Getting Started 
Misconception/Error The student does not choose measures based on the shapes of the distributions. 
Examples of Student Work at this Level The student does not make any reference to the shapes of the distributions in his or her explanation. In addition, the student may:
 Describe the shapes of the distributions without choosing a measure of center or spread.
 Choose a different measure of center and/or spread for each histogram without any justification.
 Select one histogram to represent center and the other to represent spread.

Questions Eliciting Thinking I agree that both distributions are symmetric, but did you answer the question asked? How would the symmetry of the distributions influence your choice of a measure of center or spread?
How would you describe the shape of each histogram? How would the shapes of the distributions influence your choice of a measure of center or spread? 
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. Explain the relationship between the shape of a distribution (and the presence of outliers) and the choice of an appropriate measure of center. 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 five data points that are clustered close together in value and ask the student to find the median and the mean. Next, replace one of the data points with an extreme outlier. Again, ask the student to identify the median and compute the mean. Have the student compare the means and medians of the two related sets of data. Explain that the mean may not represent a typical score in the data set with an outlier, 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.
Explain to the student that typically when a distribution of data is symmetric or approximately normal, the mean and the standard deviation are the measures of choice since they allow for an interpretation of the data in terms of known proportions within numbers of standard deviations of the mean. Additionally, the calculation of the standard deviation requires the use of the mean, so the two measures are compatible. If a distribution of data is skewed or contains outliers, typically the median along with a measure of spread such as the interquartile range (IQR) are the measures of choice for comparing distributions. Explain that the IQR gives an indication of the spread of the data around the median, so it is not unduly influenced by outliers. Additionally, the calculation of the IQR is compatible with the calculation of the median since the calculation of the quartiles is an extension of the calculation of the median.
Provide additional opportunities to select measures of center and spread to represent and compare distributions. Ask the student to justify all choices. 
Making Progress 
Misconception/Error The student chooses a measure of spread that is not consistent with the chosen measure of center. 
Examples of Student Work at this Level The student:
 Chooses the mean and the interquartile range.

Questions Eliciting Thinking Why did you choose the mean? Why did you choose the interquartile range?
What measure of spread requires the use of the mean in its calculation?
What measure of center is included in the calculation of the quartiles? 
Instructional Implications Explain to the student that typically when a distribution of data is symmetric or approximately normal, the mean along with the standard deviation are the measures of choice since they allow for an interpretation of the data in terms of known proportions within numbers of standard deviations of the mean. Additionally, the calculation of the standard deviation requires the use of the mean, so the two measures are compatible. If a distribution of data is skewed or contains outliers, typically the median along with a measure of spread such as the interquartile range (IQR) are the measures of choice. Explain that the IQR gives an indication of the spread of the data around the median, so it is not unduly influenced by outliers. Also, the calculation of the IQR is compatible with the calculation of the median since calculations of the quartiles are an extension of the calculation of the median.
Provide additional opportunities to select measures of center and spread to represent and compare distributions. Ask the student to justify all choices. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student chooses the mean and the standard deviation and explains that both distributions are symmetric and contain no outliers. Therefore, the mean and the standard deviation are appropriate.

Questions Eliciting Thinking Why did you choose the standard deviation along with the mean? Why do the mean and standard deviation go together?
If the distributions were not symmetric, which statistics would you use to compare them?
Do you think these distributions look normal? What are the advantages of using the mean and standard deviation when a distribution is normal? 
Instructional Implications Introduce the student to the properties of a normal distribution. Provide opportunities for the student to calculate proportions of data of a normal distribution greater than or less than a given value. 