Standard deviation

• Describe variability by calculating deviations from the mean (S-ID.A.2).
• Compare two data sets with the same means but different variabilities, and contrast them by calculating the deviation of each data point from the mean (S-ID.A.2).
• Interpret sets with greater deviations as having greater variability (S-ID.A.2).
• Calculate a standard deviation by hand for a small data set, and understand standard deviation as an indicator of a typical deviation from the mean of an element of the data set (S-ID.A.2).

In the previous section, students interpreted the meaning of the various measures of center. Measures of center are important because they are single numbers that show what value is typical for a data set. However, data involves variation. How much data varies is an important question. In this section, students examine variability, the other major feature of measurements taken to answer a statistical question.

In grades 6 to 8, students learned that interquartile range (IQR) and mean absolute deviation (MAD) are ways to describe spread. In this section, they learn to calculate MAD’s more sophisticated cousin, standard deviation. They start by looking at how much each data point deviates from the mean, and use these calculations to describe different data sets as more or less variable. Then, they go through the procedure for calculating standard deviation. (Although the Standards do not insist that students do such calculations or learn this procedure, doing the calculation a few times can help to illustrate what the standard deviation measures.) They come to understand standard deviation as “typical distance from the mean,” and that higher values for standard deviation imply that a distribution is more spread out, whereas lower values imply that data are more closely clustered about the mean.