https://file.aiquickdraw.com/mathbot/file/1729545594627uwo6qxf3.jpg

Let's go through the questions related to the data from the image, one by one.

### (a) Which measures of central tendency do not exist for this data set?
For the dataset given, the **mean**, **median**, and **mode** all exist:

- **Mean**: This is simply the average of all the data points.
- **Median**: The middle value when the data points are sorted.
- **Mode**: The value that appears most frequently. In this case, there are repeated values (e.g., 44 appears three times), so the mode exists.

Thus, **all of these measures exist** for the dataset.

### (b) Suppose that the measurement 22 (the smallest measurement) were replaced by 8. Which measures of central tendency would be affected?
- **Mean**: The mean would definitely change because the sum of the values would decrease due to replacing 22 with 8, affecting the average.
- **Median**: The median would remain the same because the replacement of a single value at the lower end won't affect the central value of the sorted dataset.
- **Mode**: The mode remains unaffected because the replacement does not change the most frequent value(s) in the data.

So, **the mean** is affected by the change.

### (c) Suppose that, starting with the original data set, the smallest measurement (22) were removed. Which measures of central tendency would be changed?
- **Mean**: The mean would change, as removing a value changes the sum of the data points and the total number of values, thus changing the average.
- **Median**: The median could change because removing one of the lowest values may shift the middle of the data.
- **Mode**: The mode would not change because removing 22 does not affect the frequency of the most repeated values (44, in this case).

Thus, **mean** and **median** would be affected by the removal.

Would you like further details on any specific part of this? Here are 5 questions to expand the topic:

1. How is the mean calculated for this data set?
2. How does the removal of an outlier affect the mean and median differently?
3. Can there be more than one mode for a data set?
4. What is the impact on central tendency if the data set had more variation in values?
5. How can the mean be used to interpret the distribution of data?

**Tip:** The **mean** is highly sensitive to extreme values (outliers), whereas the **median** is more resistant to such changes.

An intelligence test was administered to a group of 22 people. Their respective times for completion (in minutes) were as follows: 22, 29, 31, 31, 32, 32, 36, 37, 37, 38, 39, 39, 39, 41, 41, 42, 42, 43, 44, 44, 44. Analyze the measures of central tendency based on three scenarios.

This example covers how to calculate the mean, median, and mode of a dataset from an intelligence test administered to 22 people. It also explores how changing or removing the smallest data point impacts these measures. Suitable for students in Grades 9-12.

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