reading.
ANSWER KEY:
Normal distribution is a mathematical term that is used to describe single peaked, symmetric data. Often times, this is called the bell curve. An example is pictured below:
Using a normal distribution curve, you can describe the data to others without giving them a list of all the data, and you can make predictions based on the data. Using our data from the visor example from update 5, let’s use the calculator to draw in the normal distribution.
VISORS FOR RUNNERS, Part 2
The Style and Comfort 5K Race for Charity is coming up. The Style and Comfort Headwear Company expects 775 racers. Based on a model for hat size distributions for all women, they will need to order hats for the event.
1. On your calculator, recreate the relative frequency histogram for the 40 women’s hat sizes in update 5. Use an interval from 50 cm to 57 cm with a bin width of 1.
2. Use the mean and standard deviation of hat sizes that you calculated in update 5 to make a model of the data using a normal distribution. To model data with a normal probability density function, press Y1 and enter [DISTR] normalpdf(X, mean, standard deviation). Your screen should look like the one below, on the left. Press graph to display your histogram and the bell-shaped normal distribution that is a model of the data appear, as shown below on the right.
3. How many size 51 hats should Style and Comfort order for the anticipated 775 racers at the Style and Comfort 5K Race for Charity?
4. Use the model to predict how many of the racers at the Style and Comfort 5K Race for Charity are expected to have a hat size below 51. How many racers would you expect to have a hat size over size 56? Between 51 and 56?
GOAL: To model bell-shaped data with a normal distribution and use the model to calculate proportions.
STANDARD: S-ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S-IC.2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. S-IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
MATH PRACTICE STANDARDS: use tools strategically
Normal distribution is the last new skill in the Statistics module. Students will learn how to create and read a normal distribution curve from a given data set. Emphasize that these are predictions and approximations and that normal distribution does not provide an exact answer. The point is to get an estimate to use for future extrapolation of data. Be sure you have practiced the calculator functions and be ready to troubleshoot data entry errors.
ANSWER KEY:
You will take what you have learned in this module and complete a survey based on your research question. The following items must be included on your website and you must use multimedia elements within your description to illustrate and highlight important parts.
Use the following questions to guide you in creating this website:
After you submit your project, your final task will be to review two of your peers' web page. Using the rubric below, you will give feedback on the work they have done. Once you get your peer reviews back, you may revise and resubmit for your final grade for this unit.
GOAL: Combine all ideas to create and administer a survey, including sorting and analyzing data.
STANDARDS: all listed previously
MATH PRACTICE STANDARD: model with mathematics
This final update pulls together all the things learned in the module into a project. Student will take the research question and survey and execute their study. Final project will be submitted as a web page with multimedia elements.