Which of the following best describes the effect of outliers on the $standarddeviation$ of a data set?

• A. The $standarddeviation$ is unaffected by the presence of outliers.
• B. The $standarddeviation$ always decreases when outliers are present.
• C. The $standarddeviation$ is not resistant to outliers; outliers can greatly increase its value.
• D. The $standarddeviation$ is resistant to outliers; outliers have little to no effect on its value.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand what standard deviation measures. It quantifies the amount of variation or dispersion in a data set by calculating the average distance of each data point from the mean.
2. Step 2: Recall the formula for standard deviation: \(\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2}\), where \(x_i\) are data points, \(\mu\) is the mean, and \(n\) is the number of data points.
3. Step 3: Consider the effect of an outlier, which is a data point significantly different from the rest. Because the formula squares the difference \((x_i - \mu)^2\), large deviations (outliers) contribute disproportionately to the sum.
4. Step 4: Recognize that this large contribution from outliers increases the sum inside the square root, thus increasing the standard deviation value.
5. Step 5: Conclude that standard deviation is sensitive (not resistant) to outliers, meaning outliers can greatly increase the standard deviation of a data set.