There’s been a lot of statistics being shown and talked about for the past year, and especially in the past week, so I am hoping math teachers are taking advantage of this incredible learning opportunity to spend some time exploring statistical measures and visuals to help students make sense. Because, as we know, statistics can be manipulated, misconstrued, misinterpreted, and used to sway opinion, especially if the people viewing or hearing the statistics don’t understand what the numbers and the statistical graphs actually represent or how they are scaled.
I am not even going to pretend that I am a statistical expert - far from it! But I can recognize if a statistical representation has been scaled in a way to supposedly show one thing, when in fact, it really doesn’t show that, IF you understand what the numbers represent. I think math teachers should take this opportunity, with so much data readily available, to help students become better analyzers of data so they can interpret what is being said or shown correctly, and can make sense of the world around them instead of just believing the statistics they see and hear.
With that in mind, I am going to spend the next few posts exploring some basic statistics and data representations using the fx-991CW and ClassPad.net workspace to provide you with some tools and examples you could use to start helping students understand. I am using some simple data to model, but my suggestion to you is to use real data (like the recent election, to name one great source) to start helping students become more informed and gain some “number sense” about the world around them.
Today’s post I am focused on frequency and histograms, and am using an activity from Casio’s Essential Materials to model how to create a frequency table using a basic scientific calculator, the fx-991CW, and then using it’s QR capabilities to represent the data in a Histogram using ClassPad.net workspace. Histograms are a wonderful way to visually represent frequency (i.e. the number of times a specific value appears within a given set of data). I am also going to extend this to discuss relative frequency (i.e. the proportion of a specific data value occurring compared to the entire data set), as well as cumulative frequency (i.e. a running total of relative frequencies in a data set) as a way to analyze the data.
As an example of these three types of frequencies, let’s look at the data table to the right that shows the record of side-step tests done by students in a single homeroom and then for all students in the grade. The table shows the number of side-steps in a given amount of time and how many students fell within the range.
Frequency - that is shown in the table, where you see how many students had that specific number of steps. In the table, for the single homeroom, the 45-50 range had the highest frequency, or the ‘most’ students. But, in the grade overall, the 40-45 range had the highest frequency.
Relative Frequency - if I want to look at the proportion of the highest frequency occurring in each table, I would take the frequency and divide by the total number of occurrences. In table one, the 45-60 relative frequency would be 7/20 or .35, and in table 2, the 45-50 relative frequency would be 27/80 or .34.
Cumulative Relative Frequency - so let’s say now, I want to look at the two ranges, 40-45 and 45-50 from both tables and see what their cumulative relative frequencies are, so I find the relative frequency of both ranges in both tables, and add those. For table 1: 5/20 or .25 + .35 = .65. For table 2, 24/80 or .3 + .34 = .64. So the cumulative frequency for both tables for steps falling in the range of 40-50 is about the same.
What’s that look like if we were to compare it visually? Visualizing data often helps clarify what the numbers are saying and helps us make sense of what the data shows. The video below explores how to use the fx-991CW calculator to enter frequency, use the QR code to see the data in a histogram, and then use the free web-based dynamic workspace, ClassPad.net, to explore the relative frequency, cumulative frequency, and make sense of the data. If your students have the hand-held fx-991CW, the QR code provides a fantastic exploration tool. Or, you can just use the free fx-991CW emulator and ClassPad.net workspace to model and lead a whole-class discussion.
I have also included below two activities that help explain and provide how-to’s for creating Histograms and finding relative and cumulative frequencies.
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