Question

1. which one of the following statements is correct? a. the median and iqr are generally more affected by extreme outliers than the mean and standard deviation. b. a standard deviation equal to 0 will occur if and only if all values in a data set are equal to 0. c. because the sample mean always falls between the minimum and maximum values, the sample mean can never take on a negative value unless all values in the data set are negative. d. if at least 75% of the observations in a data set have the same value, the iqr must be equal to 0. e. none of the above statements are correct.

Explanation:

Step1: Recall properties of statistics

The median and IQR are resistant to outliers, while the mean and standard - deviation are sensitive to outliers. So, option A is incorrect.

Step2: Analyze standard - deviation formula

The standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n - 1}}$. It is 0 when all $x_{i}$ are equal (not necessarily 0), so option B is incorrect.

Step3: Understand the nature of the sample mean

The sample mean $\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}$ can be negative even if not all values are negative, for example, when some negative values have a larger magnitude than positive values, so option C is incorrect.

Step4: Recall the definition of IQR

The inter - quartile range (IQR) is $Q_{3}-Q_{1}$. If at least 75% of the observations have the same value, then $Q_{1}=Q_{3}$, and $IQR = 0$. So, option D is correct.

Answer:

D. If at least 75% of the observations in a data set have the same value, the IQR must be equal to 0.