Cumulative Distribution Function (CDF) Calculator

Online calculator for the empirical distribution function of a data series

CDF Calculator

Cumulative Distribution Function

The cumulative distribution function (CDF) computes the proportion of data values that are less than or equal to a comparison value.

Enter Data

Data Series

Input

Enter numbers separated by semicolons, spaces, or one per line. Can be copied from Excel.

Parameters

Comparison Value

Find proportion of values ≤ this value

Decimal Places

CDF Result

Distribution:

Proportion of values ≤ comparison value

CDF Properties

Range: 0 ≤ F_n(t) ≤ 1, Monotonically increasing function

Empirical Distribution Monotonically Increasing 0 to 1 Range

CDF Visualization

The empirical CDF shows cumulative proportions of data.
Step function that increases from 0 to 1.

CDF Chart would be displayed here with step function visualization

What is the Cumulative Distribution Function?

The cumulative distribution function (CDF) describes the distribution of data:

Definition: F_n(t) = proportion of values ≤ t
Range: 0 ≤ F_n(t) ≤ 1
Interpretation: Probability that a value ≤ t

Application: Data analysis, distribution description
Type: Empirical distribution from sample data
Shape: Non-decreasing step function

Empirical Distribution Properties

The cumulative distribution function has important properties:

Mathematical Properties

Monotonicity: Non-decreasing function
Bounds: 0 ≤ F_n(t) ≤ 1
Starting point: F_n(-∞) = 0
End point: F_n(+∞) = 1

Practical Applications

Percentiles: Find values at specific percentages
Data comparison: Compare different datasets
Outlier detection: Identify unusual values
Quality control: Monitor data characteristics

Applications of the Cumulative Distribution Function

The CDF is essential for data analysis and statistics:

Data Analysis

Distribution shape analysis
Percentile calculations
Data summarization
Comparison of datasets

Quality Control

Outlier detection
Process monitoring
Specification compliance
Statistical testing

Educational Use

Teaching statistics
Distribution visualization
Probability concepts
Data literacy

Scientific Research

Empirical research analysis
Experimental data evaluation
Hypothesis testing
Data exploration

Definition of the Empirical Distribution Function

Empirical Distribution Function

\[\displaystyle F_n(t)=\frac{\text{Number of elements} \leq t}{n} = \frac{1}{n}\sum_{i=1}^{n} \mathbf{1}_{x_i \leq t}\]

Proportion of sample values less than or equal to t

Interpretation

\[\text{CDF}(t) = \text{Probability}(X \leq t) = \frac{\text{Count}(X \leq t)}{n}\]

Empirical probability of values ≤ t

Properties

\[0 \leq F_n(t) \leq 1\] \[F_n \text{ is monotonically increasing}\] \[\lim_{t \to -\infty} F_n(t) = 0, \quad \lim_{t \to +\infty} F_n(t) = 1\]

Key mathematical properties

Percentile Interpretation

\[\text{If } F_n(t) = p, \text{ then } t \text{ is the } p \cdot 100\% \text{ percentile}\]

CDF value equals percentile rank / 100

Example Calculation

Example: CDF of Data Series

Data: 2, 5, 4, 8, 3, 7, 9, 3, 1, 6

Given Data

Series: 2, 5, 4, 8, 3, 7, 9, 3, 1, 6

Total values: n = 10

Comparison value: t = 5

Question: Find F_n(5)

Solution

Values ≤ 5: 2, 5, 4, 3, 3, 1 = 6 values

\[F_n(5) = \frac{6}{10} = 0.6\]

Result: 60% of values are ≤ 5

Interpretation: The CDF value of 0.6 means that 60% or 6 out of 10 data values are less than or equal to the comparison value of 5.

Step-by-Step Breakdown

Sorted Data:

1, 2, 3, 3, 4, 5, 6, 7, 8, 9

Count ≤ 5:

1, 2, 3, 3, 4, 5 (6 values)

CDF Values for Different Thresholds

Comparison Value (t)	Count ≤ t	F_n(t)	Percentage	Interpretation
1	1	0.10	10%	10th percentile
3	4	0.40	40%	40th percentile
5	6	0.60	60%	60th percentile
7	8	0.80	80%	80th percentile
10	10	1.00	100%	All values

Mathematical Foundations of the Empirical Distribution Function

The empirical cumulative distribution function (ECDF) provides a non-parametric way to describe the distribution of a dataset. It is fundamental to descriptive statistics and serves as the basis for many statistical tests and analyses.

Key Characteristics

The empirical CDF has several important properties:

Non-decreasing: F_n(t₁) ≤ F_n(t₂) when t₁ ≤ t₂
Right-continuous: Continuity from the right at all points
Step function: Jumps at observed data values
Unbiased estimator: Consistent with theoretical CDF
Convergence: Converges to true CDF as n increases (Glivenko-Cantelli)

Relationship to Theoretical Distributions

The empirical CDF approximates theoretical probability distributions:

Connection to Theory

Sample CDF: Empirical F_n(t) from data
Population CDF: Theoretical F(t)
Convergence: F_n(t) → F(t) as n → ∞
Rate: √n[F_n(t) - F(t)] → N(0, F(t)(1-F(t)))

Applications

Goodness of fit: Kolmogorov-Smirnov test
Bootstrap methods: Resampling from empirical CDF
Quantile estimation: Percentiles from empirical CDF
Distribution testing: Comparing distributions

Practical Advantages

Data Description

No distribution assumptions required
Captures actual data distribution
Easy to compute and understand
Useful for exploratory analysis

Interpretation

Directly shows percentage of data below threshold
Useful for percentile calculations
Facilitates outlier detection
Enables data comparison across datasets

Summary

The empirical cumulative distribution function is a fundamental tool in descriptive statistics and data analysis. It provides an intuitive and non-parametric way to describe how data is distributed and serves as the foundation for many statistical methods. From percentile calculations to outlier detection, the CDF enables practical understanding and analysis of empirical data.

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