Easily analyze your data’s distribution with our advanced Normality Calculator a powerful tool that checks whether your dataset follows a normal distribution. Perform multiple normality tests such as Shapiro Wilk, Anderson Darling, Jarque Bera, and D’Agostino Pearson in one place. Instantly visualize your results with an interactive histogram and make confident statistical analysis decisions.
📊 Normality Test Calculator
Easily assess if the normality assumption can be applied to your data using Shapiro-Wilk, Shapiro-Francia, Anderson-Darling, and other statistical tests.
📈 Test Results
What is a Normality Test?
A Normality Test is a fundamental statistical analysis method used to determine whether a dataset follows a normal distribution — the classic bell-shaped curve that many parametric tests assume. It helps researchers and analysts confirm whether the data is appropriate for tests such as t-tests, ANOVA, and regression models. In simple terms, a normality test checks if your sample data behaves like it was drawn from a normally distributed population.
Different normality tests like the Shapiro–Wilk test, Anderson–Darling test, Cramér–von Mises test, and D’Agostino–Pearson test use various statistical approaches, but all compare the observed data distribution to a theoretical normal curve. The outcome usually includes a test statistic and a p-value, where a p-value greater than 0.05 typically indicates that the data is normally distributed.
General Concept Formula for Normality:
Z = (X − μ) / σ
Where:
- X = individual data point
- μ = population mean
- σ = standard deviation
This formula standardizes data into Z-scores to compare how far each value is from the mean in standard deviation units. In a normal distribution, approximately 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.
Our powerful Normality Calculator automatically applies multiple normality tests — including Shapiro–Wilk, Anderson–Darling, and D’Agostino–Pearson — to quickly determine if your dataset meets the normality assumption. It also generates clear visualizations such as histograms and Q–Q plots to help users easily interpret the results in data analysis and research applications.
Interpreting the Outcome of Tests for Normality
When performing a normality test, the goal is to determine whether your dataset follows a normal distribution. Every test — such as the Shapiro–Wilk, Anderson–Darling, or D’Agostino–Pearson test — produces two key values: a test statistic and a p-value. These results guide whether to accept or reject the null hypothesis (H₀: the data is normally distributed).
Decision Rule Formula:
If p-value > α → Fail to reject H₀ (Data is normally distributed)
If p-value ≤ α → Reject H₀ (Data is not normally distributed)
Where:
- p-value = probability that the observed data fits the normal distribution
- α = significance level (commonly 0.05)
- H₀ = null hypothesis (data follows normal distribution)
In most cases, a p-value greater than 0.05 suggests that the data does not significantly deviate from a normal distribution, meaning it’s safe to use parametric tests such as t-tests or ANOVA. On the other hand, if the p-value is less than or equal to 0.05, the data may be non-normal, and alternative non-parametric tests (like Mann–Whitney or Kruskal–Wallis) are recommended.
Our intelligent Normality Calculator automatically interprets your test outcomes, highlighting whether your dataset passes or fails the normality assumption. It presents clear visual summaries (histograms, Q–Q plots, and results tables) so users can confidently decide the next steps in their statistical analysis.
Supported Tests
Our advanced Normality Calculator supports a comprehensive suite of normality tests that allow you to check whether your data follows a normal distribution. Each test is designed for different sample sizes, data characteristics, and precision levels, ensuring accurate results for every type of statistical analysis.
| Test Name | Best For | Key Statistic | Sample Size Range |
|---|---|---|---|
| Shapiro–Wilk Test | Small to medium datasets | W statistic | n ≤ 50 |
| Shapiro–Francia Test | Large datasets | W′ statistic | n > 50 |
| Anderson–Darling Test | All sample sizes; focuses on tails | A² statistic | Any n |
| Cramér–von Mises Test | Balanced overall fit | W² statistic | Any n |
| Jarque–Bera Test | Skewness and kurtosis analysis | JB statistic | Medium to large n |
| D’Agostino–Pearson Test | Detecting skewness and kurtosis jointly | K² statistic | Medium to large n |
Each of these normality tests is automatically performed in our Normality Calculator to ensure accuracy and reliability. You can view the test statistics, p-values, and graphical outputs instantly, making it easier to interpret your data’s normality status for any type of statistical analysis.
