## Introduction

“Skewness essentially is a commonly used measure in descriptive statistics that characterizes the asymmetry of a data distribution, while kurtosis determines the heaviness of the distribution tails.”

Understanding the shape of data is crucial while practicing data science. It helps to understand where the most information lies and analyze the outliers in a given data. In this article, we’ll learn about the shape of data, the importance of skewness, and kurtosis in statistics. The types of skewness and kurtosis and Analyze the shape of data in the given dataset. Let’s first understand what skewness and kurtosis is.

Learning Objectives

• You will learn how to calculate the Skewness Coefficient.

## What Is Skewness?

Skewness is a statistical measure that assesses the asymmetry of a probability distribution. It quantifies the extent to which the data is skewed or shifted to one side.

Positive skewness indicates a longer tail on the right side of the distribution, while negative skewness indicates a longer tail on the left side. Skewness helps in understanding the shape and outliers in a dataset.

If the values of a specific independent variable (feature) are skewed, depending on the model, skewness may violate model assumptions or may reduce the interpretation of feature importance.

In statistics, skewness is a degree of asymmetry observed in a probability distribution that deviates from the symmetrical normal distribution (bell curve) in a given set of data.

A skewed data set, typical values fall between the first quartile (Q1) and the third quartile (Q3).

The normal distribution helps to know a skewness. When we talk about normal distribution, data symmetrically distributed. The symmetrical distribution has zero skewness as all measures of a central tendency lies in the middle.

When data is symmetrically distributed, the left-hand side, and right-hand side, contain the same number of observations. (If the dataset has 90 values, then the left-hand side has 45 observations, and the right-hand side has 45 observations.). But, what if not symmetrical distributed? That data is called asymmetrical data, and that time skewness
comes into the picture.

## Types of Skewness

#### Positive Skewed or Right-Skewed  (Positive Skewness)

In statistics, a positively skewed or right-skewed distribution has a long right tail. It is a sort of distribution where the measures are dispersing, unlike symmetrically distributed data where all measures of the central tendency (mean, median, and mode) equal each other. This makes Positively Skewed Distribution a type of distribution where the mean, median, and mode of the distribution are positive rather than negative or zero.

In positively skewed, the mean of the data is greater than the median (a large number of data-pushed on the right-hand side). In other words, the results are bent towards the lower side. The mean will be more than the median as the median is the middle value and mode is always the most frequent value.

Extreme positive skewness is not desirable for a distribution, as a high level of skewness can cause misleading results. The data transformation tools are helping to make the skewed data closer to a normal distribution. For positively skewed distributions, the famous transformation is the log transformation. The log transformation proposes the calculations of the natural logarithm for each value in the dataset.

#### Negative Skewed or Left-Skewed (Negative Skewness)

A negatively skewed or left-skewed distribution has a long left tail; it is the complete opposite of a positively skewed distribution. In statistics, negatively skewed distribution refers to the distribution model where more values are plots on the right side of the graph, and the tail of the distribution is spreading on the left side.

In negatively skewed, the mean of the data is less than the median (a large number of data-pushed on the left-hand side). Negatively Skewed Distribution is a type of distribution where the mean, median, and mode of the distribution are negative rather than positive or zero.

Median is the middle value, and mode is the most frequent value. Due to an unbalanced distribution, the median will be higher than the mean.

## How to Calculate the Skewness Coefficient?

Skewness can be calculated using various methods, whereas the most commonly used method is Pearson’s coefficient.

Pearson’s first coefficient of skewness
To calculate skewness values, subtract the mode from the mean, and then divide the difference by standard deviation.

As Pearson’s correlation coefficient differs from -1 (perfect negative linear relationship) to +1 (perfect positive linear relationship), including a value of 0 indicating no linear relationship, When we divide the covariance values by the standard deviation, it truly scales the value down to a limited range of -1 to +1. That accurately shows the range of the correlation values.

Pearson’s first coefficient of skewness is helping if the data present high mode. But, if the data have low mode or various modes, Pearson’s first coefficient is not preferred, and Pearson’s second coefficient may be superior, as it does not rely on the mode.

Pearson’s second coefficient of skewness
subtract the median from the mean, multiply the difference by 3, and divide the product by the standard deviation.

