Calculator Form
Example Data Table
This example uses the sample set: 12, 7, 9, 9, 14, 18, 6, 11, 14, 10.
| Sorted Unique Value | Frequency | Cumulative Count | Empirical CDF |
|---|---|---|---|
| 6 | 1 | 1 | 0.10 |
| 7 | 1 | 2 | 0.20 |
| 9 | 2 | 4 | 0.40 |
| 10 | 1 | 5 | 0.50 |
| 11 | 1 | 6 | 0.60 |
| 12 | 1 | 7 | 0.70 |
| 14 | 2 | 9 | 0.90 |
| 18 | 1 | 10 | 1.00 |
Formula Used
The empirical cumulative distribution function estimates the proportion of sample points that are less than or equal to a chosen threshold.
Primary formula
Fn(x) = (1 / n) × Σ I(Xi ≤ x)
Where:
- n is the total number of observations.
- Xi is the i-th sample value.
- I(Xi ≤ x) equals 1 when the condition is true, otherwise 0.
Interpretation: If Fn(14) = 0.90, then 90% of the observed sample is at or below 14.
How to Use This Calculator
- Enter the sample values into the main data field.
- Separate values using commas, spaces, semicolons, or line breaks.
- Optionally add query points for threshold evaluation.
- Choose the number of decimal places for display.
- Click Calculate Empirical CDF.
- Review the summary metrics, ECDF table, and chart.
- Download the result report as CSV or PDF if needed.
Frequently Asked Questions
1) What does an empirical CDF show?
It shows the proportion of observed sample values that are less than or equal to each x value. It is a data-driven cumulative distribution without assuming a theoretical model.
2) Can I use repeated values?
Yes. Repeated observations are counted in the frequency column and increase cumulative probability at their shared value. Duplicates are important for an accurate step function.
3) Why does the graph look like steps?
An ECDF changes only when the sample reaches an observed value. Between observed values, the cumulative probability stays constant, which creates the staircase shape.
4) What is the difference between count < x and count ≤ x?
Count < x excludes the query value itself, while count ≤ x includes every observation equal to that value. The empirical CDF uses the less-than-or-equal form.
5) Can I enter negative numbers or decimals?
Yes. The parser accepts integers, decimal values, repeated observations, and negative values. The calculation works as long as entries are numeric.
6) Does this calculator estimate a theoretical distribution?
No. It summarizes the observed sample directly. It does not fit normal, exponential, or other theoretical distributions unless you compare them separately.
7) Why are query points useful?
Query points help answer threshold questions quickly, such as the fraction of observations below a score, measurement, cost, or risk boundary.
8) When should I use an empirical CDF instead of a histogram?
Use an ECDF when you want exact cumulative proportions and threshold comparisons. Histograms are better for binned shape impressions, while ECDFs keep cumulative precision.