Is a Correlation of 0.4 Strong? Size and p, 30 Pairs

Pearson correlation coefficient with p-value, interval and a rank check

A correlation of 0.571 sounds like a solid relationship, yet on nine pairs it gives p = 0.1082 and a 95% interval running from -0.150 to 0.896. That gap between how big r looks and what the data can actually support is what this calculator is built to show. Paste two lists of paired numbers, or two columns straight from a spreadsheet, and it returns Pearson's r, the t statistic, the p-value for a two-sided or one-sided test, a confidence interval by Fisher's z, r squared, the Spearman rank correlation and the critical r for your sample size. Already have r from a paper or a spreadsheet? Switch to the second option, type r and the number of pairs, and get the same test and interval without the raw data.

The form, box by box

What do you have - paired data, or a correlation r that is already computed. The boxes below change with the choice.
X values - the first variable, separated by spaces, commas or new lines. To paste two spreadsheet columns at once, put them here, one pair per line, and leave the Y box empty; a header row such as "Hours Score" becomes the axis names.
Y values - the second variable in the same order, so the third Y belongs to the third X. Use a dot for decimals and no thousands separators.
Correlation r and number of pairs - only for the second option: r between -1 and 1, and the n it was computed from.
Significance level - 0.05 is the usual choice, 0.01 when a false alarm is costly, 0.10 for a first look. It also sets the confidence level of the interval.
Alternative hypothesis - two-sided asks whether there is any linear relationship. Positive or negative is a one-sided test, and only honest when the direction was fixed before you saw the data.
Read the result: r in the main tile, then p, t, the interval, r squared, Spearman's rho, the critical r and any warning about a single pair or a small sample.

Problem: one pair turns nothing into r = 0.940

Take nine pairs with no pattern at all: X from 1 to 9, Y equal to 5, 3, 6, 2, 5, 4, 3, 6, 4. Pearson's r is 0.033 and p = 0.9334. Now add a tenth pair, (30, 30). The coefficient jumps to 0.940, p falls below 0.0001, and a quick report would call it a very strong relationship. Nothing about the first nine points changed. The whole result sits on one reading.

Pearson's r is built from products of distances to the mean, so one point far out on both axes can dominate the sum. The Spearman coefficient on the same ten pairs is only 0.295, because ranks cap how far any one point can pull. A plain correlation tool prints 0.940 and stops.

Three checks a plain correlation tool skips

Names the pair that matters. For the ten pairs above it reports that without pair 10 (X = 30, Y = 30) r would be 0.033 instead of 0.940. The note appears when one pair moves r by 0.2 or more, or when dropping it alone flips the significance verdict.

Puts Spearman next to Pearson. When the two differ by 0.2 or more, the result says so, since the usual causes are an outlier or a curved relationship.

Takes the direction of a one-sided test seriously. If you test for a positive correlation and r is negative, p is close to 1, not half of the two-sided value, and a note explains why the result cannot be significant.

Pearson, Spearman or Kendall: which coefficient to report

The shaded row is the case this calculator is designed for. The Spearman figure is always in the result, so the second row costs nothing extra.

Your data	Report	Why
Two measurements, a roughly straight cloud of points, no extreme pair	Pearson's r	uses the actual distances, so it has the most power
Pearson and Spearman differ by 0.2 or more	Spearman's rho, and plot the points	ranks are not pulled by one extreme value
Rising or falling, but along a curve	Spearman's rho	it measures any steady ordering, not only a straight line
Ratings or ranks with many ties, small sample	Kendall's tau	counts agreeing and disagreeing pairs directly

Data sets to paste in, and what comes back

The textbook example from R's cor.test help page. X: 44.4 45.9 41.9 53.3 44.7 44.1 50.7 45.2 60.1; Y: 2.6 3.1 2.5 5.0 3.6 4.0 5.2 2.8 3.8.
Result: r = 0.571, t = 1.841 with 7 df, p = 0.1082, interval -0.150 to 0.896, Spearman 0.600, the same figures R prints. The calculator adds that without pair 9 (60.1, 3.8) r would be 0.852 and the result would become significant.

