Common Core Standards: Math
3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
If we've got one motto, it's this: Give a student a fish, and you give him a single data point; teach a student to fish, and you give him a data distribution to condense into one number. Oh, and he'll probably hate you for it, too.
Now that students understand the ideas of center and spread, they can start describing them with numbers. Student should know how to calculate the mean, median, mean absolute deviation, and interquartile range. But it's more than that; they should also understand what these measures, well, mean. (Pun intended.)
Students should know that the center of a distribution is a one-number summary of that entire set of data. If data points were jewelry, the center would be the One Ring. A measure of center, whether it's the mean or median, is considered the "typical" value for that set of data.
Median is probably easier to explain, since it's just a matter of arranging all our data points in order and picking the one that's smack-dab in the middle. Mean, on the other hand, requires a bit of calculatin'. Whether students think of the mean as equally sharing values across all data points, as the balance point of the distribution on a dot plot, or as the average, they should be able to find means and medians of distributions.
Measures of spread are a little more difficult to visualize, but students should understand them as numbers that express how far away the data points are from the center. (We'd use the range, but it's based only on two values, which makes it…well, not the most descriptive.) Instead, we'll have students use a measure of spread that, like the mean, takes into account the values of all the data. Namely, the mean absolute deviation (MAD).
But what good is a center-spread couple if it can't double date? Enter the median and the interquartile range (IQR), another inseparable center-spread pair. Insert montage of the two happy couples going to drive-ins and sharing root beer floats.
Basically, students should know two things:
- The measure of center (mean, median) is a number that represents the entire data set.
- The measure of spread (MAD, IQR) is a number that represents the variation in the data set.
And if students can calculate all these measures, then even better. (They'll have to for 6.SP.5b anyway.) And if we've got one motto, it's this: Always be prepared.