Normality & Distribution Analysis User Guide

1. INTRODUCTION

The Normality & Distribution Analysis module provides comprehensive statistical validation of distributional assumptions across scientific disciplines (biology, chemistry, physics, medicine, psychology, engineering, and environmental science).

Most standard parametric statistical procedures—such as Analysis of Variance (ANOVA), Student's t-tests, Pearson correlation, and linear regression—require the underlying population error terms to be normally distributed. Running parametric tests on severely non-normal data can result in inflated Type I error rates or reduced statistical power. The Normality module tests continuous variables across experimental groups, provides skewness and kurtosis metrics, generates diagnostic plots (Q-Q plots, histograms, density curves, box plots, violin plots), and offers automated data transformation to achieve normality.

When to use this module:

Verify distributional assumptions prior to conducting ANOVA, t-tests, or regression modeling.
Evaluate skewness, kurtosis, and symmetry of continuous sensor readings, physiological measurements, or laboratory assays.
Compare 10 standard statistical normality tests to confirm consistency across sample sizes (small to large samples).
Automate variance-stabilizing transformations (Log, Square Root, Arcsine, Box-Cox) directly when normality is rejected.

2. AVAILABLE OPTIONS & SETTINGS

The sidebar control panel and top action bar provide settings for test selection, alpha significance thresholds, decimal precision, and automated transformations:

Control / Option	What it does	Why it is used	When to use / select
Test Selection Dropdown	Selects a specific statistical normality test (e.g., `Shapiro-Wilk`, `Anderson-Darling`, `Kolmogorov-Smirnov`) or runs `All Tests` simultaneously.	Chooses the hypothesis testing algorithm used to calculate test statistics and p-values.	Select All Tests for comprehensive multi-test confirmation, or pick a specific test suited to your sample size.
Alpha Level Selector	Toggles the significance threshold between `5% (0.05)` and `1% (0.01)`.	Sets the critical alpha cutoff for rejecting the null hypothesis of normality.	Select 5% for standard scientific studies; select 1% for stringent clinical or industrial quality bounds.
Decimal Rounding	Sets output numerical precision selector (0, 1, 2, 3, or 4 decimal places).	Formats calculated test statistics, p-values, skewness, and kurtosis in result tables.	Located in the top header bar; adjust to match target publication standards.
Transformation Toggle	Toggles automated transformation mode `On` or `Off` and opens the transformation config modal.	Automatically applies mathematical transformations if data fails normality at the chosen alpha level.	Turn On when preparing non-normal datasets for downstream parametric models.
Variables (Traits) Selection	Pill selectors choosing quantitative continuous columns to analyze for normality.	Specifies target continuous measurement variables for distribution testing.	Select one or more continuous numeric variables from your uploaded dataset.

3. INPUT DATA FORMAT

The module accepts tabular spreadsheets (Excel .xlsx, .xls, or CSV .csv) with the following structure:

Header Row: Line 1 must contain clear header names for factor columns and continuous measurement variables.
Data Types: Categorical grouping variables (e.g., Group, Condition, Site) must contain text labels; quantitative variables must contain numeric values.
Sample Size Rules: Individual sample sizes should contain at least 3 numeric observations per group to perform statistical testing.

Sample Input Table (Continuous Measurements across Experimental Groups)

Below is a representative dataset containing generalized grouping factors (Group, Condition) and quantitative measurement variables (Response_Value, Concentration):

Group	Condition	Replicate	Response_Value	Concentration
Group-01	Control	R1	45.20	12.40
Group-01	Control	R2	46.80	13.10
Group-01	Control	R3	44.90	12.80
Group-01	Treated	R1	58.40	25.60
Group-01	Treated	R2	61.20	28.10
Group-01	Treated	R3	59.10	26.40
Group-02	Control	R1	41.50	10.20
Group-02	Control	R2	43.10	11.00
Group-02	Control	R3	42.00	10.80

4. METHODS / MODES

The Normality module provides 10 standard statistical test algorithms to evaluate distributional symmetry and tail behavior:

