Of the mean, median, and mode, which measure of center is most affected by outliers? Explain your reasoning including an example with at least 10 data points.
The measure of center most affected by outliers is the mean. This is because the mean takes into account every data point in the set and is therefore highly sensitive to outliers, which can drastically affect the overall average.
For example, consider the following data set of 10 students' test scores:
80, 85, 90, 92, 93, 95, 98, 99, 100, 120
The median score is 94 (the middle value when the scores are listed in order), and the mode is also 94 (since 94 appears twice, more than any other score). However, the mean score is 94.2, which is significantly influenced by the outlier score of 120.
If we remove the outlier from the set, the mean drops to 93.1, which is much closer to the median and mode. This illustrates how outliers can heavily skew the mean and make it a less reliable measure of central tendency.
Of the mean, median, and mode, the measure of center that is most affected by outliers is the mean.
The mean is calculated by adding up all the data points and then dividing by the total number of data points. If there is an outlier, which is a value that is significantly larger or smaller than the other data points, it can greatly impact the mean.
Let's consider the following example with 10 data points: 2, 4, 6, 8, 10, 12, 14, 16, 18, 40.
The mean of this data set is (2+4+6+8+10+12+14+16+18+40) / 10 = 10.
Now, we introduce an outlier of 100 into the data set: 2, 4, 6, 8, 10, 12, 14, 16, 18, 40, 100.
The new mean of this data set is (2+4+6+8+10+12+14+16+18+40+100) / 11 = 14.55.
As we can see, the outlier value of 100 greatly influenced the mean, increasing it significantly from 10 to 14.55. In this case, the outlier pulled the mean towards itself, making it higher than what it would have been without the outlier.
In contrast, the median and mode are less affected by outliers. The median is the middle value when the data points are arranged in order, and the mode is the value that appears most frequently. These measures are not influenced by extreme values as they only consider the middle or most common values.