The Shapiro–Wilk Test / Shapiro–Francia Test
The Shapiro–Wilk test and Shapiro–Francia test are powerful normality tests designed to determine whether a dataset follows a normal distribution. The Shapiro–Wilk test is highly accurate for small to medium sample sizes, while the Shapiro–Francia test is optimized for larger datasets (n > 50). Both tests evaluate the relationship between ordered sample values and their corresponding expected normal values, making them essential tools in data analysis, statistics, and scientific research.
Shapiro–Wilk Test Formula:
W = [ ( Σ ai x(i) )² ] / Σ ( xi − x̄ )²
Where: x(i) = ordered sample values, ai = weights derived from the expected normal order statistics, x̄ = sample mean.
A W value close to 1 suggests the data is normally distributed, while smaller values indicate deviations from normality.
Shapiro–Francia Test Formula:
W′ = [ ( Σ bi x(i) )² ] / Σ ( xi − x̄ )²
Where: bi are simplified coefficients derived from expected normal order statistics. The test operates similarly to Shapiro–Wilk but is computationally simpler and faster for larger datasets.
A W′ value close to 1 indicates the data follows a normal distribution; lower values point to non-normality.
Our Normality Calculator supports both the Shapiro–Wilk and Shapiro–Francia tests to deliver quick, accurate, and detailed results. With automatic computation of the W statistic and p-value, plus visual charts like histograms and Q–Q plots, it simplifies the process of assessing data normality for researchers, students, and professionals.
The Cramér–von Mises Test
The Cramér–von Mises test is a powerful normality test that evaluates how closely a dataset’s empirical distribution matches a specified normal distribution. It focuses on the overall fit of the data rather than just the tails or center, making it a balanced and robust choice for detecting deviations from normality. This test is commonly used in statistical analysis, data modeling, and scientific research to confirm that the assumption of normality holds true before performing further analysis.
Cramér–von Mises Test Formula:
W² = (1 / 12n) + Σi=1n [F(x(i)) − (2i − 1) / (2n)]²
Where:
- n = sample size
- F(x) = cumulative distribution function (CDF) of the normal distribution
- x(i) = ordered sample values
A smaller W² statistic indicates the sample data closely follows a normal distribution, while larger values signal stronger deviations. The associated p-value determines the statistical significance of the deviation from normality.
Our advanced Normality Calculator includes the Cramér–von Mises test as part of a comprehensive suite of normality assessments. It automatically computes the W² statistic and p-value, providing instant results and intuitive visualizations like histograms and Q–Q plots to help users accurately interpret their data for statistical analysis.
The Anderson–Darling Test
The Anderson–Darling test is a highly sensitive normality test used to determine how well a dataset follows a normal distribution. Unlike other methods, it gives more weight to the tails of the data, making it especially effective in detecting deviations that occur at the extremes. This test is widely used in data analysis, statistical modeling, and quality control where precision and reliability are crucial for interpreting real-world datasets.
Anderson–Darling Test Formula:
A² = -n – (1/n) Σi=1n (2i – 1) [ln F(x(i)) + ln(1 – F(x(n+1−i)))]
Where:
- n = number of observations
- F(x) = cumulative distribution function (CDF) of the normal distribution
- x(i) = ordered sample values
A smaller A² statistic suggests the data closely follows a normal distribution, while larger values indicate stronger deviations from normality. The corresponding p-value helps confirm whether the null hypothesis of normality can be accepted.
Our Normality Calculator supports the Anderson–Darling test alongside other leading normality assessments to deliver accurate, data-driven insights. It automatically computes the A² statistic and p-value, providing users with clear visual feedback through histograms and Q–Q plots for easy interpretation in statistical analysis.
The D’Agostino–Pearson Test
The D’Agostino–Pearson test is a comprehensive normality test that evaluates whether a dataset follows a normal distribution by combining measures of skewness and kurtosis. This dual-approach method provides a more balanced assessment, making it particularly effective for detecting asymmetry and tail deviations in both small and large datasets. It is widely used in data analysis, scientific research, and statistical modeling where the assumption of normality is essential for accurate interpretation.
