We have previously identified a normal distribution to be symmetrical in shape. But when you're dealing with real-world data you'll often come across asymmetric distributions as well. In this lesson, you'll learn how to measure asymmetry (or skewness) in a distribution. Additionally, you'll learn about kurtosis. Kurtosis defines whether a distribution is truly "normal" or whether it may have so-called "fatter" or "thinner" tails than you would observe when data are normally distributed.

You will be able to:

• Understand the concept of symmetrical distribution
• Calculate and describe skewness and kurtosis as measures of non-symmetry and diverging from normality

A distribution is symmetric if the relative frequency or probability of certain values are equal at equal distances from the point of symmetry. The point of symmetry for normal distributions is the mean (and at the same time median and mode!)

Have a look at following histogram:

This distribution meets all of the conditions of being symmetrical.

The most common symmetric distribution is the normal distribution, however, there are a number of other distributions that are symmetric. Here is a good article that looks into all sorts of symmetrical distributions. We'll focus on normal distributions (by far the most common group) here, and see how these can lose symmetry!

Skewness is the degree of distortion or deviation from the symmetrical normal distribution. Skewness can be seen as a measure to calculate the lack of symmetry in the data distribution.

Skewness helps you identify extreme values in one of the tails. Symmetrical distribution has a skewness of 0.

Distributions can be positively or negatively skewed.

A distribution is positively skewed when the tail on the right side of the distribution is longer (also often called "fatter"). When there is positive skewness, the mean and median are bigger than the mode.

Distributions are negatively skewed when the tail on the left side of the distribution is longer or fatter than the tail on the right side. The mean and median are smaller than the mode.

This behavior is shown in the images below:

Skewness can have implications for data analysis and usage of certain models. The "normality assumption" seen before does not hold when data is skewed. When data is skewed, you'll need to transform the data first.

For univariate data $Y_1, Y_2, ..., Y_n$ the formula for skewness is:

$$\dfrac{\dfrac{\displaystyle\sum^n_{i=1}(Y_i-Y)^3}{n}}{s^3}$$

where $Y$ is the mean, $s$ is the standard deviation, and $n$ is the number of data points. This formula for skewness is referred to as the Fisher-Pearson coefficient of skewness. There are also other ways to calculate skewness, yet this one is the one that is used most commonly.

The rule of thumb seems to be:

• A skewness between -0.5 and 0.5 means that the data are pretty symmetrical
• A skewness between -1 and -0.5 (negatively skewed) or between 0.5 and 1 (positively skewed) means that the data are moderately skewed.
• A skewness smaller than -1 (negatively skewed) or bigger than 1 (positively skewed) means that the data are highly skewed.

Example

Imagine you have house values ranging from 200,000 USD to 1,500,000 USD with an average of 800,000 USD.

If the peak of the distribution is left of the average value, the house prices are positively skewed. This means that more than half of the houses were sold for less than the average value 800,000 USD, and that there are a limited number of houses that were sold for a much higher value than 800,000 USD, leading to a long tail in the higher price ranges.

If the peak of the distributed data is on the right hand side of the average value, this means there is negative skewness, meaning that more than half of the houses were sold for more than the average value of 800,000 USD. Additionally, this means that there is a long tail in the lower price ranges.

Kurtosis deals with the lengths of tails in the distribution.

Where skewness talks about extreme values in one tail versus the other, kurtosis aims at identifying extreme values in both tails at the same time!

You can think of Kurtosis as a measure of outliers present in the distribution.

The distribution denoted in the image above has relatively more observations around the mean, then a steep decline and longer tails compared to the normal distribution.

For univariate data $Y_1, Y_2, \dots, Y_n$ the formula for kurtosis is:

$$\dfrac{\dfrac{\displaystyle\sum^n_{i=1}(Y_i-Y)^4}{n}}{s^4}$$

If there is a high kurtosis, then you may want to investigate why there are so many outliers. Presence of outliers could be indications of errors on the one hand, but they could also be some interesting observations that may need to be explored further. For banking transactions, for example, an outlier may signify a fraudulent activity. How we deal with outliers mainly depends on the domain.

Low kurtosis in a data set is an indication that data has light tails or lack of outliers. If we get low kurtosis, then also we need to investigate and trim the dataset of unwanted results.

A mesokurtic distribution has kurtosis statistics that lie close to the ones of a normal distribution. Mesokurtic distributions have a kurtosis of around 3. According to this definition, the standard normal distribution has a kurtosis of three.

When a distribution is platykurtic, the distribution is shorter and tails are thinner than the normal distribution. The peak is lower and broader than Mesokurtic, which means that the tails are light, and that there are fewer outliers than in a normal distribution.

When you have a leptokurtic distribution, you have a distribution with longer and fatter tails. The peak is higher and sharper than the peak of a normal distribution, which means that data have heavy tails, and that there are more outliers.

Outliers stretch your horizontal axis of the distribution, which means that the majority of the data appear in a narrower vertical range. This is why the leptokurtic distribution looks "skinny".

In this lesson, you learned about skewness and kurtosis. In the next lab, you'll learn how to measure skewness and kurtosis in python.