Guide to Skewness and Kurtosis

Realizing the shape of data distribution is key in many areas, from finance to quality control. Skewness and Kurtosis are two pivotal measures of distribution shape and features that give a picture of distribution workings. Skewness and kurtosis of normal distribution measures do not just replace basic statistics like mean and variance but describe how unbalanced and long-tailed data distributions could be characterized.

In statistics, skewness and kurtosis are two vital concepts conveying a pattern of data. Both are supplements to simple measures like mean and variance, but their interests diverge as we delve into the data distribution. This article will explain all the aspects, which involve the definitions, formula for skewness and kurtosis, skewness and kurtosis examples, moments of skewness and kurtosis in statistics, descriptive statistics skewness and kurtosis interpretations, and applications

Real World Examples of Skewness & Kurtosis

Here are some real-world examples of how skewness and kurtosis are applied in different fields like Finance & Engineering:

1. Finance

Skewness

Stock Returns: The kurtosis of stock returns is studied in the field of finance as well. A negatively skewed distribution shows that a greater number of instances of negative market returns are likely to occur; if so, higher risk exists for the investors. For example, considering the market downturn and the returns distribution might have a negative skew in a situation where the returns have more extreme negative returns than positive on the other hand.

Kurtosis

Portfolio Risk: The high kurtosis in the returns distribution means that a portfolio is exposed to high risk and is therefore likely to experience gains or losses in extreme values. This is important for risk estimation as it points out how a few extreme events may lead to a poor performance of the portfolio.

2. Engineering

Skewness:

Quality Control: Skewness in manufacturing: measuring the global shape of processes. For instance, when machined parts are measured, the dimensions might be skewed. It might be beneficial to increase the size of the parts if positive skewness occurs due to the increased size of the parts produced by the machinery.

Kurtosis:

Vibration Analysis: In mechanical engineering, kurtosis is applied in condition monitoring as a tool to detect the defects of the rotating machines. The kurtosis coefficient is a fairly reliable method of identifying the wear in the bearings and gears by the value of a high coefficient.

Definition of Skewness

Skewness is a statistical measure that describes how the total mass of data is not distributed equally around the central point of the histogram or how the data is not symmetrically distributed around the mean. It tells whether the data points are lumped mostly on one side of the distribution mean, which divulges the move and scale of the skewness. Skew lets us know that the distribution tends to the left or the right (negative or positive skew, respectively).

Mathematically, skewness is measured by the third standardized frequency of the distribution. For a sample of 𝑛n observations with mean 𝑥ˉxˉ and standard deviation 𝑠s, the skewness 𝛾γ is given by:

𝛾=𝑛(𝑛−1)(𝑛−2)∑𝑖=1𝑛(𝑥𝑖−𝑥ˉ𝑠)3γ=(n−1)(n−2)n​∑i=1n​(sxi​−xˉ​)3

Where 𝑥𝑖xi presents​ individual observations in the dataset.

Types of Skewness

There are three main types of skewness:

1. Positive Skewness (Right-Skewed Distribution)

• Description: The right side of the latter tails in the positively skewed distribution is longer or fatter than that of the left side.

• Implication: The majority of the data points can be found to the left of the distribution with few extreme values in the range.

• Example: Income distribution in many countries is very uneven so that only small groups of people get a lot more wealth while most of the population gets only a little.

• Skewness Value: The skewness greater than 0

2. Negative Skewness (Left-Skewed Distribution)

• Description: For a negatively skewed distribution with a long or “fat” tail on the left side of the curve (see image).

• Implication: The majority of data is mainly gathered on the right, with few outliers on the left.

• Example: Age at retirement; a majority of people are still active by retirement because they retire at a typical age, but a few retire much earlier.

• Skewness Value: Skewness is < 0

3. Zero Skewness (Symmetric Distribution)

• Description: In a symmetrical distribution of data, its distribution is evenly around the mean value with no skewness.

• Implication: In the distribution, the right and the left sides of the distribution are mirror images of each other.
• Example: The normal distribution (which we rarely observe in the real-life data).

• Skewness Value: Skewness = 0

Interpretation of Skewness

• Symmetric Distribution (Skewness ≈ 0): The data is more or less evenly distributed around the mean, suggesting no preference for either end.

• Right-Skewed Distribution (Skewness > 0): The data is skewed to the right, which is shown by the long tail of the line graph.

• Left-Skewed Distribution (Skewness < 0): The data pattern is skewed on the left side, with a longer tail on the right, indicating that there are more values on the right side.

Definition of Kurtosis

Kurtosis is a statistical parameter that identifies the shape of the distribution's tails concerning the whole distribution. Explicitly, kurtosis means how much of a distribution’s tail is concerned with the occurrence of extreme values (outliers). A normal distribution or a baseline is considered when comparing the presence and intensity of these deviations. Higher kurtosis represents more outliers and is sharper, while lower kurtosis will have fewer outliers and a flatter peak.

Types of Kurtosis

Kurtosis can be categorized into three main types based on how the distribution compares to a normal distribution in terms of tail weight and peak sharpness:

1. Mesokurtic

• Description: A distribution with kurtosis is the same as one of a normal distribution.
• Characteristics: The tails are not too heavy and not too light as well. Excess kurtosis roughly equals zero. Presents the minor standouts.
• Examples: The Gaussian distribution, which constitutes the most common case of mesokurtic distribution, is the paradigm.

