Of the mean, median, and mode, which measurements of the center is most likely to be affected by outliers? Explain your reasoning including an example with 10 data points
The mean is most likely to be affected by outliers because it takes into consideration all the data points and can be heavily skewed by extreme values.
For example, let's say we have 10 data points for the salaries of employees at a small company:
$30,000, $35,000, $40,000, $45,000, $50,000, $55,000, $60,000, $65,000, $70,000, $100,000
If we calculate the mean salary, we get:
Mean = (30,000 + 35,000 + 40,000 + 45,000 + 50,000 + 55,000 + 60,000 + 65,000 + 70,000 + 100,000) / 10
Mean = $53,000
However, we can see that the outlier of $100,000 is heavily skewing the mean. If we calculate the median instead (which is the middle value when the data is arranged in numerical order), we get:
Median = $52,500
The median value is less affected by the outlier and is a better representation of the typical salary at the company.
Of the mean, median, and mode, the mean is the measurement of the center that is most likely to be affected by outliers.
The mean is calculated by summing up all the values in a data set and then dividing by the total number of values. Outliers, which are extreme values that are significantly higher or lower than the majority of the data points, can have a strong impact on the mean because they pull the average towards themselves.
Let's consider an example with 10 data points: 2, 3, 3, 4, 5, 6, 7, 8, 9, and 100. Here, the majority of the data points are relatively close to each other, but we have one outlier, 100, which is much larger than the others.
If we calculate the mean of these data points: (2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 100) / 10 = 48. This mean of 48 is significantly higher than the values of the data set, primarily due to the influence of the outlier, 100.
In comparison, the median is the middle value when all the data points are arranged in ascending or descending order. In this case, the median would be 6, which is a good representation of the central tendency of the data set regardless of the outlier.
The mode is the value(s) that occur most frequently in the data set. In our example, there are no repeated values, so there is no mode. The absence of a mode is not affected by outliers in this case.
Therefore, when there are outliers in a data set, the mean is the measurement of the center most likely to be affected, while the median and mode are more resistant to the influence of outliers.