**2.MD.9**

*Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.*Recently, I was working on a project in which the above standard (2.MD.9) was one of the foci. Although I had a couple of different questions about this standard, I want to focus on the last part of the standard.

As I grew up in Japan and going through the Japanese elementary and secondary schools, I never came across a line plot as a way to represent data. Granted that the Japanese school mathematics curriculum back then placed a very limited emphasis on data analysis. However, I did learn bar graphs, broken line graphs, histograms, circle graphs, etc. which are pretty standard statistical graphs. In the CCSS are mentioned several times in Grades 3 through 5, and I wondered why. So, I read the progression document on K-5 Measurement & Data. In the document, the authors discuss that there are two paths in the K-5 MD (data analysis portion) standards. One path deals with categorical data and the other with measurement data. The authors state that in the categorical data path, they focus on bar graphs to represent and analyze such data. On the other hand, measurement data are represented on a line plot.

Although I understand the importance of students understanding the nature of data, I found their focus on bar graphs for categorical data and line plot for measurement data to be rather strange. Students should definitely understand that the nature of data influences what analysis is possible. For example, with categorical data, it is not possible to calculate the mean or the median. However, do we always represent measurement data on a line plot? Are there any cases where we might use a bar graph to represent measurement data?

Suppose a second grade class collected the data on the height of the students in the classroom (in nearest cm). In that situation, doesn't it make sense to represent the data (heights) using a bar graph - or anything other than a line plot? In fact, the mean of the data set is obtained when we even out the height of those bars (although second graders won't be calculating the mean). So, saying that a line plot is the primary representation of measurement data seems to be rather strange.

It seems like the distinction being discussed is more about whether or not we are looking at the actual measurements or the frequencies (counts) of measurement data. A line plot is used to represent the frequencies of measurement data - how many students are 134cm tall, how many are 135cm tall, etc.. Eventually, students may create intervals and represent the frequency distribution as a histogram. Or, if we collect data to see how (air) temperature changes during the school day. It seems perfectly reasonable to represent the data using a broken line graph. In a way, the distinction may be whether the measurement data are represented on the horizontal axis or the vertical axis. I am not sure if the distinction is categorical or measurement data. When I asked a statistician, she told me that when we create intervals to make a histogram, we are "categorizing" the data. So, in a certain sense, an interval is like a category, like "strawberry" as the favorite ice cream flavor.

Anyway, I do think it is important that students understand the distinction between categorical data and measurement data. I also think it is important for students to be mindful about whether or not we are interested in the actual values of measurement data or frequencies. However, when we represent frequencies of measurement data, there may be some things we shouldn't do. For example, with categorical data (like favorite ice cream flavors), it is perfectly appropriate to order the data from most frequent to the least frequent ones as we draw a bar graph. However, with measurement data on a horizontal axis (a line plot), such a manipulation is not appropriate.