Question

which one of the following statements is correct? the median and iqr are generally more affected by extreme outliers than the mean and standard deviation. a standard deviation equal to 0 will occur if and only if all values in a data set are equal to 0. 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. if at least 75% of the observations in a data set have the same value, the iqr must be equal to 0. none of the above statements are correct.

Explanation:

1. The median and IQR are robust statistics and less affected by extreme outliers compared to the mean and standard - deviation. So the first statement is false.
2. A standard deviation equal to 0 occurs if and only if all values in a data - set are equal (not necessarily 0). So the second statement is false.
3. The sample mean can be negative even if not all values in the data - set are negative. For example, in the data - set \{- 1,1,0\}, the mean is 0. So the third statement is false.
4. If at least 75% of the observations in a data - set have the same value, the third quartile \(Q_3\) and the first quartile \(Q_1\) will be equal, and \(IQR = Q_3 - Q_1=0\). So the fourth statement is true.

Answer:

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