Oh Massachusetts!

Sorry to keep picking on you. I’m sure other states that made tweaks to the CCSS fell into similar traps. It’s just that I know your standards because we’re forced to consider them now that we’re using MCAS. There was so much care in crafting the CCSS that some of your edits have damaged what was a precise, almost fragile structure, and this one I’ll call attention to today is more like the first that I mentioned in that it introduces a mathematical error into your framework.

It’s June, so it must be time to talk about Statistics and Probability, and I want to focus on **6.SP.2**. This is one of the standards where you added some exposition to make the standard more clear and explicit for teachers. While I think this was a good move, the changes you made here belie a misunderstanding of statistics that has become really embedded into the curriculum by decades of instruction based on a fundamentally wrong mental shortcut endemic to grammar school statistics. You added two parenthetical explanations to the CCSS text, clarifying that a distribution “can be described by its center (median, mean, and/or mode), spread (range, interquartile range), and overall shape.”

Comparing this language with what CCSS already had in **6.SP.5c** – “… quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation) …” – in this case, you replaced “mean absolute deviation” with “interquartile range”, which is fine, but you left the note on measures of center untouched.

Why, then, did you include “mode” in **6.SP.2**? Mode is not a measure of center. This is a common misconception that needs to be eradicated.

Let me say it again:

**Mode is not a measure of center**.

Consider the set of ten numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 9. The median is 5.5 (right in the center). The mean is 5.4 (again, right near the center). The mode is 9 (way out at the end).

Suppose the second 9 in that set now becomes a 1. The new median is 4.5 (right in the center, not a big change). The mean is 4.6 (again, not a big change). And the mode is now 1. How can we claim that the mode is in any way the center of the data set if it’s jumping from one end to the other based on a change in one value?

And now suppose the data set is: 1, 1, 2, 3, 4, 6, 7, 8, 9, 9. Now our mean and median are both 5, and our mode is … well we have 2 modes – 1 and 9. How can a so-called measure of center take on two values?

Granted these are contrived sets of numbers and not actual data, but the fact remains that Mode is not a measure of center. It just happens to begin with an M. If it was called something else, we probably wouldn’t have this problem.

Mode is a better tool for telling us about the overall shape of the data, which is elsewhere in **6.SP.2**. Think of our first list of numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 9). The middle is around 5 and it’s big at the high end. We surely aren’t worried about Grade 6 students getting into skewness and kurtosis, so this way of talking about shape could be helpful.

Mode isn’t useless. (Though I’ll note it doesn’t appear in the CCSS at all.) It’s just not a measure of center.