# Guide: Skewness and Kurtosis

In the area of data analysis and statistical modelling, understanding the shape and characteristics of your data distribution can be as crucial as knowing the data points themselves. Enter Skewness and Kurtosis: two statistical measures that provide deep insights into the asymmetry and “tailedness” of your data distribution. While they may sound complex, these metrics are essential tools for anyone involved in data-driven decision-making, from supply chain managers to financial analysts.

This comprehensive guide aims to demystify skewness and kurtosis, breaking down their definitions, importance, and calculation methods. Whether you’re a seasoned statistician or a beginner in the field, this guide will equip you with the knowledge to interpret these metrics effectively, thereby enabling more robust analyses and better decisions.

## Table of Contents

## What is Skewness?

Understanding skewness is akin to uncovering the hidden tendencies of your data. Skewness tells us a lot about where values are concentrated, which, in turn, helps us make sense of the data’s overall behavior. Let’s dig deeper into its definition and the different types of skewness.

### Definition

Skewness is a statistical metric that quantifies the degree and direction of asymmetry in a data distribution relative to a perfect normal distribution. In simpler terms, it tells you how much and in which direction your data set “leans” or “skews.”

### Mathematical Aspect

Mathematically, skewness is calculated using the third standardized moment. The formula for skewness ($$) is:

Where $$ is the sample size, ${}_{}$ represents each data point, $\stackrel{}{}$ is the mean, and $$ is the standard deviation. A skewness value can range from negative infinity to positive infinity.

#### Types of Skewness

Understanding the types of skewness is crucial for interpreting the metric meaningfully. The skewness types can be broadly classified into three categories:

##### Positive Skewness

In a positively skewed distribution, the tail on the right side (larger values) is longer or fatter than the left tail. In such distributions, a large number of data points are concentrated on the lower side, while fewer and more variable data points exist on the higher side. The mean is typically greater than the median, and the skewness value will be greater than zero. This is common in scenarios like income distribution where most people earn average or below-average incomes, but a few earn significantly more.

##### Negative Skewness

A negatively skewed distribution is characterized by a long tail extending toward the left. In this situation, the mean is less than the median, and the skewness value will be less than zero. A classic example is the age at retirement; most people retire between 60 and 70, but some retire much earlier, skewing the distribution negatively.

##### Zero Skewness

When a data distribution is perfectly symmetrical, it exhibits zero skewness. This means that the values are evenly distributed on both sides of the mean, which equals the median. Zero skewness is an ideal characteristic often attributed to a perfect normal distribution, but it’s essential to note that zero skewness does not necessarily imply normality.

#### Summary

Understanding skewness helps analysts and decision-makers get a clearer picture of their data distribution’s shape. By knowing whether your data is positively skewed, negatively skewed, or exhibits zero skewness, you can apply the most appropriate statistical methods for analysis and make more informed decisions.

## What is Kurtosis?

Kurtosis is a fascinating concept that delves into the nuances of data distribution, going beyond the central tendency and dispersion metrics like mean and standard deviation. Understanding kurtosis can be an excellent asset for identifying outlier behavior, making it a valuable tool for analysts in various fields. Let’s break down this intriguing concept further.

#### Definition

Kurtosis quantifies how “tailed” a distribution is compared to the normal distribution. In layman’s terms, it measures the “extreme-ness” of data points, or how much the distribution deviates from the bell curve in its tails.

#### Mathematical Aspect

The formula for kurtosis ($$) can be expressed as:

Here, $$ represents the sample size, ${}_{}$ is each data point, $\stackrel{}{}$ is the mean, and *$$* is the standard deviation. The “- 3” at the end is often added to make the kurtosis of the normal distribution zero for easier interpretation.

### Types of Kurtosis

Kurtosis can be classified into three major types:

#### Leptokurtic

A leptokurtic distribution has “heavy” tails and a sharper peak than a normal distribution. This kind of distribution is more prone to outliers and extreme values. Financial returns often exhibit leptokurtic behavior, where most returns are close to the mean but extreme returns (either gains or losses) occur more frequently than one would expect in a normal distribution.

#### Platykurtic

A platykurtic distribution has “light” tails and a flatter peak compared to a normal distribution. This type indicates that outliers are less frequent. In a platykurtic distribution, data points are generally more evenly spread out and not as concentrated around the mean.

