Asymmetrical distribution

Asymmetrical distribution
See also
Continuous distribution Log-normal distribution Autocorrelation Multicollinearity Probability density function Quantitative variable Statistical significance Measurement uncertainty Statistical methods

Asymmetrical distribution is a type of probability distribution where the mean, median, and mode are not equal. The distribution is skewed to one side and is not symmetrical. It can be either positively skewed, where the tail extends to the right of the mean, or negatively skewed, where the tail extends to the left of the mean.

Asymmetrical distribution is common in many areas such as economics and finance, where income and wealth often follow a positively skewed distribution. In summary, asymmetrical distribution is a type of probability distribution where the mean, median, and mode are not equal and the distribution is skewed to one side.

Example of Asymmetrical distribution

An example of an asymmetrical distribution is the normal distribution, which is an important probability distribution in statistics. It is a bell-shaped curve with a slightly longer right tail, making it slightly positively skewed. The mean, median, and mode are all equal, but the right tail of the distribution is slightly longer, which makes it asymmetrical. The formula for the normal distribution is: ${\begin{aligned}f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {1}{2\sigma ^{2}}}(x-\mu )^{2}}\end{aligned}}$ Where

/mu; is the mean,

/sigma; is the standard deviation,

and x is the random variable.

In summary, an example of an asymmetrical distribution is the normal distribution, which is a slightly positively skewed bell-shaped curve. The mean, median, and mode are all equal, but the right tail of the distribution is slightly longer.

Formula of Asymmetrical distribution

${\begin{aligned}\gamma _{1}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {(x_{i}-{\overline {x}})^{3}}{s^{3}}}\end{aligned}}$ Where

\overline{x} is the mean, n is the number of observations,

and s is the standard deviation.

Asymmetrical distribution is common in many areas such as economics and finance, where income and wealth often follow a positively skewed distribution.

When to use Asymmetrical distribution

Asymmetrical distribution can be used to analyze a wide variety of data sets where the mean, median, and mode are not equal. It is commonly used in economics and finance, where income and wealth often follow a positively skewed distribution. Asymmetrical distribution can also be used to analyze data sets with outliers, where the outliers cause the mean to be different from the median and mode.

Types of Asymmetrical distribution

Positively skewed distributions have the following characteristics:

• The mean is greater than the median
• The mode is less than the median and the mean
• The tail is longer on the right side

Negatively skewed distributions have the following characteristics:

• The mean is less than the median
• The mode is greater than the median and the mean
• The tail is longer on the left side

There are several types of asymmetrical distributions, including the following:

• Cauchy Distribution: This type of distribution has a heavy tail on one side, and is used to model extreme events.
• Log-normal Distribution: This type of distribution is often seen in natural phenomena, and is used to model stock prices that experience large swings.
• Gamma Distribution: This type of distribution is used to model random events that occur with a certain frequency.

Steps of Asymmetrical distribution

Asymmetrical distribution consists of 3 steps:

• Step 1: Calculate the mean, median, and mode of the data set.
• Step 2: Calculate the skewness using the formula above.
• Step 3: Determine whether the distribution is positively or negatively skewed based on the results.

Advantages of Asymmetrical distribution

The advantages of asymmetrical distributions include:

• The ability to represent real-world data that may not follow a symmetrical distribution
• The ability to more accurately identify outliers
• The ability to more easily identify trends and patterns in data

Disadvantages of Asymmetrical Distribution

The disadvantages of asymmetrical distributions include:

• The difficulty of making predictions from the data due to the potential for extreme values
• The difficulty of using traditional statistical methods such as t-tests and ANOVAs
• The potential for misinterpretation of the data due to its skewed nature

In summary, asymmetrical distributions have the ability to represent real-world data that may not follow a symmetrical distribution and can more easily identify trends and patterns in data. However, they also have the potential for misinterpretation of the data due to its skewed nature and the difficulty of making predictions from the data.

Limitations of Asymmetrical distribution

Asymmetrical distribution has several limitations which include:

• The mean, median, and mode can be difficult to interpret
• The variance is not equal for the two sides of the distribution
• The distribution is not easily modelled using traditional methods
• Outliers can have a large effect on the distribution

Other approaches related to Asymmetrical distribution

There are a few other approaches used to measure asymmetry in a distribution.

• The Pearson’s median skewness coefficient (PMC) is a measure of skewness which is based on the median and the quartiles of a dataset.
• The Jarque-Bera test is a measure of asymmetry which uses the skewness and kurtosis of a dataset to determine if it is normally distributed.
• The Gini coefficient is a measure of asymmetry which is used to measure the inequality of wealth distribution in a society.

In summary, there are other approaches to measure asymmetry in a distribution besides the formula for skewness. These include the Pearson’s median skewness coefficient, the Jarque-Bera test, and the Gini coefficient.

Suggested literature

• Mühlradt, P. F., & Golecki, J. R. (1975). Asymmetrical distribution and artifactual reorientation of lipopolysaccharide in the outer membrane bilayer of Salmonella typhimurium. European journal of biochemistry, 51(2), 343-352.
• Sutton, C. D. (1993). Computer-intensive methods for tests about the mean of an asymmetrical distribution. Journal of the American Statistical Association, 88(423), 802-810.
• Kozubowski, T. J., & Podgórski, K. (1999). A class of asymmetric distributions. Actuarial Research Clearing House, 1, 113-134.