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Critical r table (significance of a correlation)

--- --- Methodology

A table of critical values for Pearson's correlation coefficient: the smallest |r| that is significant at your α, given the number of pairs n. Use the lookup box to check an observed r, or read the printed table below, from n = 3 to n = 100.

0.05 is the usual choice.

The number of (x, y) pairs in the sample.

Critical r at α = 0.05, n = 20 pairs 0.444
Number of pairs20
Degrees of freedom18
Equivalent t critical value2.101
Show the working, step by step

  1. The test of H₀: ρ = 0 on n pairs has n − 2 degrees of freedom.

    df = 20 − 2 = 18

  2. Take the two-tailed t critical value and convert it to the r scale:

    r* = 2.101 / √(2.101² + 18) = 0.444

  3. Small samples need a large |r| to clear the bar; the table shows the drop-off as n grows.

    At n = 25 the same α gives r* = 0.396

Same tCriticalAlpha as the t-table page; the r scale is just the t critical value run through r* = t*/√(t*² + df).

How to read the table

The rows are the number of pairs n and the columns the significance levels, two-tailed. A cell is the smallest absolute r whose correlation with zero is still significant at that α. Compare it with your sample r; the sign of r is irrelevant to significance.

Worked lookup for n = 20, α = 0.05:

  1. Find the row n = 20.
  2. Find the column 0.05.
  3. The cell reads 0.444 (highlighted below).
  4. A sample correlation with |r| ≥ 0.444 rejects H₀: ρ = 0 at the 5% level; r = 0.40 does not.
Critical values of Pearson's r, two-tailed test of H₀: ρ = 0, by number of pairs n
n 0.10.050.020.01
3 0.9880.9971.0001.000
5 0.8050.8780.9340.959
10 0.5490.6320.7150.765
15 0.4410.5140.5920.641
20 0.3780.4440.5160.561
25 0.3370.3960.4620.505
30 0.3060.3610.4230.463
40 0.2640.3120.3670.403
50 0.2350.2790.3280.361
60 0.2140.2540.3000.330
80 0.1850.2200.2600.286
100 0.1650.1970.2320.256

Where the numbers come from

Testing H₀: ρ = 0 on n pairs reduces to a t-test: the statistic t = r √(n − 2) / √(1 − r²) is t-distributed with n − 2 degrees of freedom under the null. Solving that for r gives the critical value used by this page,

r* = t* / √(t*² + (n − 2)), t* the two-tailed t critical value at α, df = n − 2

Two consequences are visible in the table. First, r* falls with n: with n = 10 the 0.05 bar is 0.632, with n = 100 it is 0.197, and with n = 1000 it is 0.062. Second, the large-n limit is the z-based value z*/√(z*² + n − 2) with z* = 1.96, because the t distribution becomes normal; the lookup box computes that value for any n, printed or not. All values are computed by the same t critical value routine this site uses for its t-test calculators, rounded to 3 decimal places.

Critical value versus power

The table answers "is this r big enough to be significant at α?", not "is my sample big enough to detect a correlation of a given size?". A true ρ = 0.2 is invisible at n = 15 (critical 0.514) and routine at n = 200. Use the power analysis calculator to work out the n for the ρ you want to be able to see, and the correlation coefficient calculator for the exact p-value and confidence interval of an observed r — which is the modern replacement for the table's "between 0.05 and 0.01" verdicts.

R critical value table: the worked example on this page, with its result and chart
R critical value table: the worked example above, at a glance.

Common questions

What is the critical value of r?

The smallest |r| that counts as a real correlation at your significance level, for the number of pairs n you have. With n = 20 pairs at α = 0.05 (two-tailed) the value is 0.444: sample correlations of 0.444 or larger in absolute value reject H₀: ρ = 0, and anything smaller is consistent with chance. It is row n = 20, column 0.05 of the table above.

How is the critical r computed?

From the t distribution: a two-sided test of H₀: ρ = 0 on n pairs has the statistic t = r √(n − 2) / √(1 − r²), t-distributed with n − 2 degrees of freedom. Inverting that for r gives r* = t* / √(t*² + (n − 2)), where t* is the two-tailed t critical value at α from the t table. The table above is that t value re-expressed on the r scale, which is why the critical r only falls as n grows.

Why do I need more pairs to see small correlations?

Because the sampling variability of r shrinks with n, and the price of the test is in critical value, not power. At n = 10 pairs, α = 0.05, correlations must clear 0.632; at n = 30, 0.361; at n = 100, 0.197. A true correlation of 0.2 is practically undetectable with 15 pairs (critical 0.514) but easy to detect with 200. The power analysis calculator tells you how many pairs you need for a target power.

Is the test one-tailed or two-tailed?

The table above is two-tailed: it answers "is r different from zero?" in either direction. If the research question is directional ("is r positive?") the one-tailed critical value is smaller, and the lookup box above switches tails. As with any t-test, the tail choice must be fixed before the data are seen.

What about Spearman’s rho and other non-parametric correlations?

Spearman's rho is tested with the same t form — t = ρ √(n − 2) / √(1 − ρ²) — for n roughly above 20, so this table applies with ρ in place of r. For very small samples, exact tables or permutation tests are preferred. The correlation coefficient calculator reports the p-value directly, which does not need a table at all.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.

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