Let's work through the questions based on the data provided in the image.

(a) Which measures of central tendency do not exist for this data set?

You are asked to identify whether any of the three measures of central tendency—mean, median, mode—do not exist.

From the frequency chart, we can infer:

• Mean: The mean is always calculable, as it represents the average of all data points.
• Median: The median, the middle value when the data is ordered, also exists.
• Mode: The mode is the most frequent data point. From the chart, the most frequent value is in the 0 to 10 range (with a frequency of 8). Therefore, a mode exists.

Answer: None of the measures are missing, so all of these measures exist.

(b) If the value 35 (largest measurement) is replaced by 68, which measures would be affected?

• Mean: Yes, the mean would change because the sum of all values increases, thus changing the average.
• Median: The median will not change because only the largest number is changed, and the middle of the data set remains unaffected.
• Mode: The mode won't change as it depends on the most frequent value, which isn't altered by replacing the largest number.

Answer: The mean is affected.

(c) If the smallest measurement (−6) were removed, which measures would be affected?

• Mean: Yes, the mean would be affected because the sum of the data changes when the −6 is removed.
• Median: Removing the smallest value may affect the median because the order of the data would shift.
• Mode: The mode would not change, as removing −6 doesn't impact the most frequent value.

Answer: Mean and Median would be affected.

Let me know if you would like a more detailed explanation on any of these concepts or calculations!

Here are 5 related questions for further exploration:

1. How is the mean calculated for this specific data set?
2. Why does changing the largest value not affect the median?
3. Could removing a different number besides −6 affect the mode?
4. How would adding more negative values change the mean?
5. In what scenarios would the mode change in a dataset?

Tip: The mode is useful when identifying the most common event, while the mean gives you an overall average, and the median provides a middle-ground perspective.