Measure paired variable movement with clean statistical summaries. Compare sample and population covariance values instantly. Visualize probability weighted outcomes using flexible exports and charts.
Paste paired values with commas, spaces, new lines, semicolons, or vertical bars.
This sample shows a probability-weighted paired dataset used for covariance interpretation.
| Pair | X | Y | Probability | Meaning |
|---|---|---|---|---|
| 1 | 2 | 1 | 0.10 | Low X and low Y outcome |
| 2 | 4 | 3 | 0.15 | Early positive movement |
| 3 | 6 | 4 | 0.20 | Moderate paired behavior |
| 4 | 8 | 7 | 0.25 | Higher weighted co-movement |
| 5 | 10 | 9 | 0.30 | Strong positive weighted pair |
These formulas calculate the weighted mean of each variable using the entered probability distribution.
A positive result means the variables tend to increase together. A negative result means one tends to rise when the other falls.
Use this when the entered values represent the full population of interest.
Use this when the entered values are a sample drawn from a larger population.
Correlation standardizes covariance and makes the relationship easier to compare across different scales.
Covariance measures whether two variables move together. Positive values suggest they rise together, negative values suggest opposite movement, and values near zero show weak linear co-movement.
Probabilities let you weight each paired outcome by likelihood. This is useful in risk models, scenario analysis, finance, reliability studies, and probabilistic machine learning workflows.
The calculator assigns equal probability to every pair. That turns the weighted result into an evenly weighted covariance analysis across all observations.
Population covariance divides by n and assumes you entered the full dataset. Sample covariance divides by n minus 1 and is better for estimating from sampled observations.
Covariance depends on the scale of the variables. Correlation removes scale effects and makes the strength of the linear relationship easier to compare across datasets.
Yes. When the normalization option is checked, the tool rescales entered probabilities so their total becomes 1.0 before calculating weighted expectations and covariance.
It usually means there is little or no linear co-movement. However, nonlinear relationships can still exist, so a zero covariance does not always mean independence.
The Plotly graph displays the X and Y pairs as a scatter plot. Marker size reflects probability, and mean reference lines help you see directional co-movement visually.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.