About Correlation Coefficient Calculator
This correlation coefficient calculator is free, needs no sign-up and runs in your browser, so the pairs you paste are never uploaded. It computes Pearson r, Spearman rank correlation ρ and Kendall τ-b together, each with a p-value, and gives a confidence interval for r.
Paste two columns copied from Excel or Google Sheets, or type one x, y pair per line. A text first line becomes the axis names. The scatter plot with its trend line, a plain-English reading of the strength and direction, and the Pearson calculation step by step all update as you edit.
Choose the coefficient that matches your data. Pearson measures a straight-line relationship between numeric variables. Spearman and Kendall use ranks, so they suit ordinal data, curved but steadily rising relationships and data with outliers.
How to use Correlation Coefficient Calculator
- Paste your data as two columns from a spreadsheet, or type one x, y pair per line. Use the "Two columns" tab if your x and y values are in separate lists.
- Pick the headline coefficient: Pearson r, Spearman ρ or Kendall τ-b. All three are always shown in the table.
- Choose a confidence level for the interval of r. 95% is the usual choice.
- Read the strength and direction sentence, the p-value and the confidence interval. A small p-value means the correlation is unlikely to be zero in the population.
- Check the scatter plot for curves, clusters or outliers, because one number can hide them.
- Open the step-by-step panel to see the means, the sums of squares, r, the t statistic and the Fisher z interval.
Features
- Pearson r with a t test for H0: ρ = 0 and a Fisher z confidence interval.
- Spearman rank correlation with tied values given their average rank.
- Kendall τ-b, corrected for ties, with a normal approximation for p.
- Paste from Excel or Google Sheets: tab, comma, semicolon and space separated pairs all work, and decimal commas are understood.
- Scatter plot with a least squares trend line, a values table for screen readers and a PNG download.
- Strength and direction in plain words, plus r² for Pearson.
- Worked Pearson steps in KaTeX that you can copy as LaTeX.
- Four bundled examples, including Anscombe set I and a data set with an outlier.
The formulas
Pearson r divides the covariance of x and y by the product of their spreads. Spearman ρ is Pearson r applied to the ranks. Kendall τ-b compares every pair of observations and counts the pairs that move together (concordant) against those that move apart (discordant).
To test whether r differs from zero, the calculator uses t = r√(n − 2) ÷ √(1 − r²) with n − 2 degrees of freedom. The confidence interval transforms r with the Fisher z, adds and subtracts the margin, and transforms back.
r = Sxy / sqrt(Sxx × Syy)
Sxy = Σ(x − mean x)(y − mean y)
t = r × sqrt(n − 2) / sqrt(1 − r²)
tau-b = (concordant − discordant) / sqrt((n0 − n1)(n0 − n2))Worked examples
Hours studied and score, 8 students gives r = 0.9842, p = 9.66e-6
Spearman ρ = 0.9762 and Kendall τ-b = 0.9286. A very strong positive relationship.
Anscombe's quartet, set I gives r = 0.8164
All four Anscombe sets share r ≈ 0.816, yet only this one looks like a straight line. That is why the scatter plot matters.
Steadily rising readings with one huge last value gives Pearson r = 0.549, Spearman ρ = 1
The ranks are in perfect order, so Spearman is 1, while the outlier drags Pearson down.
Reading the size of a correlation
| Absolute value | Strength |
|---|---|
| Below 0.1 | Negligible |
| 0.1 to 0.3 | Weak |
| 0.3 to 0.5 | Moderate |
| 0.5 to 0.8 | Strong |
| 0.8 to 1 | Very strong |
Which coefficient to use
| Coefficient | Best for | Sensitive to outliers |
|---|---|---|
| Pearson r | Numeric data, straight-line relationship | Yes |
| Spearman ρ | Ordinal data or a steady curve | Little |
| Kendall τ-b | Small samples, many ties, ranks | Little |
Good to know
Correlation is not causation
A strong correlation says two variables move together. It does not say that one causes the other. A third factor can drive both, as when ice cream sales and drowning rise together because of hot weather.
Limits of this calculator
The Spearman and Kendall p-values use the t and normal approximations, which are reliable from about 10 pairs. For very small samples, use an exact table. At most 5000 pairs are accepted. The tool tests one pair of variables at a time and does not adjust for others.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Your data is not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- Always look at the scatter plot. A curve, two clusters or one outlier can make r misleading.
- If r is near zero but the plot shows a clear curve, the relationship is real but not linear.
- With many pairs, even a tiny r can be significant. Judge the size of r, not only the p-value.
- Do not include totals or averages of the same variable twice. Repeated measures break the independence that the p-value assumes.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your pairs are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your pairs are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is a correlation coefficient?
A number from -1 to +1 that measures how closely two variables move together. +1 means a perfect rise together, -1 a perfect opposite movement and 0 no relationship of the kind measured.
How do I calculate Pearson r by hand?
Find the means of x and y, then add up (x - mean x)(y - mean y), (x - mean x)² and (y - mean y)². Divide the first sum by the square root of the product of the other two. The steps panel shows each number for your data.
When should I use Spearman instead of Pearson?
Use Spearman when the data are ranks or ordinal scores, when the relationship rises or falls steadily but not in a straight line, or when outliers would distort Pearson.
What does the p-value mean here?
It is the chance of seeing a correlation this far from zero if the true correlation were zero. A small value, usually below 0.05, is taken as evidence that a real correlation exists.
Can I paste data straight from Excel?
Yes. Copy two columns and paste them into the box. Tabs, commas, semicolons and spaces are all accepted, and a text first row is used as the axis names.
What is a good correlation value?
It depends on the field. Above 0.8 is usually called very strong and below 0.3 weak. In physics 0.9 can be disappointing, while in social science 0.3 can be meaningful.
Popular Correlation Coefficient Calculator pages
- Pearson Correlation CalculatorPaste x, y pairs to get Pearson r, r², a p-value and a confidence interval.
- Spearman Rank Correlation CalculatorPaste x, y pairs to get Spearman ρ, which resists outliers and measures any steady rise or fall.
- Kendall Tau Correlation CalculatorPaste x, y pairs to get Kendall τ-b, built from concordant and discordant pairs.