Kolmogorov-Smirnov Uniform Calculator

Understanding the Kolmogorov-Smirnov Uniform Test

The Kolmogorov-Smirnov (K-S) test is a powerful non-parametric goodness-of-fit test used to determine whether a given sample of data matches a hypothesized theoretical distribution, such as the continuous uniform distribution. Unlike standard parametric tests, it makes no assumptions about the underlying distribution normality, focusing purely on comparing cumulative distribution functions.

Formula Used

The test statistic $D$ is defined as the supremum of the absolute distance between the empirical cumulative distribution function ($F_n(x)$) and the theoretical cumulative distribution function ($F_0(x)$):

$$D = \sup_x |F_n(x) - F_0(x)|$$

For a fully specified uniform distribution on the interval $[a, b]$, the theoretical CDF $F_0(x)$ is calculated as:

$$F_0(x) = \begin{cases} 0 & \text{for } x < a \\ \frac{x - a}{b - a} & \text{for } a \le x \le b \\ 1 & \text{for } x > b \end{cases}$$

The empirical distribution function for ordered sample points $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$ is evaluated at each step using index evaluations to capture maximum deviations both from above and below.

How to Use This Calculator

• Step 1: Insert your raw data values inside the text area in the first column, separated by spaces, commas, or line breaks. Alternatively, use the helper button.
• Step 2: Specify your uniform distribution boundaries parameters ($a$ and $b$) within the middle column configuration settings.
• Step 3: Select your preferred significance level threshold ($\alpha$) and alternative test hypothesis style in the final column.
• Step 4: Click the calculate action button to review immediate empirical evaluations, critical metrics, and tables right above the form.

Frequently Asked Questions (FAQs)