**Questions Eliciting Thinking**What are the shapes of these two distributions? What term describes this shape?
What is the difference between the mean and the median? Why would you choose one over the other?
Would you expect the mean and median to be the same for either of these distributions?
How does the distribution of the data affect the mean? Median? |

**Instructional Implications**If needed, review terminology associated with the shapes of distributions such as symmetric, normal, skewed, and uniform. Show examples of each type and introduce the concept of an outlier.
Provide opportunities for the student to calculate both the mean and the median of a variety of distributions. Include symmetric, normal, skewed, and uniform data sets. For example, give the student a set of data that is perfectly symmetric and ask the student to find the median and the mean. Emphasize that when a distribution is symmetric, the mean and the median are the same. Next, give the student a set of data that is skewed and again, ask the student to compute both the mean and the median. Guide the student to observe that the mean and median of a skewed distribution are not the same. Explain that the mean may not represent a typical score in the data set that is skewed, so the median may be a better choice of a measure of center. Make clear that the shape of a distribution and the presence of outliers should be taken into account when selecting an appropriate measure of center.
Discuss how the calculations of the mean and the median are influenced by outliers. Explain that since the calculation of the mean involves summing all of the values in the distribution, outliers contribute directly to its calculation. On the other hand, the median of a distribution such as {1, 2, 3, 4, 5} is unchanged if the final value, 5, is replaced by 100 since the median only takes into account the order of the data rather than the actual values.
Provide additional opportunities to select measures of center to represent and compare distributions. Ask the student to justify all choices. |

**Questions Eliciting Thinking**I agree that the distributionsÂ are skewed but is the mean the best choice for comparing skewed distributions?
You correctly selected the median to compare the two distributions, but what is your justification for that choice?
Can you explain why you described these distributions as symmetric? |

**Instructional Implications**Give the student a set of data that is perfectly symmetric and ask the student to find the median and the mean. Emphasize that when a distribution is symmetric, the mean and the median are the same. Next, give the student a set of data that is skewed and again, ask the student to compute both the mean and the median. Guide the student to observe that the mean and median of a skewed distribution are not the same. Explain that the mean may not represent a typical score in the data set that is skewed, so the median may be a better choice of a measure of center. Make clear that the shape of a distribution and the presence of outliers should be taken into account when selecting an appropriate measure of center.
Provide additional opportunities to select measures of center to represent and compare distributions. Ask the student to justify all choices. |

**Instructional Implications**Challenge the student to create three small sets of data: one in which the mean is less than the median, one in which the mean is greater than the median, and one in which the mean is equal to the median. Have the student relate the shapes of the distributions to each of these outcomes. |