Calculator
Example Data Table
| Observation | Value | Sorted Position | Cumulative Proportion |
|---|---|---|---|
| 1 | 12 | 1 | 0.0833 |
| 2 | 15 | 3 | 0.2500 |
| 3 | 18 | 4 | 0.3333 |
| 4 | 21 | 8 | 0.6667 |
| 5 | 24 | 9 | 0.7500 |
| 6 | 34 | 12 | 1.0000 |
This example shows how repeated values affect the step structure. Each step rises when the cumulative count increases. The ECDF reaches 1.0000 at the largest observation.
Formula Used
The empirical cumulative distribution function measures the proportion of observations less than or equal to a chosen value. For any value x, the ECDF is:
F(x) = Number of observations less than or equal to x / Total number of observations
If the sorted sample is x(1), x(2), ..., x(n), then each step increases by 1/n at observed values. Repeated values create larger jumps because multiple observations accumulate at the same point.
This tool also reports an inverse ECDF estimate. For probability p, it returns the smallest observed value whose cumulative proportion is at least p.
How to Use This Calculator
- Enter the sample values in the data box.
- Add an x value if you want F(x).
- Add a probability p from 0 to 1.
- Choose the number of decimal places.
- Press Calculate ECDF to generate results.
- Review the summary cards, table, and graph.
- Download the ECDF table as CSV if needed.
- Download a PDF report for documentation or sharing.
About This ECDF Tool
This calculator helps analyze distribution shape without assuming a normal model. It is useful for reliability analysis, quality control, environmental monitoring, classroom statistics, operational reporting, and exploratory data analysis.
Because the ECDF uses actual observations, it preserves repeated values, outliers, and irregular spacing. That makes it a practical choice when you want transparent, sample-based cumulative probabilities.
The included graph displays the ECDF as a step plot. The table shows unique values, frequencies, cumulative counts, and cumulative proportions. Summary outputs provide count, range, quartiles, variance, and standard deviation for deeper review.
FAQs
1. What does the ECDF show?
It shows the share of observations less than or equal to each value. The curve rises in steps and reaches one at the largest sample value.
2. Why does the graph look like steps?
The ECDF changes only at observed sample values. Between observations, the cumulative proportion stays constant, so the graph forms horizontal segments with jumps.
3. How are repeated values handled?
Repeated values increase the jump size at that point. If a value appears three times, the ECDF rises by three divided by the total sample size.
4. What does F(x) mean here?
F(x) is the empirical probability that a randomly selected observation from the sample is less than or equal to x.
5. What is the inverse ECDF result?
It returns the smallest observed value where the cumulative proportion meets or exceeds the selected probability. It is useful for sample-based percentile lookup.
6. Is this the same as a theoretical CDF?
No. A theoretical CDF comes from a probability model. An ECDF is built directly from the observed sample without assuming a distribution.
7. When should I use an ECDF?
Use it when you want a direct view of cumulative probabilities, especially for small datasets, irregular distributions, outliers, or quick comparison of samples.
8. Can I export the results?
Yes. This page includes a CSV export for the ECDF table and a PDF export for the visible results section.