Which of the following summary measures is affected least by outliers? a) The median b) The variance c) The range d) The mean e) None of these
The correct answer and explanation is:
Correct Answer: a) The median
The median is the summary measure that is least affected by outliers. The median is the middle value in a data set that has been ordered from smallest to largest. If there is an odd number of observations, the median is the exact middle number. If there is an even number, the median is the average of the two middle numbers. This measure is based purely on position rather than the value of data points, making it resistant to extreme values.
On the other hand, the mean is calculated by summing all the values and dividing by the number of observations. Since every value contributes to the total, an extremely high or low outlier can significantly skew the result. For example, in the data set {2, 3, 4, 5, 100}, the mean is much higher than the majority of the numbers due to the presence of the outlier 100.
The range is the difference between the highest and lowest values in the data set. Outliers can greatly increase the range because a single very high or very low value stretches the spread of the data.
The variance measures the average squared distance of each value from the mean. Since it involves squaring the deviations from the mean, outliers have a disproportionately large impact on the variance. Just a single extreme value can cause the variance to rise sharply.
In contrast, the median remains stable unless the number or position of extreme values changes drastically. For this reason, the median is often used in situations where the data may contain outliers or is skewed. It provides a more accurate representation of the center of a data set in such cases.
Therefore, among all the listed measures, the median is the least affected by outliers.