Spearman Rank Correlation Calculator

Measure	Value	p	Strength
Pearson r	0.5486	0.101	strong positive
Spearman ρ	1	0	very strong positive
Kendall τ-b	1	5.7e-5	very strong positive
95% CI for r	-0.1238 to 0.8757
Sample	10 pairs, 8 degrees of freedom

Measure

Value

Strength

Pearson r

0.5486

0.101

strong positive

Spearman ρ

very strong positive

Kendall τ-b

5.7e-5

very strong positive

95% CI for r

-0.1238 to 0.8757

Sample

10 pairs, 8 degrees of freedom

Pearson r, step by step

1Means
$\bar{x}=\frac{\sum x}{n},\ \bar{y}=\frac{\sum y}{n}$
$\bar{x}=5.5,\ \bar{y}=47\ (n=10)$
2Sums of squares
$S_{xy}=\sum (x-\bar{x})(y-\bar{y}),\ S_{xx}=\sum (x-\bar{x})^2,\ S_{yy}=\sum (y-\bar{y})^2$
$S_{xy}=1855,\ S_{xx}=82.5,\ S_{yy}=138590$
3Pearson r
$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$
$r=\frac{1855}{\sqrt{82.5\cdot 138590}}=0.54859$
r² = 0.301, so about 30.1% of the variation in y is linearly related to x.
4Test statistic
$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$
$t=1.8559,\ df=8$
Two-sided p = 0.1006 for H0: the true correlation is 0.
595% confidence interval
$z=\operatorname{artanh}(r),\ z\pm\frac{z_{\alpha/2}}{\sqrt{n-3}}$
$\rho\in[-0.1238,\ 0.8757]$
The Fisher z interval is back-transformed with tanh.

About Spearman Rank Correlation Calculator

This Spearman rank correlation calculator turns each variable into ranks and then correlates the ranks. It opens on a series with one extreme value, where Spearman reports a perfect relationship that Pearson understates. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

Spearman ρ measures a monotonic relationship: one that always rises or always falls, though not necessarily in a straight line. Tied values get the average of the ranks they span.

Use it for ordinal data such as survey scales and rankings, for curved relationships like growth, and whenever outliers would dominate Pearson. It is Pearson r on the ranks.

How to use Spearman Rank Correlation 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.

Spearman rho

Without ties, ρ can be computed from the rank differences d between each pair. With ties the calculator uses Pearson on the average ranks, which is exact.

rho = 1 − 6 × Σd² / (n × (n² − 1))   (no ties)
rho = Pearson r of rank(x) and rank(y)   (with ties)

Worked examples

Ten readings that rise every day, the last one 400 gives Spearman ρ = 1, Pearson r = 0.549
Every rank agrees, so ρ is 1. One huge value pulls Pearson well below that.

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 the difference between Spearman and Pearson?

Pearson measures straight-line association in the raw values. Spearman measures whether the ranks agree, so it captures any steadily rising or falling pattern and is less affected by outliers.

How are ties handled?

Tied values each receive the average of the ranks they would have taken, for example two values tied for 3rd and 4th both get 3.5.

Is the p-value exact?

It uses the t approximation, which is dependable from about 10 pairs. For fewer pairs, compare ρ with an exact critical value table.

Measure	Value	p	Strength
Pearson r	0.5486	0.101	strong positive
Spearman ρ	1	0	very strong positive
Kendall τ-b	1	5.7e-5	very strong positive
95% CI for r	-0.1238 to 0.8757
Sample	10 pairs, 8 degrees of freedom

Measure

Value

Strength

Pearson r

0.5486

0.101

strong positive

Spearman ρ

very strong positive

Kendall τ-b

5.7e-5

very strong positive

95% CI for r

-0.1238 to 0.8757

Sample

10 pairs, 8 degrees of freedom

Scatter plot of reading against day with a trend line
Point	Data	Trend line
1	(1, 2)	(1, -54.18181818181817)
2	(2, 3)	(10, 148.1818181818182)
3	(3, 5)
4	(4, 6)
5	(5, 8)
6	(6, 9)
7	(7, 11)
8	(8, 12)
9	(9, 14)
10	(10, 400)

Point

Data

Trend line

(1, 2)

(1, -54.18181818181817)

(2, 3)

(10, 148.1818181818182)

(3, 5)

(4, 6)

(5, 8)

(6, 9)

(7, 11)

(8, 12)

(9, 14)

(10, 400)

Pearson r, step by step

1Means
$\bar{x}=\frac{\sum x}{n},\ \bar{y}=\frac{\sum y}{n}$
$\bar{x}=5.5,\ \bar{y}=47\ (n=10)$
2Sums of squares
$S_{xy}=\sum (x-\bar{x})(y-\bar{y}),\ S_{xx}=\sum (x-\bar{x})^2,\ S_{yy}=\sum (y-\bar{y})^2$
$S_{xy}=1855,\ S_{xx}=82.5,\ S_{yy}=138590$
3Pearson r
$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$
$r=\frac{1855}{\sqrt{82.5\cdot 138590}}=0.54859$
r² = 0.301, so about 30.1% of the variation in y is linearly related to x.
4Test statistic
$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$
$t=1.8559,\ df=8$
Two-sided p = 0.1006 for H0: the true correlation is 0.
595% confidence interval
$z=\operatorname{artanh}(r),\ z\pm\frac{z_{\alpha/2}}{\sqrt{n-3}}$
$\rho\in[-0.1238,\ 0.8757]$
The Fisher z interval is back-transformed with tanh.

About Spearman Rank Correlation Calculator

Spearman ρ measures a monotonic relationship: one that always rises or always falls, though not necessarily in a straight line. Tied values get the average of the ranks they span.

Use it for ordinal data such as survey scales and rankings, for curved relationships like growth, and whenever outliers would dominate Pearson. It is Pearson r on the ranks.

Spearman Rank Correlation Calculator