Interpreting Correlations

An Interactive Visualization

Created by Kristoffer Magnusson · Version 0.8.0

Correlation is one of the most widely used tools in statistics. The correlation coefficient summarizes the association between two variables. In this visualization I show a scatter plot of two variables with a given correlation. The variables are samples from the standard normal distribution, which are then transformed to have a given correlation by using Cholesky decomposition. By moving the slider you will see how the shape of the data changes as the association becomes stronger or weaker. You can also look at the Venn diagram to see the amount of shared variance between the variables. It is also possible to drag the data points to see how the correlation is influenced by outliers.

Correlation

Correlation: —

Shared variance: —

FAQ

How do I use this visualization?

Use the slider or the correlation input to set the correlation. The same sample is re-transformed each time, so the points move smoothly as the association strengthens or weakens; the Venn diagram shows the corresponding shared variance.

Edit the data points

With Edit points on (the pencil above the chart), choose Drag to move points, Add to insert a point wherever you click, or Delete to remove points. All statistics — including the slider — follow the observed sample, so dragging a single point into the corner is a quick way to see how sensitive the correlation is to outliers.

Presets

Small/medium/large snap the correlation to the conventional benchmarks (0.1, 0.3, 0.5). Anscombe I–IV load Anscombe's quartet — four datasets sharing nearly identical summary statistics but wildly different shapes — and Anscombosaurus loads the datasaurus. Both are reminders to always plot your data.

Load your own data (CSV)

The settings drawer can load a CSV file with this structure:

x,y
10.0,11.0
12.0,12.0
...,...

The first row is the column names, which become the axis labels; the first column is taken as x and the second as y — all other columns are ignored. The file is parsed in your browser and not uploaded anywhere: this site performs no server-side calculations.

Probability ellipses

Toggle the ellipses in the settings drawer; they mark the regions expected to hold 50, 80, 95, and 99% of the data under a bivariate normal with the sample's moments. Protip: click an ellipse to highlight it and read off its probability (except in Add mode, where clicks place new points).

Slope chart

The slope-chart toggle redraws each (x, y) pair as a Pre–Post line — the same data viewed as change scores, where the correlation governs how much the lines cross.

The settings drawer also draws new samples of any size, sets the population means and SDs, re-frames the axes (Rescale), and saves the current state in your browser's localStorage.

What are the formulas?

The sample product-moment correlation between x and y is

r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{(n - 1)\, s_x s_y}

where $s_x$ and $s_y$ are the sample standard deviations. The shared variance shown next to the Venn diagram is $r^2$ , and the regression line uses the ordinary least-squares fit,

b_1 = r\,\frac{s_y}{s_x}, \qquad b_0 = \bar{y} - b_1 \bar{x}

To display a chosen correlation $\rho$ , the sample is first standardized (z-scored). Writing $c$ for the observed correlation of the standardized pair, whitening removes the sampled association,

z_{y \perp x} = \frac{z_y - c\, z_x}{\sqrt{1 - c^2}}

which is the inverse Cholesky factor of the observed 2×2 correlation matrix applied to the data. The target correlation is then imposed with the Cholesky factor of $\begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix}$ and the result is scaled back to the target moments:

y^{*} = \left(\rho\, z_x + \sqrt{1 - \rho^2}\, z_{y \perp x}\right) s_y^{\text{target}} + \bar{y}^{\text{target}}

Because the standardized sample has unit variances by construction, the transformed points have a sample correlation of exactly $\rho$ — no simulation error.

The probability ellipses are level curves of the fitted bivariate normal: the ellipse containing probability $p$ has half-axes

s_x \sqrt{\chi^2_2(p)} \quad \text{and} \quad s_y \sqrt{\chi^2_2(p)}

along the principal directions, where $\chi^2_2(p)$ is the chi-square quantile with 2 degrees of freedom.

How do I cite this page?

Cite this page according to your favorite style guide.

The current version is 0.8.0 (changelog).

APA 7

Magnusson, K. (2026). Interpreting correlations: An interactive visualization (Version 0.8.0) [Web App]. R Psychologist. https://rpsychologist.com/correlation/

BibTeX

@software{magnussonCorrelation,
  author = {Kristoffer Magnusson},
  title = {Interpreting correlations: An interactive visualization},
  url = {https://rpsychologist.com/correlation/},
  version = {0.8.0},
  date = {2026-10-24},
}

I found a bug/error/typo or want to make a suggestion!

Please report errors or suggestions by opening an issue on GitHub; if you want to ask a question, use GitHub Discussions.

I'm gonna ask a large number of students to visit this site. Will it crash your server?

No, it will be fine. The app runs in your browser, so the server only needs to serve the files.

Can I include this visualization in my book/article/etc?

Yes, go ahead! This visualization is dedicated to the public domain (CC0), which means you can copy, modify, and distribute it, even for commercial purposes, without asking permission. Attribution is not required but always appreciated. The source code is licensed MIT, and the text on this page is CC-BY 4.0.

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