Learning Math: Data Analysis, Statistics, and Probability
Data Organization and Representation Part E: Bar Graphs and Relative Frequencies (30 Minutes)
In This Part: Frequency Bar Graphs
The line plot is a useful graph for examining small sets of data. It’s especially helpful as a device for learning basic statistical ideas. But for larger data sets, it can be awkward to create, since for each data value there is a corresponding dot. That’s a lot of dots for data sets with hundreds or thousands of values! You can, however, replace a line plot with a frequency bar graph.
Let’s look at the transition from line plot to frequency bar graph.
We start with the line plot we’ve been using. Remember that the number of dots over each value on the horizontal axis corresponds to the frequency of that data value:
Now draw a rectangle over each value, with a height corresponding to the frequency of that value:
Now remove the dots, and add a vertical scale that indicates the frequency of each value on the horizontal scale:
The frequency bar graph contains the same information as the line plot for the counts of raisin boxes, but it doesn’t indicate the raisin count for each individual box. The height of each bar or rectangle tells us the frequency for the corresponding raisin count.
In This Part: Relative Frequency
Although the frequency bar graph is useful in many ways, it, like the line plot, can be an awkward graph for large data sets, since the vertical axis corresponds to the frequency of each data value. For large data sets, some data values occur many times and have a high frequency. Consequently, the vertical axis would have to be scaled according to the largest frequency. Imagine the sheet of paper you’d need for the economy-size box of raisins!
An alternative is to use relative frequency, or frequency as a proportion of the whole set. A relative or proportional comparison is usually more useful than a comparison of absolute frequencies. For example, the statement “Five of the 17 boxes have 28 raisins” is more useful than the statement “Five boxes have 28 raisins.”
In this case, the relative frequency of the count 5 is 5/17, which can also be written in decimal form as .294 (rounded to three digits). To find the percentage, multiply the decimal by 100 to obtain 29.4%. This means that 29.4% of the raisin boxes contain 28 raisins.
Here is a frequency table for the raisin count, with the corresponding relative frequencies written as fractions, decimals, and percentages:
The next transition in the representation is to replace frequencies with relative frequencies. The relative frequency bar graph looks exactly the same as the frequency bar graph. It is important to note that the sizes of the bars remain the same whether you use frequencies or relative frequencies.
If you are working with actual raisins, draw a frequency bar graph and a relative frequency bar graph with your data. Do your graphs look the same?
Groups: Small groups can present their bar graphs to the whole group at this time.
Session 1 Statistics As Problem Solving
Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced.
Session 2 Data Organization and Representation
Explore different ways of representing, analyzing, and interpreting data, including line plots, frequency tables, cumulative and relative frequency tables, and bar graphs. Learn how to use intervals to describe variation in data. Learn how to determine and understand the median.
Session 3 Describing Distributions
Continue learning about organizing and grouping data in different graphs and tables. Learn how to analyze and interpret variation in data by using stem and leaf plots and histograms. Learn about relative and cumulative frequency.
Session 4 Min, Max and the Five-Number Summary
Investigate various approaches for summarizing variation in data, and learn how dividing data into groups can help provide other types of answers to statistical questions. Understand numerical and graphic representations of the minimum, the maximum, the median, and quartiles. Learn how to create a box plot.
Session 5 Variation About the Mean
Explore the concept of the mean and how variation in data can be described relative to the mean. Concepts include fair and unfair allocations, and how to measure variation about the mean.
Session 6 Designing Experiments
Examine how to collect and compare data from observational and experimental studies, and learn how to set up your own experimental studies.
Session 7 Bivariate Data and Analysis
Analyze bivariate data and understand the concepts of association and co-variation between two quantitative variables. Explore scatter plots, the least squares line, and modeling linear relationships.
Session 8 Probability
Investigate some basic concepts of probability and the relationship between statistics and probability. Learn about random events, games of chance, mathematical and experimental probability, tree diagrams, and the binomial probability model.
Session 9 Random Sampling and Estimation
Learn how to select a random sample and use it to estimate characteristics of an entire population. Learn how to describe variation in estimates, and the effect of sample size on an estimate's accuracy.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in this course can be applied through a case study of a K-2 teacher, Ellen Sabanosh, a former course participant who has adapted her new knowledge to her classroom.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in this course can be applied through case studies of a grade 3-5 teacher, Suzanne L'Esperance and grade 6-8 teacher, Paul Snowden, both former course participants who have adapted their new knowledge to their classrooms.