Heavily Skewed Distributions: Measure of Central Tendency

B

The mean is affected severely by outliers and hence, should not be used as a measure of central tendency with skewed distributions.

When a distribution is heavily right or left skewed, it means that the data is concentrated towards one tail of the distribution, causing a long tail on the opposite side. In such cases, the shape of the distribution is asymmetric, and the traditional measures of central tendency may not accurately represent the typical or central value of the data.

The measures of central tendency commonly used to describe a distribution are the mean, median, and mode. Let's examine each option in relation to skewed distributions:

A. Skewness: Skewness is a measure of the asymmetry of a distribution. It quantifies the degree and direction of skewness. If a distribution is heavily right-skewed, the skewness value will be positive, indicating a longer tail on the right side. Similarly, if a distribution is heavily left-skewed, the skewness value will be negative, indicating a longer tail on the left side. Therefore, skewness is a useful measure for identifying and quantifying the skewness of a distribution, and it can be used even in heavily skewed distributions.

B. Mean: The mean is the most commonly used measure of central tendency. It is the arithmetic average of all the values in a dataset. However, the mean can be significantly influenced by outliers or extreme values, especially in skewed distributions. When a distribution is heavily skewed, the mean tends to be pulled towards the direction of the long tail. Consequently, using the mean as a measure of central tendency in heavily skewed distributions may not accurately represent the central value of the data.

C. Median: The median is another measure of central tendency. It represents the middle value of a dataset when the data is arranged in ascending or descending order. Unlike the mean, the median is not affected by outliers or extreme values. In heavily skewed distributions, the median provides a more robust measure of central tendency since it is less influenced by extreme values. Therefore, the median is a suitable measure to use when dealing with heavily skewed distributions.

D. Mode: The mode is the value or values that appear most frequently in a dataset. In heavily skewed distributions, the mode can still be a useful measure of central tendency since it represents the peak or the most common value(s) in the distribution. Therefore, the mode can be used as a measure of central tendency, even in heavily skewed distributions.

Considering the explanations above, the measure of central tendency that should not be used when a distribution is heavily right or left skewed is option B: the mean. While the mean is widely used and valuable in symmetric or approximately symmetric distributions, it can be misleading in heavily skewed distributions, as it tends to be influenced by outliers or extreme values. In such cases, the median and mode are more appropriate measures to represent the central value of the data.