The measure of center that is most affected by outliers is the mean. The reason for this is that the mean is calculated by summing all the values and then dividing by the number of values. Therefore, an outlier—a value that is significantly higher or lower than the other values—can skew the mean substantially.
Example with 10 Data Points:
Consider the following set of data points:
\[ 2, 3, 5, 7, 8, 10, 12, 14, 15, 100 \]
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Calculating the Mean: \[ \text{Mean} = \frac{(2 + 3 + 5 + 7 + 8 + 10 + 12 + 14 + 15 + 100)}{10} = \frac{166}{10} = 16.6 \]
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Calculating the Median: To find the median, we first arrange the data in ascending order: \[ 2, 3, 5, 7, 8, 10, 12, 14, 15, 100 \] Since there are 10 data points (an even number), the median is the average of the 5th and 6th values. \[ \text{Median} = \frac{8 + 10}{2} = 9 \]
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Calculating the Mode: In this dataset, there is no repeated number, which means there is no mode.
Summary of Results:
- Mean: 16.6
- Median: 9
- Mode: None
Observations:
In this example, the presence of the outlier (100) significantly affected the mean, pulling it up to 16.6. In contrast, the median remained unaffected by the outlier and stayed at 9, reflecting the central tendency of the majority of the data points much better. The mode doesn't apply here since there are no repeating values.
This illustrates why the mean is the most affected by outliers compared to the median and mode, and highlights the importance of considering the median in datasets where outliers may skew the results.