# Skewness in Statistics: Definition, Formula & Example

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• 0:00 Definition of Skewness
• 1:53 Properties of Skewed…
• 2:41 Formula for Skewness
• 3:35 Examples of Skewness
• 5:43 Lesson Summary
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Lesson Transcript
Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you'll learn about skewness in statistics, including what data distribution and bell curves look like with and without skew. After that, you'll learn a formula to calculate skew, and then you can test your knowledge with a brief quiz.

## Definition of Skewness

Skewness in statistics represents an imbalance and asymmetry from the mean of a data distribution. If you look at a normal data distribution using a bell curve, the curve will be perfectly symmetrical. Now, this doesn't happen all that often! In order to fully understand when a data distribution is imperfect and skewed, let's look at a normal data distribution and symmetrical bell curve.

First, let me remind you of a few basic terms

• Mean is the average of the numbers in the data distribution
• Median is the number that falls directly in the middle of the data distribution
• Mode is the number that appears most frequently in the data distribution

In a normal data distribution, the mean is directly in the middle (and top point) of the bell curve. Imagine that Mrs. Thomas wanted to teach her high school statistics class on the first day about data distributions, standard deviations, and bell curves. She asks her 16 student class to secretly divulge their summer job incomes. Each student provides Mrs. Thomas with a piece of paper with their income. She rounds each income level to the nearest 500 and makes a chart.

Now that we see the data on a chart, we can see that four of the students made about \$2,000 in total over the summer. If we find the mean, we see that it is \$2,000. The mode and median in this data distribution also happen to be \$2,000. In a normal data distribution and perfectly symmetrical bell curve, the median and mean are always the same value. Take a look at the graph of the data which represents a normal bell curve (no skewness at all!).

## Properties of Skewed Bell Curves

In a symmetric bell curve, the mean, median, and mode are all the same value. How easy is that? But in a skewed distribution, the mean, median, and mode are all different values. You can see this represented in this image:

A skewed data distribution or bell curve can be either positive or negative. A positive skew means that the extreme data results are larger. This skews the data in that it brings the mean (average) up. The mean will be larger than the median in a skewed data set. A negative skew means the opposite: that the extreme data results are smaller. This means that the mean is brought down, and the median is larger than the mean.

## Formula for Skewness

The formula to find skewness manually is this:

skewness = (3 * (mean - median)) / standard deviation

In order to use this formula, we need to know the mean and median, of course. As we saw earlier, the mean is the average. It's the sum of the values in the data distribution divided by the number of values in the distribution. And if the data distribution was arranged in numerical order, the median would be the value directly in the middle.

Now, you may be asking: What is standard deviation? Standard deviation tells you how different and varied your data set really is. Standard deviation shows you how far your numbers spread out from the mean and median. Here is the formula to find standard deviation:

## Examples of Skewness

#### Example 1: Zero Skewness

Taking the example from earlier (student summer income), we have the following 16 values in our data set (all are in dollars):

500, 1000, 1000, 1500, 1500, 1500, 2000, 2000, 2000, 2000, 2500, 2500, 2500, 3000, 3000, 3500

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