# Mean and Median for Skewed Distribution

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Statistics for long tailed distribution

Introduction

In this article, I am going to show skewed distribution statistics. Most importantly I am going to talk about mean and median which are the basic statistics for skewed distributions. I guess mostly we are familiar with normal distribution. Normal distribution is ideal. Normal distribution looks like a bell shaped curve like below which is symmetric on the both side. Here, we will discuss about non-symmetric distributions.

Skewed Distribution Statistics

Skewed distributions are those distributions which are non-symmetric and can have longer tails on either left or right side. For normally distributed data, mean. median and mode all point to a single value whereas the values shift differently in case of skewed data.

Distributions having a longer tail on right side are also called positively skewed distributions and distributions having a longer tail on left side are also called negatively skewed distributions. For normal data, the median (2nd quartile or Q2) is located at the center of the box as shown in the boxplots below.

When we have longer tail on right or left side, the mean gets shifted towards the tail or outlier. The median also get shifted but the mean gets more shift. That’s why median based statistics are less susceptible than mean based statistics. The figure above demonstrates a clear picture of this type of movement of mean and median for skewed data.

The same phenomenon is true for outliers. If there are outliers instead of longer tail, the mean also shifts more than the median. A single outlier possessing extreme value, has the potential to significantly relocate the mean. Therefore, mean based statistics extracted from data which has an…