Question

Which measures are sensitive to extreme values?

Answer 1

Robust measures such as the median and interquartile range (IQR) are sensitive to extreme values and provide more accurate representations of central tendency and spread.

Step-by-step explanation:

In statistics, measures that are sensitive to extreme values are called robust measures. Robust measures are less affected by extreme values and provide a more accurate representation of central tendency. Examples of robust measures include the median and interquartile range (IQR).

The median is the middle value in a dataset when arranged in ascending order. It is less influenced by extreme values compared to the mean. The IQR is a measure of dispersion that represents the range between the first quartile (25th percentile) and the third quartile (75th percentile). It is less affected by extreme values than the range or standard deviation.

For example, consider a dataset of test scores: 80, 85, 90, 95, and 120. The mean would be heavily influenced by the extreme value of 120, giving an inaccurate representation of the typical score. However, the median and IQR would not be affected, providing a more reliable measure of central tendency and spread.

Answer 2

The mean and standard deviation are measures sensitive to extreme values, which can cause them to misrepresent the data set's general trend. Using a graphing calculator or statistical calculations, outliers can be identified and assessed for their impact on the dataset.

Step-by-step explanation:

Measures that are sensitive to extreme values include the mean and standard deviation, as they both take into account every value in a data set. Extreme values can significantly shift the mean and inflate the standard deviation, making them less representative of the general trend within the data. In contrast, the median and interquartile range (IQR) are more robust measures of central tendency and spread, respectively, because they are less affected by outliers.

To identify outliers using a TI-83, 83+, or 84+ graphing calculators, one can utilize a graphical approach by plotting the data and drawing an extra pair of lines two standard deviations above and below the best-fit line. Alternatively, calculate each residual to see which ones exceed twice the standard deviation. The outlier can be determined numerically or graphically, and its effect on the data set can be assessed by removing it and analyzing the new best-fit line.

In some cases, data far from two standard deviations from the mean—especially in mound-shaped and symmetric distributions—can be considered outliers. These methods help distinguish whether a data point is truly an anomaly or representative of a real, but extreme variation within the data set.