Today, let’s explore the fascinating combination of AI and statistics through visual examples generated by the AI analysis tool Bayeslab, with a focus on the most common statistical concept: the normal distribution.

Following the five questions about the normal distribution introduced in the previous template,

1.why normal distribution is important?

2.When normal distribution is used?

3.Normal distribution looks like

4. Normal distribution where is the mean?

5.Normal distribution with standard deviation

6. What is the density function?

7. Normal distribution with percentages

8. What is the 68–95–99.7 rule?

9. Normal distribution to standard normal distribution

10. Normal distribution with z-scores

we will continue to introduce the following issues in this template:

11. Which test for normal distribution?

12.Can normal distribution be skewed?

11. Which test for normal distribution?

To determine if data follows a normal distribution, graphical methods like the Q-Q plot and statistical tests such as the Shapiro-Wilk test can be used.

▶︎Q-Q Plot: Visually assesses how data points deviate from a reference line representing a normal distribution.

▶︎Shapiro-Wilk Test: Provides quantitative statistical results to evaluate normality.

Additionally, the following tests can be used:

▶︎ D’Agostino & Pearson Test: Combines skewness and kurtosis to assess normality, especially for large samples. It calculates the K² statistic and p-value; a p-value less than 0.05 indicates significant deviation from normality.

▶︎ Anderson-Darling Test: An enhanced version of the K-S test, it focuses on tail deviations and is suitable for various sample sizes. It calculates the A² statistic and p-value; a very small p-value (<0.0001) signifies significant deviation from normality.

▶︎ Kolmogorov-Smirnov Test: Compares the sample’s cumulative distribution function with the theoretical normal distribution and is suitable for large samples. A large p-value (>0.1000) indicates that the data fits a normal distribution.

In statistical analysis, such as examining the “Soybean Feed vs. Iron-Deficiency Anemia” dataset, tests for normality are conducted on columns like “Regular feed,” “10% soybean feed,” and “15% soybean feed.” All test results are displayed by data column and saved in the file “Test for normal distribution.txt.”

Result File :

12.Can normal distribution be skewed?

An ideal normal distribution is symmetrical and unbiased. Although actual data may theoretically show normal distribution characteristics, it may skew due to external factors or sample diversity.

Skewness results in the distribution tilting to the left or right, indicating deviation from normal characteristics.

At this point, the skewness coefficient can be used to quantify the degree of skewness in the distribution.

If the skewness is severe, the distribution is considered non-normal, and other statistical methods may be needed for processing or analysis.

If a dataset is skewed and deviates from the ideal normal distribution, this may affect the effectiveness and accuracy of statistical analysis.

In such cases the following methods can be employed for handling and analysis:

(1) Data Transformation:

Data transformation can reduce skewness by altering the data’s shape, making it closer to a normal distribution. Common transformation methods include:

(2) Using Other Distribution Models:

Consider alternative statistical distribution models for data analysis when normality assumption is not met.

If transformation is ineffective, consider using other more suitable distribution models.

For example:

• Log-normal Distribution: Data that follows a normal distribution after log transformation.

• Gamma Distribution: Commonly used for data with positive skewness.

• Exponential Distribution: Used for modeling asymmetric variability.

(3) Non-Parametric Methods:

When data significantly deviates from a normal distribution, non-parametric methods do not require assumptions about the data’s distribution form and offer a more robust analysis approach. For example:

Mann-Whitney U Test: Used for comparing two independent samples.

Wilcoxon Signed-Rank Test: Used for comparing two paired samples.

Kruskal-Wallis Test: Used for comparing multiple independent samples.

(4) More Robust Statistical Methods (Effect Size):

Robust statistical methods are less sensitive to skewness and outliers, allowing for more effective analysis of skewed data. For example:

Median: Serves as an alternative to the mean, less affected by outliers.

Interquartile Range (IQR): Serves as an alternative to standard deviation, based on the middle 50% of data points.

(5) Mixture Models: For more complex data distributions, consider using Mixture Models, such as Gaussian Mixture Models (GMMs), which treat data as a combination of several different normal distributions.

These methods effectively handle and analyze skewed datasets, improving the precision and reliability of statistical analysis. These techniques offer diversified solutions for addressing common skewness and outliers in real-world data.

The statistical methods for skewed distributions will be discussed in detail later; for now, this provides a brief overview.

About Bayeslab :

• Bayeslab: Website

The AI First Data Workbench

• X: @BayeslabAI

• Documents: https://bayeslab.gitbook.io/docs

• Blogs:https://bayeslab.ai/blog

Bayeslab is a powerful web-based AI code editor and data analysis assistant designed to cater to a diverse group of users, including :

👥 data analysts ，🧑🏼‍🔬experimental scientists, 📊statisticians, 👨🏿‍💻 business analysts, 👩‍🎓university students, 🖍️academic writers, 👩🏽‍🏫scholars, and ⌨️ Python learners.