Outliers are observations that fall well above or below the overall bulk of the data. Consider a set of 50 (univariate) data points with a single outlier. Suppose the outlier is removed from the data set, which of the following is/are always true? Select all that apply.

Group of answer choices

The removal will cause the range to change.

The removal will cause the mean to decrease.

The removal will cause the standard deviation to decrease.

The removal will cause the interquartile range to decrease.

Let's analyze each option in the context of removing an outlier from a data set:

1. **The removal will cause the range to change.**  
   - The **range** is the difference between the maximum and minimum values in a data set. If the outlier is either the maximum or minimum value, removing it will change the range. Thus, this statement is **true** if the outlier is one of these extreme values.  
   **This is true.**

2. **The removal will cause the mean to decrease.**  
   - If the outlier is **above** the bulk of the data, removing it will decrease the mean, but if the outlier is **below** the bulk of the data, removing it will cause the mean to increase. Hence, it is not always true that the mean will decrease after removing an outlier.  
   **This is false (not always true).**

3. **The removal will cause the standard deviation to decrease.**  
   - The **standard deviation** measures the spread of the data. Outliers typically increase the spread of the data. Therefore, removing an outlier will almost always reduce the standard deviation because it reduces the extreme deviation from the mean.  
   **This is true.**

4. **The removal will cause the interquartile range (IQR) to decrease.**  
   - The **interquartile range (IQR)** measures the spread of the middle 50% of the data (the difference between the 75th and 25th percentiles). Since the IQR focuses on the central part of the data, removing an outlier (which is outside this central range) is unlikely to affect the IQR significantly.  
   **This is false (not always true).**

### Conclusion:
- The statements that are **always true** are:
  - The removal will cause the range to change.
  - The removal will cause the standard deviation to decrease.

Would you like further clarification or more details on any of these points?

Here are 5 related questions to expand your understanding:
1. How does the presence of multiple outliers affect the standard deviation?
2. In what cases would removing an outlier increase the interquartile range?
3. How does skewness change when an outlier is removed?
4. What is the impact of outliers on the median?
5. Can removing an outlier ever increase the range?

**Tip:** When analyzing data, it's important to visually inspect your data (e.g., using a boxplot) to easily identify potential outliers.

Outliers are observations that fall well above or below the overall bulk of the data. Consider a set of 50 (univariate) data points with a single outlier. Suppose the outlier is removed from the data set, which of the following is/are always true? Select all that apply.

Group of answer choices:
- The removal will cause the range to change.
- The removal will cause the mean to decrease.
- The removal will cause the standard deviation to decrease.
- The removal will cause the interquartile range to decrease.

Explore how removing an outlier from a data set affects statistical measures such as the range, mean, and standard deviation. Understand which outcomes are always true, including changes to the range and standard deviation, and learn more about the impact of outliers on data spread. Suitable for high school statistics students.

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