The problem asks to name two measures of the variation of a distribution and describe the conditions under which each measure is preferred for measuring variability in a single data set. Below is an analysis of each option, with explanations for the correct ones:

1. A. The interquartile range is preferred when there are few data points.

   • This is incorrect. The interquartile range (IQR) is typically preferred for larger data sets and when data has outliers or is skewed, as it focuses on the middle 50% of the data, making it robust to extreme values.

2. B. The mean is preferred when the data is relatively symmetric.

   • This is correct. The mean is a good measure of central tendency when the data is symmetric because it is sensitive to every data point, and in symmetric distributions, all data points are evenly spread.

3. C. The standard deviation is preferred when there are many data points.

   • This is correct. Standard deviation measures the spread of the data and becomes more meaningful as the sample size increases. It's commonly used when data is more symmetric and without outliers.

4. D. The z-score is preferred when there are many data points.

   • This is incorrect as z-scores are used to measure how many standard deviations an element is from the mean, rather than measuring variability directly.

5. E. The median is preferred when the data is strongly skewed or has outliers.

   • This is correct. The median is not affected by extreme values (outliers), making it a better choice when data is skewed or has outliers.

6. F. The standard deviation is preferred when the data is relatively symmetric.

   • This is correct. Standard deviation works well for symmetric data sets because it is sensitive to outliers, which typically distort results in skewed distributions.

7. G. The interquartile range is preferred when the data is strongly skewed or has outliers.

   • This is correct. The IQR, as mentioned earlier, focuses on the central portion of the data and is less affected by skewness and outliers.

8. H. The median is preferred when the data is relatively symmetric.

   • This is incorrect. While the median is a robust measure of central tendency, the mean is generally preferred for symmetric data sets.

Correct answers:

• B. The mean is preferred when the data is relatively symmetric.
• C. The standard deviation is preferred when there are many data points.
• E. The median is preferred when the data is strongly skewed or has outliers.
• F. The standard deviation is preferred when the data is relatively symmetric.
• G. The interquartile range is preferred when the data is strongly skewed or has outliers.

Would you like further clarification or additional details?

Here are 5 related questions to expand your understanding:

1. Why is the mean sensitive to outliers in a data set?
2. In what scenarios would you use the mode as a measure of central tendency?
3. How does skewness affect the choice between median and mean?
4. Can you calculate standard deviation for a very small sample size? What considerations should be made?
5. Why is the IQR considered robust to outliers?

Tip: When dealing with skewed data or outliers, always consider using the median or interquartile range to minimize the influence of extreme values.