#### Mesokurtic

A mesokurtic distribution has a kurtosis value similar to that of the normal distribution. In a mesokurtic distribution, the frequency and likelihood of extreme values or outliers are almost the same as those in a normal distribution.

Kurtosis is an invaluable metric that offers insights into the “tailedness” of a distribution. By understanding whether your distribution is leptokurtic, platykurtic, or mesokurtic, you can better prepare for the likelihood of encountering extreme values or outliers, thus making more robust statistical models and business decisions.

## Why Skewness and Kurtosis Matter

Understanding the shape and characteristics of a data distribution is crucial in fields ranging from finance to healthcare, logistics to marketing. Two often-overlooked but incredibly informative measures are skewness and kurtosis. These metrics offer valuable insights into the “behavior” of data, far beyond central tendencies like mean or median. Let’s delve into why understanding skewness and kurtosis is so essential.

### Detect Non-Normality in Data

The assumption of normality is prevalent in many statistical methods and machine learning algorithms. However, real-world data often deviates from this idealized normal distribution. Skewness and kurtosis serve as early warning signals, enabling you to identify whether your data follows a normal distribution or not. This identification is crucial for selecting the most appropriate statistical tests and models for your data.

### Make Better Predictions and Models

The efficiency of predictive models like regression, ANOVA, and machine learning algorithms can be significantly affected by the underlying data distribution. Knowing the skewness and kurtosis helps in choosing the right transformation techniques (e.g., log-transform for positively skewed data) or alternative non-parametric methods. Consequently, this improves the accuracy and reliability of your models.

### Identify Potential Outliers

Both skewness and kurtosis are sensitive to extreme values. A high positive or negative skewness may indicate the presence of outliers pulling the mean away from the median. Similarly, a high kurtosis value could signify an abnormal concentration of values at the tails. Identifying these outliers is crucial, as they can dramatically impact your analyses and conclusions.

### Improve Decision-Making in Various Fields

**Finance:**Skewness and kurtosis are critical in assessing investment risks. For instance, a leptokurtic distribution of stock returns may indicate a higher risk of extreme price changes.**Healthcare:**Understanding the distribution of medical data like patient age or blood pressure levels can help in identifying atypical cases that require further investigation.**Supply Chain:**In logistics, knowing the skewness of delivery times can help in optimizing inventory levels, thereby reducing costs.**Marketing:**In customer behavior analytics, understanding the kurtosis can help in identifying niche customer segments that exhibit extreme behavior, either very loyal or very dissatisfied.

Understanding skewness and kurtosis is not just a statistical nicety; it’s a necessity for anyone who works with data. These measures provide you with the tools to make more informed decisions, build more robust models, and ultimately contribute to the success of your analytical endeavors across various domains.

## Calculating Skewness and Kurtosis:

Understanding skewness and kurtosis is crucial for anyone working with data, as these metrics provide valuable insights into the distribution of the data set. However, to make use of these insights, one needs to know how to calculate them. Below are the formulas and detailed explanations for calculating skewness and kurtosis.

### Skewness Formula

The skewness of a data set is calculated using the following formula:

Here,

- $n$ is the sample size.
- $x_{i}$ is each data point.
- $xˉ$ is the sample mean.
- $s$ is the sample standard deviation.

The formula adjusts for sample size and ensures that the skewness of a normal distribution is zero. The sum of the cubed deviations from the mean is normalized by the sample size and the sample standard deviation to give a dimensionless skewness value.

## Skewness Calculation Example with 30 Data Points

## Example Input Data

Here are 30 sample data points generated from an exponential distribution:

$$\begin{array}{r}{\displaystyle 0.94,6.02,2.63,1.83,0.34,0.34,0.12,4.02,1.84,2.46,}\\ {\displaystyle 0.04,7.01,3.57,0.48,0.40,0.41,0.73,1.49,1.13,0.69,}\\ {\displaystyle 1.89,0.30,0.69,0.91,1.22,3.08,0.45,1.44,1.80,0.10}\end{array}$$## Calculations

## Step 1: Calculate the Sample Mean ($\stackrel{}{}$$xˉ$)

- Add up all the 30 data points.
- Divide the sum by 30 (the number of data points).
The sample mean turns out to be approximately 1.61.

## Step 2: Calculate the Sample Standard Deviation ($$$s$)

- Subtract the sample mean from each data point and square the result.
- Add up all these squared differences.
- Divide this sum by 29 (which is $n−1$).
- Take the square root of the result.
The sample standard deviation is approximately 1.70.