2. Leptokurtic

• Description: The mentioned distribution is characterized by fatter tails and sharper peaks in comparison to the normal distribution.
• Characteristics: Values of kurtosis coefficient that are higher (excess kurtosis > 0). This implies that there is an increase in heightened extreme deviations (outliers). Polarity points are more on the mean and in the tail.
• Examples: The asymmetric nature of financial returns is also attributed to them being leptokurtic by nature owing to the presence of extreme events.

3. Platykurtic

• Description: A dispersion with a less pronounced slope towards the tails and a wider peak compared to a normal distribution.
• Characteristics: Low degree of kurtosis (kurtosis excess < 0). There are fewer outliers and deviations. Data scatters evenly.
• Examples: The uniform distribution stands for one of the platykurtic curve examples.

Fact to Know: Data analysis, risk quantification, and decision-making procedures need to be familiar with outliers and their kurtosis types so that the effects of outliers can be identified.

Implications of Skewness and Kurtosis in Real Life

Utilizing Excel, R, and Python simplifies the process of getting the skewness and kurtosis values. Interpreting these outcomes allows us to pinpoint the misalignment of data distributions and understand the presence of outliers, which helps in better statistical analysis and more robust decision-making. These means are not only applicable to finance, quality control, or environmental studies, but they also give a deeper understanding of data features.

Using Excel

- Steps to Calculate Skewness:

1. Put your data in the column.
2. Utilizing the formula =SKEW(range) to get skewness.
3. Example: =SKEW(A1:A10)

- Steps to Calculate Kurtosis:

1. Fill in the column with your data.
2. With the formula =KURT(range), you can compute kurtosis.
3. Example: =KURT(A1:A10)

Using R

1. Install needed packages (if not installed yet).

install.packages("e1071")

2. Take up the package and find out skewness and kurtosis.

Using Python

1. If not already installed, set up the corresponding packages (if needed).

!pip install scipy numpy

2. Importing the packages and finding the skewness and kurtosis.

Interpreting the Results

Skewness Interpretation-

1. Positive Skewness (> 0)

• Shows tail ends of the distribution are heavy.
• Example: If 3 = skewness. 2, the distribution displays a moderate degree of positive skewness.

2. Negative Skewness (< 0)

• Demonstrates a left-tail distribution with the mean lower than the median.
• Example: When skewness = -0. 9, slightly left-oriented distribution.

3. Approximate Zero Skewness (≈ 0)

• Indicates a symmetric distribution.
• Example: If the value of skewness is 0. 1, the distribution is rather balanced.

Kurtosis Interpretation-

1. Positive Excess Kurtosis (> 0)

• Implies a leptokurtic distribution with fat tails and far-reaching outliers.
• Example: Will excess kurtosis = -2. 5, a plate distribution is characterized by much bigger outliers than a normal distribution.

2. Negative Excess Kurtosis (< 0):

• It signifies the presence of a platykurtic frequency distribution using narrower tails and a lesser number of heavy outliers.
• Example: In such a form, the Kurtosis = -1. 1) there are fewer outliers in this distribution than in normal distribution.

3. Approximate Zero Excess Kurtosis (≈ 0):

• Indicates mesokurtic distribution- which is normally distributed.
• Example: If normality = 0, kurtosis is = 0. 05 --is that the distribution is close to normal.

Conclusion

The values of skewness and kurtosis serve as powerful and essential statistical tools that can unveil information regarding the form and symmetry of data distributions, as well as the length of their tails. From the practical cases of financial data analysis and quality control in manufacturing, it is obvious that these measures serve more than just traditional descriptive statistics because they provide extra information, which is important in making decisions and solving problems, among others.

As for financial data analysis, various types of skewness and kurtosis provide investors with insights into the degree of risk of different assets and hence enable improved decisions related to portfolio management and risk mitigation. These parameters play a crucial role in conformity to qualitative control processes by instant identification of deviations from the specifications, hence leading to continuous improvement programs.

FAQs

Q. What is skewness?

A. Skewness is a measure of symmetry in a probability distribution curve. It is used to determine if data is situated more on one side of the mean than the other side of the distribution.

Q. What is the use of skewness?

A. Skewness is a concept in statistics that is used to find what a distribution looks like. It tells whether the data fits normal distribution or is skewed to one side.

Q. How is kurtosis calculated?

A. Kurtosis determination is made by factoring in how the normal distribution's tails are flat or peaked. Typically, it is assessed by comparing an observed distribution to a normal distribution

Q. What is the purpose of skewness and kurtosis?

A. Skewness helps in asymmetry measurement, while kurtosis helps in tail behavior inspection.

Q. What is the symbol for kurtosis?

A. The symbol for kurtosis is often shown as "K" or "Kurt."

Q. What are the uses of kurtosis?

A. Kurtosis is one of the characteristics of populations that are applied in finance, economics, and engineering to find out how the sample data behaves. For instance, in finance, kurtosis helps in the risk assessment and the estimation of the volatility of the investments.

Q. What is the skewness and kurtosis formula?

A. The formula for skewness is typically expressed as Skewness = n(n−1)(n−2)∑i=1n(sxi−x¯s​)3 where xi are different data points.