Cumulative Distribution, CDF Calculator

On this page the empirical distribution of a data series is calculated.

To perform the calculation, enter a series of numbers. Then click the 'Calculate' button. The list can be entered unsorted.

Input format

The data can be entered as a series of numbers, separated by semicolons or spaces. You can enter the data as a list (one value per line). Or from a column from Excel spreadsheet by copy & paste

Formulas and examples

In descriptive statistics, the empirical distribution (probability distribution) is a value which describes how high the proportion of values is that are less than or equal to the comparison value. The result is between 0 and 1 inclusive.

Definition of the empirical distribution function

The empirical distribution function assigns to each real number the proportion of sample values, that are less than or equal to this number. If the observation values in the sample (the sample values) are given as \(x_1, x_2, \ldots, x_n\), then the empirical distribution function is defined as:

\(\displaystyle F_n(t)=\frac{Number \ of \ elements ≤ t}{n} \ \ = \frac{1}{n}\sum _{i-1}^{n} 1_{x_i} ≤t\)

Example

To calculate the empirical distribution function, the observation values are arranged in ascending order and the relative frequency is determined, with which they are less than or equal to a certain number.

This example searches for the distribution for the following number series with 10 numbers

\(\displaystyle 2, 5, 4, 8, 3, 7, 9, 3, 1, 6 \)

5 is assumed as the comparison value. So the numbers are searched whose value is 5 or smaller.

\(\displaystyle \color{#44F}{2, 5, 4}, 8, \color{#44F}{3}, 7, 9, \color{#44F}{3, 1}, 6 \)

Of the 10 numbers, this applies to 6 numbers.

The empirical distribution is \(\displaystyle \frac{6}{10}= 0.6 \) or \(60\%\)

The empirical distribution function helps to describe the distribution of the data and provides information about the probability that a measured value takes on at most a certain value.

More statistics functions

Arithmetic Mean • Contraharmonic Mean • Covariance • Empirical distribution CDF • Deviation • Five-Number Summary • Geometric Mean • Harmonic Mean • Inverse Empirical distribution CDF • Kurtosis • Log Geometric Mean • Lower Quartile • Median • Pooled Standard Deviation • Pooled Variance • Skewness (Statistische Schiefe) • Upper Quartile • Variance