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Let's analyze the questions one by one:

### (a) Which measures of central tendency do not exist for this data set?
- The measures of central tendency are **mean**, **median**, and **mode**.
- In most datasets, the mean and median always exist, but the mode may not exist if no number repeats.
- The correct answer depends on whether the dataset has a mode. If the dataset does not have any repeating numbers, the **mode** does not exist. If there are repeating numbers, all of the measures will exist.
  
Therefore, possible answers:
- If there are no repeating numbers: **Mode** would not exist.
- Otherwise: **All of these measures exist**.

### (b) Suppose the largest measurement (76) is replaced by 87. Which measures would be affected?
- **Mean**: The mean will definitely change because one of the data values is replaced by a different number (76 → 87).
- **Median**: The median may or may not change, depending on whether 76 was the middle value.
- **Mode**: If 76 was a mode (i.e., the most frequent value), the mode will change.
  
The most likely affected measures are **Mean** and possibly **Mode**.

### (c) Suppose the smallest measurement is removed. Which measures would change?
- **Mean**: Removing the smallest value will affect the mean, as the total sum of the values will decrease.
- **Median**: Removing the smallest value may or may not change the median, depending on how many values are in the dataset.
- **Mode**: If the smallest value was the mode (i.e., the most frequent value), then the mode will change.

Most likely, **Mean** and possibly **Mode** will be affected.

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Do you have any questions or need further clarification?

Here are 5 relative questions for deeper exploration:
1. How does the presence of outliers affect the mean compared to the median?
2. In which scenarios might the mode be a more useful measure than the mean or median?
3. How does adding or removing values affect the median in an even-sized dataset?
4. Can a dataset have more than one mode? What is this called?
5. How is the median calculated if there is an even number of data points?

**Tip:** The mean is sensitive to every data value, so even a small change in the dataset affects it directly.

Which measures of central tendency do not exist for this data set? Choose all that apply.

Explore which measures of central tendency (mean, median, and mode) exist in a dataset. Learn how the mean is calculated, how the median depends on data ordering, and how the mode reflects frequency. This problem is suitable for students in Grades 7-9.

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