Question

outliers have very little or no effect on which of the following? check all that apply.
mean
range
median
standard deviation
mode
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Explanation:

• Mean: The mean is calculated by summing all data points and dividing by the number of data points. An outlier (a very large or very small value) can significantly change the sum, thus affecting the mean. For example, if we have data \( \{1,2,3\}\) with mean \(\frac{1 + 2+3}{3}=2\), and we add an outlier \(100\), the new mean is \(\frac{1+2 + 3+100}{4}=26.5\).
• Range: The range is calculated as \(R=\text{Max}-\text{Min}\). An outlier can be the new \(\text{Max}\) or \(\text{Min}\). For instance, if data is \( \{2,3,4\}\) with range \(4 - 2=2\), and we add an outlier \(10\), the new range is \(10 - 2 = 8\).
• Median: The median is the middle - value when data is ordered. If the number of data points \(n\) is odd, \(M = x_{\frac{n + 1}{2}}\) (where \(x_i\) is the \(i\) - th ordered data point), and if \(n\) is even \(M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}\). For example, for data \( \{1,2,3,4,100\}\) (odd \(n = 5\)), the median is \(x_3=3\). For data \( \{1,2,3,100\}\) (even \(n = 4\)), the median is \(\frac{x_2 + x_3}{2}=\frac{2+3}{2}=2.5\). Outliers have little effect on the median as it only depends on the position of data points, not their actual magnitudes (as long as the relative position of non - outlier data points doesn't change much).
• Standard Deviation: The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\), and for the population standard deviation \(\sigma=\sqrt{\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}}\). An outlier, since it is far from the mean (\(\bar{x}\) or \(\mu\)), will increase \(\sum_{i = 1}^{n}(x_i-\bar{x})^2\) (or \(\sum_{i=1}^{N}(x_i-\mu)^2\)), thus affecting the standard deviation.
• Mode: The mode is the most frequently occurring value. An outlier is usually a single value (not part of the most frequent group). For example, if data is \( \{1,1,2,3\}\) (mode is \(1\)), and we add an outlier \(100\), the mode remains \(1\).

Answer:

Median, Mode