Rule of thumb :
If the skewness is between -0.5 & 0.5, the data are nearly symmetrical.
If the skewness is between -1 & -0.5 (negative skewed) or between 0.5 & 1(positive skewed), the data are slightly skewed.
If the skewness is lower than -1 (negative skewed) or greater than 1 (positive skewed), the data are extremely skewed.

## What Is Kurtosis?

Kurtosis is a statistical measure that quantifies the shape of a probability distribution. It provides information about the tails and peakedness of the distribution compared to a normal distribution.

Positive kurtosis indicates heavier tails and a more peaked distribution, while negative kurtosis suggests lighter tails and a flatter distribution. Kurtosis helps in analyzing the characteristics and outliers of a dataset.

The measure of Kurtosis refers to the tailedness of a distribution. Tailedness refers to how often the outliers occur.

Peakedness in a data distribution is the degree to which data values are concentrated around the mean. Datasets with high kurtosis tend to have a distinct peak near the mean, decline rapidly, and have heavy tails. Datasets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

In finance, kurtosis is used as a measure of financial risk. A large kurtosis is associated with a high level of risk for an investment because it indicates that there are high probabilities of extremely large and extremely small returns. On the other hand, a small kurtosis signals a moderate level of risk because the probabilities of extreme returns are relatively low.

## What Is Excess Kurtosis?

The excess kurtosis is used in statistics and probability theory to compare the kurtosis coefficient with that normal distribution. Excess kurtosis can be positive (Leptokurtic distribution), negative (Platykurtic distribution), or near zero (Mesokurtic distribution). Since normal distributions have a kurtosis of 3, excess kurtosis is calculated by subtracting kurtosis by 3.

Excess kurtosis  =  Kurt – 3

## Types of Excess Kurtosis

1. Leptokurtic or heavy-tailed distribution (kurtosis more than normal distribution).
2. Mesokurtic (kurtosis same as the normal distribution).
3. Platykurtic or short-tailed distribution (kurtosis less than normal distribution).

#### Leptokurtic (Kurtosis > 3)

Leptokurtic has very long and thick tails, which means there are more chances of outliers. Positive values of kurtosis indicate that distribution is peaked and possesses thick tails. Extremely positive kurtosis indicates a distribution where more numbers are located in the tails of the distribution instead of around the mean.

#### Platykurtic (Kurtosis < 3)

Platykurtic having a thin tail and stretched around the center means most data points are present in high proximity to the mean. A platykurtic distribution is flatter (less peaked) when compared with the normal distribution.

#### Mesokurtic (Kurtosis = 3)

Mesokurtic is the same as the normal distribution, which means kurtosis is near 0. In Mesokurtic, distributions are moderate in breadth, and curves are a medium peaked height.

## Conclusion

The skewness is a measure of symmetry or asymmetry of data distribution, and kurtosis measures whether data is heavy-tailed or light-tailed in a distribution. Data can be positive-skewed (data-pushed towards the right side) or negative-skewed (data-pushed towards the left side).

When data is skewed, the tail region may behave as an outlier for the statistical model, and outliers unsympathetically affect the model’s performance, especially regression-based models. Some statistical models are robust to outliers like Tree-based models, but it will limit the possibility of trying other models. So there is a necessity to transform the skewed data to be close enough to a Normal distribution.

Key Takeaways

• Skewness is a statistical measure of the asymmetry of a probability distribution. It characterizes the extent to which the distribution of a set of values deviates from a normal distribution.
• Skewness between -0.5 and 0.5 is symmetrical.
• Kurtosis measures whether data is heavily heavy-tailed or light-tailed.
• Data sets with high kurtosis have heavy tails and more outliers, while data sets with low kurtosis tend to have light tails and fewer outliers.
• Excess kurtosis can be positive (Leptokurtic distribution), negative (Platykurtic distribution), or near zero (Mesokurtic distribution).

Q1. Is kurtosis a measure of shape?

A. Kurtosis describes the shape of the distribution tale in relation to its overall shape. Distribution can be sharply peaked with low kurtosis, and distribution can have a lower peak with high kurtosis.

Q2. What does negative kurtosis indicate?

A. A distribution with a negative kurtosis value indicates that the distribution has lighter tails than the normal distribution.

Q3. What is the shape of a data distribution?

A. A distribution of data item values may be symmetrical or asymmetrical. Two common examples of symmetry and asymmetry are the ‘normal distribution’ and the ‘skewed distribution.’

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