A curve that r cannot see. X: 1 2 3 4 5; Y: 2 4 5 4 2, a perfect arch.
Result: r = 0.000 and p = 1.0000, although Y is completely determined by X. Zero correlation means no straight-line trend, not no relationship.

Sample data pasted from a spreadsheet. Ten lines of study hours and test scores under the header "Hours Score", Y box left empty.
Result: the axes are named Hours and Score, r = 0.974, t = 12.224, 95% interval 0.891 to 0.994, r squared 0.949.

A one-sided test in the wrong direction. X: 1 to 8; Y: 9 8 8 6 5 5 3 1, alternative set to positive.
Result: r = -0.975, p above 0.9999, not significant. Switched to negative, the same data give p below 0.0001 and a bound from -1 to -0.895.

Critical values of r: a table to keep

An r counts as significant in a two-sided test once its absolute value reaches the figure in the table. The calculator prints the exact value for your n, from the t distribution with n - 2 degrees of freedom.

Pairs (n)	alpha 0.10	alpha 0.05	alpha 0.01
5	0.805	0.878	0.959
8	0.621	0.707	0.834
10	0.549	0.632	0.765
15	0.441	0.514	0.641
20	0.378	0.444	0.561
30	0.306	0.361	0.463
50	0.235	0.279	0.361
100	0.165	0.197	0.256

Read it both ways. With 100 pairs an r of 0.2 already passes at 0.05, which is why large samples make weak relationships significant. With 8 pairs you need 0.707, and the R example above falls short at 0.571 with nine.

Is a correlation of 0.3 or 0.5 strong? It depends on n

With r and n alone the calculator gives the same test and interval as from the data, plus the fewest pairs at which that r would pass at 0.05.

r and pairs	p, two-sided	95% interval	Fewest pairs
0.3, 30	0.1072	-0.068 to 0.596	44
0.3, 100	0.0024	0.110 to 0.469	44
0.5, 30	0.0049	0.170 to 0.729	16
0.7, 10	0.0242	0.126 to 0.923	9
0.2, 1,000	below 0.0001	0.140 to 0.259	97

Questions people bring to a correlation

What counts as a strong correlation?

The benchmarks most often quoted are Cohen's: 0.10 small, 0.30 medium, 0.50 large, for either sign. They were written for research on people, where relationships are naturally weak. For two instruments measuring the same quantity, 0.9 can still be disappointing. Size and significance are separate questions: 0.571 is large by Cohen yet not significant on nine pairs.

How do I calculate Pearson's r in Excel or Google Sheets?

=CORREL(A2:A11, B2:B11) or =PEARSON(A2:A11, B2:B11) return the same r. Neither gives a p-value; =RSQ gives r squared. Copy the two columns and paste them into the X box of this calculator to get p, the interval and the checks.

Why is the p-value from the t statistic?

If the true correlation is zero and the pairs are roughly normal, t = r√(n - 2) / √(1 - r²) follows a Student t distribution with n - 2 degrees of freedom. For the R example that is 0.571 × √7 / √(1 - 0.326) = 1.841, and the two tails beyond it hold 0.1082 of the distribution.

Why is the interval not symmetric around r?

The coefficient cannot go past 1, so its sampling distribution is squeezed on one side. Fisher's transformation z = atanh(r) spreads it back into a near-normal shape with standard error 1/√(n - 3), the interval is built there and transformed back. That is why 0.571 gets -0.150 to 0.896, a longer reach downward than upward. It needs at least 4 pairs.

Can two data sets have the same r and look nothing alike?

Yes. Anscombe's quartet is four small data sets with r of about 0.816 each: one a noisy line, one a clean curve, one a line with a single outlier, one a vertical column with a single far point. The r is identical and the pictures are not, which is the case for always plotting the pairs.

Does a significant correlation mean X causes Y?

No. Both can follow a third variable, the causation can run the other way, or the pattern can come from how the sample was chosen. A correlation tells you the two move together in a straight line and how closely; the reason has to come from the design of the study.

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Is a correlation of 0.4 strong? Cohen's size and significance with 30 pairs

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