Statistical Normality Test	Optimal Sample Size Range	Null Hypothesis (H0) & Decision Rule	Recommended Science Application
Shapiro-Wilk	Small to Medium (n = 3 to 50)	H0: Data is normally distributed. Reject H0 if p-value < Alpha.	Gold standard test for biological, agricultural, and small-sample experimental data.
Anderson-Darling	All sample sizes (n ≥ 5)	H0: Data follows normal distribution. Gives higher weight to distribution tails.	Ideal when tail behavior and extreme values are critical (environmental, engineering trials).
Kolmogorov-Smirnov	Large samples (n > 50)	H0: Sample distribution matches continuous normal baseline.	General goodness-of-fit comparison against theoretical cumulative distribution.
Lilliefors	Medium to Large samples	Modifies K-S test when population mean and variance are unknown.	Useful for continuous physical and chemical measurements with estimated parameters.
Jarque-Bera	Large samples (n > 100)	H0: Sample skewness and excess kurtosis match a normal distribution (skewness = 0, kurtosis = 3).	Commonly applied in financial modeling, econometrics, and large sensor streams.
D'Agostino-Pearson	Medium to Large (n ≥ 20)	Combines skewness and kurtosis tests into an omnibus chi-square statistic.	Versatile test for clinical, physiological, and industrial quality control data.
Cramér-von Mises	Medium to Large samples	Evaluates distance between empirical and theoretical cumulative distributions.	High statistical power for symmetric, continuous physical measurements.
Pearson Chi-Square	Large samples binned into classes	H0: Frequency counts across binned intervals match expected normal counts.	Applied to binned discrete or continuous industrial measurements.
Shapiro-Francia	Medium to Large samples (n = 5 to 5000)	Modification of Shapiro-Wilk optimized for platykurtic and leptokurtic data.	Effective for heavy-tailed biological and ecological measurement series.
Ryan-Joiner	Small to Medium samples	Calculates correlation between observed values and normal scores (similar to Shapiro-Wilk).	Widely used alternative in engineering quality control and quality assurance studies.

5. RESULTS

Upon clicking RUN ANALYSIS, the module displays a multi-tab analytical suite comprising main statistical summary tables, interactive visualization plots, and automated text interpretation.

Sample Result 1: Main Normality Test Summary Table

The main summary table reports sample size, mean, median, standard deviation, skewness, kurtosis, test statistics, p-values, and automated conclusion badges:

Variable	Group	Sample Size (n)	Mean	Std Dev	Skewness	Kurtosis	Test Statistic	p-Value	Normality Status
Response_Value	Group-01	12	45.850	2.140	0.124	-0.342	0.968	0.7420	Pass (Normal)
Response_Value	Group-02	12	42.100	1.980	0.085	-0.510	0.974	0.8250	Pass (Normal)
Concentration	Group-01	12	18.400	6.850	1.452	2.180	0.812	0.0084	Fail (Non-Normal)

Sample Result 2: Diagnostic Plot Suite

The Plots tab generates 5 interactive visual diagnostic plots for each variable:

Q-Q Plot (Quantile-Quantile): Plots sample quantiles against theoretical normal quantiles. Points falling closely along the diagonal line indicate normality.
Histogram: Displays frequency distribution overlaid with an ideal normal bell curve.
Density Plot: Shows kernel density estimation compared against the theoretical normal density.
Box Plot: Highlights median, interquartile range (IQR), and potential outliers.
Violin Plot: Combines box plot metrics with continuous distribution shape width.

6. QUICK WORKFLOW

Upload Dataset: Upload your .xlsx or .csv spreadsheet using the sidebar upload control.
Select Test & Alpha: Choose your desired normality test (e.g., Shapiro-Wilk or All Tests) and set the significance level (5% or 1%) in the top bar.
Select Variables: Click the pill selectors in the sidebar to choose quantitative continuous variables for analysis.
Run Analysis: Click RUN ANALYSIS to compute statistics, p-values, and plot visualizations.
Review Results & Plots: Inspect the summary table, toggle to the Plots tab to inspect Q-Q plots, and view automated text interpretations.
Export Outputs: Export summary tables as Excel (.xlsx) or DOCX, or download plot figures as publication-ready images or PowerPoint presentations.

7. IMPORTANT NOTES

Interpretation Rule for Normality Hypotheses

Unlike standard hypothesis testing where p-value < Alpha indicates a positive finding, in normality testing the null hypothesis (H0) states that the data is normal. Therefore, a p-value > Alpha (e.g., > 0.05) means you fail to reject normality (Data is Normal), whereas a p-value < Alpha indicates significant deviation from normality.

Cite DATES in Research Papers

If you use the DATES Normality & Distribution Analysis module for statistical validation in published scientific work, please cite it as follows:

@software{dates_app_2026, author = {DATES Development Team}, title = {DATES: Data Analysis and Trial Evaluation System}, year = {2026}, url = {https://dates-app.org}, note = {Data Preparation & Normality Analysis Modules} }