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 does not make sense because the mean is the true center of the data. b. 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. c. the statement makes sense because mean is not always the true average of the data. d. the statement does not make sense because the mean measures the spread of the data items.

Explanation:

The mean is calculated as the sum of all data items divided by the number of data items. Outliers (extremely high or low values) can significantly affect the mean. 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 22 does not represent the 'typical' value of the first four data items (\(1,2,3,4\)). Knowing the spread (e.g., range, standard deviation) helps in understanding the context of the mean. Option A is wrong because the mean can be skewed by outliers. Option C is incorrect as the mean is always the arithmetic average (\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)). Option D is wrong because the mean does not measure the spread (measures like range, variance measure spread).

Answer:

B. 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.