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1. **Question (a):** "Which measures of central tendency do not exist for this data set?"
   - The measures of central tendency in question are **Mean**, **Median**, and **Mode**.
     - **Mean**: Exists, calculated by summing all bids and dividing by the total number of bids.
     - **Median**: Exists, as it is the middle value when the data is ordered.
     - **Mode**: Exists, as the data has repeated values (e.g., 19 appears multiple times).
   - **Answer**: All of these measures exist.

2. **Question (b):** "If the largest measurement (23) is replaced by 38, which measures would be affected?"
   - **Mean**: Would be affected because the mean depends on the sum of the values.
   - **Median**: Would not be affected since the middle value(s) would remain the same.
   - **Mode**: Would not be affected, as replacing a single value does not change the most frequent values.
   - **Answer**: Mean.

3. **Question (c):** "If the largest measurement is removed, which measures of central tendency would be changed?"
   - **Mean**: Would be affected, as the sum of the data would decrease.
   - **Median**: Might be affected because removing a value can shift the middle of the data.
   - **Mode**: Would not be affected, as it depends on the frequency of values, and 23 is not a mode.
   - **Answer**: Mean and Median.

4. **Question (d):** "Are the relative values of the mean and median for this data set typical of a skew to the left?"
   - In a **left-skewed** distribution, the mean is typically less than the median because of lower values pulling the mean down.
   - Looking at the graph and data, the **mean is likely greater** than the median, indicating a right skew, where higher values pull the mean upward.
   - **Answer**: Mean is greater.

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Would you like a detailed explanation for any of these answers?

Here are 5 related questions for further practice:
1. How is the mean calculated for a given data set?
2. How do outliers affect the mean compared to the median?
3. What types of data distributions commonly result in the mode being equal to the mean and median?
4. How can a histogram help in determining the skewness of a distribution?
5. What are the differences between measures of central tendency and variability?

**Tip:** When analyzing a data set, always consider whether there are outliers that might affect measures like the mean.

Bids were placed in a silent auction for a sword reputed to have been used at the Battle of Hastings. The bidders placed various bids, and the question asks about measures of central tendency such as mean, median, and mode, and how these measures would change with specific alterations to the dataset.

Explore how to calculate and analyze measures of central tendency, including mean, median, and mode, in the context of a silent auction dataset. Learn how changes to the data affect these measures, focusing on skewness and outliers. Suitable for Grades 9-12.

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