D’Agostino–Pearson Test Formula:
K² = Z₁² + Z₂²
Where:
- Z₁ = standardized value of skewness
- Z₂ = standardized value of kurtosis
- K² follows a chi-square distribution with 2 degrees of freedom
A smaller K² statistic (or a p-value > 0.05) indicates that the data is likely normally distributed. Larger values suggest deviations from normality caused by skewness, kurtosis, or both.
Our advanced Normality Calculator includes the D’Agostino–Pearson test among its suite of normality assessments, automatically computing skewness, kurtosis, and the K² statistic. With visual outputs like histograms and Q–Q plots, users can easily interpret data patterns and make informed decisions in statistical analysis.
The Jarque & Bera Test
The Jarque–Bera test is a widely used normality test that determines whether sample data follows a normal distribution by analyzing its skewness and kurtosis. It measures how much the shape of your dataset differs from a perfect bell curve, making it especially valuable in econometrics and data analysis. A significant p-value from this test suggests that the dataset deviates from normality, guiding analysts on whether to apply parametric or non-parametric tests. The Jarque–Bera test is simple, efficient, and ideal for both small and large datasets when assessing distribution symmetry and tail behavior.
Jarque–Bera Formula:
JB = ( n / 6 ) × [ S² + ( (K − 3)² / 4 ) ]
Where: n = sample size, S = skewness, K = kurtosis
If the p-value > 0.05, the data is considered normally distributed; otherwise, it deviates from normality.
Our advanced Normality Calculator features the Jarque–Bera test along with other key normality assessments to provide quick and precise results. It automatically calculates the test statistic and delivers clear visual insights, enabling researchers, students, and professionals to make confident decisions in statistical analysis and data interpretation.
Practical Examples
Understanding how to interpret the results of a normality test becomes easier with real-world examples. Below are some practical scenarios showing how different datasets perform when analyzed using our Normality Calculator. These examples demonstrate how the p-value and test results can help determine whether a dataset follows a normal distribution or not.
| Example | Dataset Description | Normality Test Used | P-Value | Interpretation |
|---|---|---|---|---|
| Example 1 | Heights of 100 students | Shapiro–Wilk Test | 0.321 | Data is normally distributed |
| Example 2 | Monthly sales revenue | Anderson–Darling Test | 0.042 | Data is not normally distributed |
| Example 3 | Daily temperature readings | Jarque–Bera Test | 0.187 | Data follows normal distribution |
| Example 4 | Website traffic counts | D’Agostino–Pearson Test | 0.009 | Data deviates from normality |
These examples show how different normality tests can provide valuable insights about the structure of your data. Using our Normality Calculator, you can easily replicate these tests online to verify assumptions, improve data analysis accuracy, and ensure the reliability of your statistical models.
Normality Calculator – Frequently Asked Questions (FAQ)
1. What is a Normality Calculator?
A Normality Calculator helps determine whether your data follows a normal distribution by using statistical tests like Shapiro-Wilk, Anderson-Darling, and Jarque-Bera. It simplifies analysis by automating calculations and displaying clear results.
2. Which normality tests are supported?
Our calculator supports multiple tests: Shapiro-Wilk, Shapiro-Francia, Anderson-Darling, Jarque-Bera, Cramer-von Mises, and d’Agostino-Pearson. These tests handle different sample sizes and distribution checks to ensure accurate results.
3. How do I interpret the results?
A significant p-value indicates deviation from normality, while a non-significant p-value suggests data is approximately normal. The calculator provides clear interpretation guidance for each test to help in statistical decision-making.
4. Can I test small and large datasets?
Yes, the Normality Calculator accommodates both small (n < 50) and large (n > 50) datasets. It automatically selects the appropriate tests and provides accurate results regardless of sample size.
5. What kind of data can I use?
You can use continuous numeric data such as test scores, measurements, or survey results. The calculator generates histograms with normal distribution overlays, helping visualize data conformity to normality.
6. Why is testing for normality important?
Testing for normality is crucial because many statistical methods, like t-tests, ANOVA, and regression, assume normal data. The Normality Calculator ensures your data meets these assumptions before applying parametric tests.