# 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.

## Tasks

WHAT: The four parts of this task give different perspectives on standard deviation. In part 1, students match dot plots to standard deviations. In part 2, they create small data sets with given characteristics, e.g., standard deviation equal to 0. In part 3, students examine two histograms, say which has the greater standard deviation, and explain why (MP3). In part 4, students are asked to create sets of numbers with the same means and different standard deviations, and with different means and the same standard deviations (MP2). Parts 3 and 4 are more difficult so students are encouraged to work with partners.

WHY: The task commentary states, “the purpose of this task is to deepen student understanding of the standard deviation as a measure of variability in a data distribution.” The task provides students with an opportunity to look for and make use of structure while working with standard deviations (MP7).

WHAT: In the first part of this task, students are asked to explain the differences in calculating a MAD and a standard deviation. Then students calculate the standard deviation for a small data set: four test scores. In the last two parts of this task, students use standard deviations in comparing this set of test scores with two other sets and interpreting differences in spread.

WHY: This task is placed here to introduce standard deviation calculations because: it connects and builds on students’ prior knowledge (MAD); it provides students with a small data set, allowing calculations by hand; it provides an opportunity to compare standard deviations of several data sets in a real-world context.