## Step 3: Calculate Skewness

- Subtract the sample mean from each data point and divide by the sample standard deviation.
- Cube these results and sum them all up. Let’s call this sum “A.”
- Multiply “A” by 30 (the number of data points). This becomes a new value, “B.”
- Multiply “B” by 30 (again, the number of data points) and 29 (n minus 1) and divide by 28 (n minus 2) and 29 (n minus 1). This gives us another value, “C.”
- Finally, “C” is the skewness of the distribution.
The skewness turns out to be approximately 1.74.

## Interpretation

A skewness value of 1.74 is greater than zero, indicating that the distribution is positively skewed. This means the tail on the right side (larger values) is longer or fatter than the left tail.

## Output Graph and Results

The histogram visualizes the distribution of the 30 data points. The red dashed line indicates the mean value of the sample.

### Kurtosis Formula

The kurtosis of a data set is calculated using the following formula:

Here,

- $n$ is the sample size.
- $x_{i}$ is each data point.
- $xˉ$ is the sample mean.
- $s$ is the sample standard deviation.

The formula for kurtosis also adjusts for sample size and ensures that the kurtosis of a normal distribution is zero. Similar to skewness, the sum of the fourth powers of the deviations from the mean is normalized by the sample size and the sample standard deviation.

## Kurtosis Calculation Example with 30 Data Points

## Example Input Data

We have 30 sample data points generated from a Laplace distribution to simulate leptokurtic behavior. Here are the first 5 values for reference:

$-0.29,2.32,0.62,0.22,-1.16,\dots $## Step-by-Step Calculation

Sample Mean

($xˉ$) :$\stackrel{}{The\; sample\; mean\; turns\; out\; to\; be\; approximately\; -\; 0.23..}$Sample Standard Deviation ($s$): The sample standard deviation is approximately $1.25$Number of Data Points ($n$): The number of data points in the sample is $30.$Kurtosis Calculation: Using the given formula, the kurtosis value is approximately -3.25. ## Step 1: Calculate the Sample Mean ($xˉ$)

- Add up all the 30 data points.
- Divide the sum by 30 (the number of data points).
The sample mean is approximately−0.23.

## Step 2: Calculate the Sample Standard Deviation ($$$s$)

- Subtract the sample mean from each data point and square the result.
- Add up all these squared differences.
- Divide this sum by 29 (which is $n−1$).
- Take the square root of the result.
The sample standard deviation is approximately

$1.25$.

## Step 3: Calculate Kurtosis

- Subtract the sample mean from each data point and divide by the sample standard deviation.
- Raise these results to the power of 4 and sum them all up. Let’s call this sum “A.”
- Multiply “A” by 30 (the number of data points). This becomes a new value, “B.”
- Multiply “B” by 31 (one more than the number of data points) and divide by 29 (n minus 1), 28 (n minus 2), and 27 (n minus 3). This gives us another value, “C.”
- Subtract (3 times 29 times 28) from “C” and divide by 29 (n minus 1), 28 (n minus 2), and 27 (n minus 3). This gives us yet another value, “D.”
- Finally, “D” minus 3 is the kurtosis of the distribution.
The kurtosis turns out to be approximately − 3.25.

## Interpretation

A kurtosis value of −3.25 is less than zero, indicating that the distribution is platykurtic. This means that the distribution has “light” tails and is less prone to outliers, which is also evident from the histogram.

## Output Graph and Results

The histogram visualizes the distribution of the 30 data points, and the red dashed line indicates the mean value of the sample.

### Why These Formulas?

Both formulas are constructed to give a dimensionless number that provides a relative measure of skewness and kurtosis, independent of the units of the data set. This makes them broadly applicable across different domains and types of data.

Understanding how to calculate skewness and kurtosis is essential for anyone who wishes to delve deep into data analysis, as these metrics offer a more nuanced understanding of data distribution than central tendencies like mean or median. They are particularly useful for identifying outliers, tail behavior, and the propensity for extreme values in your data.

### Interpreting Results of Skewness and Kurtosis: A Detailed Guide

Understanding the numerical results of skewness and kurtosis calculations provides valuable insights into the characteristics of a data distribution. These metrics tell a story about the shape of the data, its tails, and its peaks. Below is a detailed guide on interpreting these values.

