Final answer:
The choice of measure of center and variability when comparing distributions depends on the shape of the distributions. For symmetric (normal) distributions, mean and standard deviation are appropriate. For skewed distributions, median and quartiles or IQR are better measures.
Step-by-step explanation:
When comparing the distributions of data sets, especially the shape of distributions, both the measure of center and the measure of variability are important to consider. The measure of center provides a central value for the data set which includes the mean, median, and mode, while the measure of variability gives an idea of the spread of the data, such as the standard deviation or the interquartile range (IQR).For symmetric distributions, especially when the distribution is normal (bell-shaped), the mean is an appropriate measure of center because it is equal to the median and the mode. The standard deviation is a useful measure of variability in this case as it reflects how the data items vary from the mean.However, in the case of skewed distributions, the median is often a better measure of center because it is not as affected by outliers as the mean. In such distributions, along with the median, quartiles or the IQR are preferred as measures of variability over the standard deviation, because they provide a clearer picture of the spread that is not influenced by the skew or extreme values.Considering the above, for symmetric distributions like a normal distribution, one would use the mean and standard deviation to compare distributions. For distributions that are not symmetric, the median along with quartiles or IQR would provide a more accurate representation.