1 -2 class periods
Ohio Standards Alignment
- Paper, pencils, rulers for bar graphs
Categorical data, measures of central tendency, relative frequencies, analysis through charts and graphs, sampling methods.
Students are asked to find the average birth month. Analysis of the data and examination of just what “average” means in the case of months leads to the mode being selected as the best way to find an "average" with categorical data. Students create appropriate graphs to illustrate what the mode is. Next, students are asked if this information is representative of the entire population, and why or why not. Questions about sampling arise in this portion of the lesson. Finally, students are asked to create a method to determine the average day of the week for birthdays. Students explore a question that engages them even as it leads to deeper understanding of basic statistical concepts.
- Understand measures of central tendency and choose the most appropriate measure for categorical data.
- Consider sampling methods as part of a well-designed survey.
- Collect and analyze data.
To engage students, you could begin with the question: "Are you born in the same month as many of your classmates?" This easily leads to:
Each month is assigned a number: January =1, February = 2, ..., December = 12. Students compile the birth month data from the class and begin their investigation.
What is the average month for births?
Students begin by collecting birthday data from the class -- month, day, day of the week (if known). Encourage creativity in finding the “average,” including any charts or graphs that may be useful. After the students have worked for a while, the question becomes:
Students should eventually agree that, for categorical data, only the mode really makes sense. They can then create further graphs that illustrate what the mode is and investigate the shape of the distribution as well.
Does the answer you have make sense in the context of the problem?
At this point, expand the question to larger populations.
Is the answer you have representative of the school? The city? The country? Why or why not?
Discussion here centers on random samples, convenience samples, and how each may affect the results of the data collection.
The closing asks students to consider a similar problem:
What is the “average” day of the week for births of students in this school?
- Students generally fall into the trap of thinking algorithmically and calculating the mean of the numbers collected. In every class where I have used this prompt, the solution comes up as a number between 6 and 7, which is to be expected given the rather uniform data set of birth months. Since birthdays are categorical data, the mean is an inappropriate way to find the center of the data. Students discuss their results and are prompted to decide what average means in this instance (the most common month) and the differences between quantitative and qualitative data.
- A bar graph allows discussion of the shape of the graph. A chart can also help analyze the data. With small classes, there may be insufficient data to make a representative graph. Additional data can be found by grouping several classes together or by getting the birth months of politicians, historical figures, celebrities, etc.
- By expanding the question to larger populations, students can start to consider why random samples are used, what a convenience sample is, and whether results of a study are transferable to the general population. The book, Life: The Odds (and How to Improve Them), gives a five-year average of month of birth that shows a nearly uniform distribution. As part of the discussion on random samples, students may want to consider whether environmental or other factors could affect their classroom sample. Many people believe that the two months that come nine months after the height of the holiday season (August and September) are more common months for births than other months. The data in Life suggest that this is not the case. Data for large populations will show an overall uniform distribution with slight variations.
- The closing question about the day of the week lets students combine all that has been discussed into one product. The students will get to discuss how to collect data, how to get a random sample, whether or not a convenience sample can be used, and factors that may affect the data. A key factor for day of the week data over the past ten years or so is that almost all scheduled births occur on Monday - Wednesday because of doctors' schedules and to keep the length of hospital stays shorter. Therefore, a bar graph of day-of-the-week data is generally skewed to have a tail on Friday - Saturday and higher bars on Monday - Wednesday.
- Depending on the school, it may be difficult to do a random sample of students, so design of the study will have to take that into account. Determining the day of the week of birth, given the birthdate, is a lesson in IMP (Interactive Mathematics Program) that can be used here.
Fendel, D., Resek, D., Alper, L., & Fraser, S. (1999). Interactive Mathematics Project (IMP). Key Curriculum Press, pp. 394, 410-413.
Helpful ideas on finding the day of the week of one's birth.
Bayer, G. (2003). Life: The Odds (and How to Improve Them). Putnam Books, pp. 162-167. This gives a five-year average of the month of birth that shows a nearly uniform distribution.
From the teaching files of Fred Dillon.