#### Interpreting Skewness

The skewness metric quantifies the degree and direction of asymmetry in a data distribution. Here’s how to interpret its value:

**Skewness > 0**: A positive skewness value indicates that the distribution is**positively skewed**. In this case, the tail on the right side (i.e., the side of larger values) is longer or fatter than the left tail. In a positively skewed distribution, the mean will be greater than the median.**Example**: Income distribution often shows positive skewness where the majority of people earn average or below-average incomes, but a small number earn exceptionally high incomes.**Skewness < 0**: A negative skewness value means that the distribution is**negatively skewed**. Here, the tail on the left side (i.e., the side of smaller values) is longer or fatter than the right tail. In a negatively skewed distribution, the mean will be less than the median.**Example**: The age at retirement can be negatively skewed, as most people retire between 60 and 70, but some might retire much earlier, skewing the distribution to the left.

#### Interpreting Kurtosis

Kurtosis measures the “tailedness” or the propensity for extreme values in a distribution. Here’s how to interpret its value:

**Kurtosis > 3**: A kurtosis value greater than 3 (note that we use 3 as the baseline because we often use “excess kurtosis” which makes the kurtosis of the normal distribution zero) suggests that the distribution is**leptokurtic**. Such distributions have “heavy” tails and are prone to outliers.**Example**: Financial markets often show leptokurtic distributions, indicating the potential for extreme changes in asset prices.**Kurtosis < 3**: A kurtosis value less than 3 indicates that the distribution is**platykurtic**. Such distributions have “light” tails, meaning they are less prone to extreme values.**Example**: Height distributions are often platykurtic, indicating that extremely tall or short individuals are less common.

## Conclusion

Understanding the concepts of skewness and kurtosis is crucial for anyone dealing with data analysis, as these metrics offer profound insights into the shape and characteristics of data distributions. Skewness helps you understand the asymmetry of your data, informing you whether the outliers are mostly on the high end or low end of the scale.

Kurtosis, on the other hand, provides you with an understanding of the “tailedness” of your distribution, thus helping you gauge the potential for outliers. Through our step-by-step examples, we demonstrated how to calculate these metrics, removing the intimidation often associated with statistical analysis. Whether you’re in manufacturing, logistics, or any other field that relies on data, mastering skewness and kurtosis will empower you to make better decisions, build more accurate models, and ultimately, achieve more reliable outcomes.

## References

- Groeneveld, R.A. and Meeden, G., 1984. Measuring skewness and kurtosis.
*Journal of the Royal Statistical Society Series D: The Statistician*,*33*(4), pp.391-399. - Joanes, D.N. and Gill, C.A., 1998. Comparing measures of sample skewness and kurtosis.
*Journal of the Royal Statistical Society: Series D (The Statistician)*,*47*(1), pp.183-189. - Mardia, K.V., 1970. Measures of multivariate skewness and kurtosis with applications.
*Biometrika*,*57*(3), pp.519-530.

##### Q: What is the significance of a positive skewness value?

**A:** A positive skewness value indicates that the distribution is positively skewed. In such a distribution, the tail on the right side (larger values) is longer or fatter than the left tail. This often means that a large number of data points are concentrated on the lower side, while fewer and more variable data points exist on the higher side.

##### Q: Can a distribution have zero skewness but still not be normal?

**A:** Yes, a distribution can have zero skewness but still not be a normal distribution. Zero skewness simply means that the distribution is symmetrical around the mean. However, other factors like kurtosis and the actual distribution of data points can differentiate it from a normal distribution.

##### Q: How does kurtosis affect the risk assessment in financial data?

**A:** In financial data, a high kurtosis (leptokurtic) indicates a higher likelihood of extreme values or outliers. This could mean a higher risk, as the asset or portfolio might experience extreme gains or losses more often than a normal distribution would predict. Conversely, a low kurtosis (platykurtic) suggests a lower risk of extreme values.

##### Q: Why do we subtract 3 in the kurtosis formula?

**A:** The subtraction of 3 in the kurtosis formula is a standardization technique that makes the kurtosis of a normal distribution equal to zero. This makes it easier to compare the kurtosis of different distributions to a normal distribution.

##### Q: What are the common real-world examples where skewness and kurtosis are used?

**A:** Skewness and kurtosis are widely used across various fields. For example, in manufacturing, they can help identify quality issues. In finance, they are used to assess the risk and return of financial instruments. In healthcare, they may be used to understand the distribution of certain medical conditions or treatment outcomes. They are essential tools for anyone involved in data analysis and decision-making based on statistical data.

## Author

#### Daniel Croft

Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

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