Question

determine whether the following statement makes sense or does not make sense, and explain the reasoning. the mean can be misleading if you dont know the spread of data items. choose the correct answer below. a. the statement makes sense because there may exist some outliers (maybe much greater or much less than the rest of the data) in the data items. b. the statement makes sense because mean is not always the true average of the data. c. the statement does not make sense because the mean measures the spread of the data items. d. the statement does not make sense because the mean is the true center of the data

Explanation:

The mean is sensitive to outliers. If there are outliers (values that are much greater or much less than the rest of the data), the mean can be pulled in the direction of those outliers. For example, if we have data set \(\{1,2,3,4, 100\}\), the mean is \(\frac{1 + 2+3+4 + 100}{5}=\frac{110}{5} = 22\). The value \(100\) (an outlier) has a large impact on the mean. Without knowing the spread (which can help identify outliers), the mean may not represent the "typical" value of the data set well.

• Option B is incorrect because the mean is the arithmetic average of the data (\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)), so it is the true average in the arithmetic - sense.
• Option C is incorrect because the mean (\(\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}\)) does not measure the spread. Measures of spread include the range (\(R=\text{max}(x)-\text{min}(x)\)), variance (\(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)) and standard deviation (\(s=\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)).
• Option D is incorrect because the mean is affected by outliers and may not be the "true" center (in a sense of representing the typical value) when there are extreme values.

Answer:

A. The statement makes sense because there may exist some outliers (maybe much greater or much less than the rest of the data) in the data items.