Question 1168183: You have learned that the median is less affected than the mean by extremely large or extremely small values. Illustrate this concept with a data set containing at least seven values.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! You are absolutely correct! The median is a more robust measure of central tendency in the presence of outliers (extremely large or small values) compared to the mean. Let's illustrate this with a data set.
**Our Initial Data Set (7 values):**
Let's consider a simple data set representing, for example, the number of books read by seven students in a month:
```
5, 6, 7, 8, 9, 10, 11
```
**Calculating the Mean:**
Mean = (5 + 6 + 7 + 8 + 9 + 10 + 11) / 7 = 56 / 7 = **8**
**Calculating the Median:**
To find the median, we first order the data set (which is already ordered). The median is the middle value. With 7 values, the middle value is the 4th value.
Median = **8**
In this initial data set, the mean and the median are the same.
**Introducing an Extremely Large Value (Outlier):**
Now, let's introduce an extremely large value into the data set. Imagine one student read a significantly higher number of books:
```
5, 6, 7, 8, 9, 10, 100
```
**Recalculating the Mean:**
Mean = (5 + 6 + 7 + 8 + 9 + 10 + 100) / 7 = 145 / 7 ≈ **20.71**
Notice how the single extremely large value (100) has drastically pulled the mean upwards. The mean is now significantly higher than most of the values in the data set, and it no longer feels like a typical representation of the "center" of the data.
**Recalculating the Median:**
The data set is still ordered. The middle value (the 4th value) remains the same:
Median = **8**
Observe that the median has remained unchanged despite the presence of the outlier. This is because the median only depends on the position of the values in the ordered data set, not on the actual magnitude of the extreme values.
**Conclusion:**
This example clearly illustrates that the **mean** is highly sensitive to extreme values. A single outlier can significantly distort its representation of the central tendency. In contrast, the **median** remains relatively unaffected by these extreme values, providing a more stable and often more representative measure of the "typical" value in a data set containing outliers.
We could perform a similar exercise by introducing an extremely small value, and we would observe the mean being pulled downwards while the median remains stable. This robustness to outliers is a key advantage of using the median in certain situations.
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