Final answer:
The range is a misleading index of variability when there are extreme outliers because it does not accurately reflect the spread of the majority of the data. Outliers can be identified using the Interquartile Range (IQR), and data can be checked against the mean by considering how many standard deviations away they are.
Step-by-step explanation:
The range can be a misleading index of variability in a dataset when there are extreme outliers. This occurs because the range is calculated as the difference between the highest and lowest values, so one or two extreme values can dramatically increase the range, not reflecting the true variability of the majority of the data.
Outliers can be identified using an appropriate numerical test involving the Interquartile Range (IQR). If a data set has a standard deviation of zero, it means all of the data have the same value. We generally consider the mean, median, and mode as measures of the center, where the mean is the average, the median is the middle value when data is ordered, and the mode is the most frequently occurring value.
Identifying Outliers with IQR
To identify outliers, we can calculate the IQR and then determine if any data values are more than 1.5 times the IQR above the third quartile or below the first quartile. When such outliers are found, they might be removed or investigated further depending on the context of the data. Additionally, if any data values are farther than two standard deviations from the mean, this can also indicate outliers. However, this criterion is most applicable to data that is symmetric or normally distributed.