Answer

What does r mean when it is reported next to a t test, and how do you turn it into d?

Inspired by a question on Cross Validated · October 3, 2026

effect sizet testcorrelation

The short answer

An r printed after a t test is almost always an effect size, not a separate correlation between two measured variables. It answers the question "how strongly is the outcome related to which group a person is in?" on the familiar −1 to 1 scale of a correlation. Authors usually compute it straight from the test statistic, using a formula popularised by Robert Rosenthal for meta-analysis:

r = √(t² / (t² + df)), where df is the degrees of freedom of the t test (n₁ + n₂ − 2 for two independent groups).

This is not a loose analogy. For Student's (equal-variance) t test, the number this formula gives is exactly the ordinary Pearson correlation you would get by coding one group as 1, the other as 0, and correlating that code with the scores. Statisticians call it the point-biserial correlation. The formula only returns the size, so the sign is set by the direction of the difference.

Seeing it in one example

The code below simulates two groups of 40 with a true difference of half a standard deviation, runs a t test, and then computes r, r², and Cohen's d in several ways so you can see they agree. The last two parts involve no randomness: they show how r depends on the group split and what Cohen's benchmarks for r mean on the d scale.

set.seed(632153)

# Two independent groups of 40, true difference of half an SD
score <- c(rnorm(40, mean = 55, sd = 10), rnorm(40, mean = 50, sd = 10))
group <- rep(c(1, 0), each = 40)               # 1 = treatment, 0 = control
n1 <- 40; n2 <- 40

round(tapply(score, group, mean), 2); round(tapply(score, group, sd), 2)
tt <- t.test(score[group == 1], score[group == 0], var.equal = TRUE)
t  <- unname(tt$statistic); df <- unname(tt$parameter)
round(c(t = t, df = df, p = tt$p.value), 3)
round(tt$conf.int, 2)                           # 95% CI for the mean difference

# 1. r from t, and the same r as a plain correlation with the 0/1 group code
r_from_t <- sqrt(t^2 / (t^2 + df))
round(c(r_from_t = r_from_t, cor_with_group = cor(score, group)), 3)

# 2. r squared = share of variance explained by group (eta squared)
ss <- summary(aov(score ~ factor(group)))[[1]][["Sum Sq"]]
round(c(r_squared = r_from_t^2, eta_squared = ss[1] / sum(ss)), 3)

# 3. Cohen's d three ways
sp <- sqrt((39 * var(score[group == 1]) + 39 * var(score[group == 0])) / 78)
d_pooled  <- (mean(score[group == 1]) - mean(score[group == 0])) / sp
d_from_t  <- t * sqrt(1 / n1 + 1 / n2)
d_from_r  <- 2 * r_from_t / sqrt(1 - r_from_t^2)  # equal-n shortcut
round(c(d_pooled = d_pooled, d_from_t = d_from_t, d_from_r = d_from_r), 3)
round(c(mean_diff = mean(score[group == 1]) - mean(score[group == 0]), d = d_pooled), 2)

# 4. Same true d = 0.5, different group splits: r shrinks as groups get unequal
p <- c(0.5, 0.3, 0.2, 0.1)                      # share of people in group 1
round(data.frame(p, r = 0.5 * sqrt(p * (1 - p)) / sqrt(1 + 0.25 * p * (1 - p))), 3)

# 5. Cohen's r benchmarks translated to d (equal groups, large samples)
r_b <- c(0.10, 0.30, 0.50)
round(data.frame(r = r_b, d = 2 * r_b / sqrt(1 - r_b^2)), 2)

import numpy as np
import pandas as pd
from scipy import stats

rng = np.random.default_rng(632153)

# Two independent groups of 40, true difference of half an SD
treat = rng.normal(55, 10, 40)
control = rng.normal(50, 10, 40)
score = np.concatenate([treat, control])
group = np.repeat([1, 0], 40)                 # 1 = treatment, 0 = control
n1, n2 = 40, 40

print("M", round(control.mean(), 2), round(treat.mean(), 2),
      "SD", round(control.std(ddof=1), 2), round(treat.std(ddof=1), 2))
tt = stats.ttest_ind(treat, control, equal_var=True)
t, df = tt.statistic, n1 + n2 - 2
ci = tt.confidence_interval(0.95)             # 95% CI for the mean difference
print("t", round(t, 3), "df", df, "p", round(tt.pvalue, 3),
      "CI", round(ci.low, 2), round(ci.high, 2))

# 1. r from t, and the same r as a plain correlation with the 0/1 group code
r_from_t = np.sqrt(t**2 / (t**2 + df))
print("r_from_t", round(r_from_t, 3), "cor_with_group", round(np.corrcoef(score, group)[0, 1], 3))

# 2. r squared = share of variance explained by group (eta squared)
ss_between = n1 * (treat.mean() - score.mean())**2 + n2 * (control.mean() - score.mean())**2
ss_total = ((score - score.mean())**2).sum()
print("r_squared", round(r_from_t**2, 3), "eta_squared", round(ss_between / ss_total, 3))

# 3. Cohen's d three ways
sp = np.sqrt((39 * treat.var(ddof=1) + 39 * control.var(ddof=1)) / 78)
d_pooled = (treat.mean() - control.mean()) / sp
d_from_t = t * np.sqrt(1 / n1 + 1 / n2)
d_from_r = 2 * r_from_t / np.sqrt(1 - r_from_t**2)  # equal-n shortcut
print("d_pooled", round(d_pooled, 3), "d_from_t", round(d_from_t, 3), "d_from_r", round(d_from_r, 3))
print("mean_diff", round(treat.mean() - control.mean(), 2), "d", round(d_pooled, 2))

# 4. Same true d = 0.5, different group splits: r shrinks as groups get unequal
p = np.array([0.5, 0.3, 0.2, 0.1])            # share of people in group 1
print(pd.DataFrame({"p": p, "r": 0.5 * np.sqrt(p * (1 - p)) / np.sqrt(1 + 0.25 * p * (1 - p))}).round(3))

# 5. Cohen's r benchmarks translated to d (equal groups, large samples)
r_b = np.array([0.10, 0.30, 0.50])
print(pd.DataFrame({"r": r_b, "d": 2 * r_b / np.sqrt(1 - r_b**2)}).round(2))

The figures below come from one seeded run of the R code. Python's random numbers differ from R's, so its simulated sample is different (its groups happen to be closer together, with r = .201 and d = 0.404), but every identity holds in the same way, and parts 4 and 5, which involve no randomness, match R exactly.

• The test. The treatment group averaged 55.06 (SD = 9.58) and the control group 49.81 (SD = 12.33), giving t(78) = 2.13, p = .037.
• r two ways. The formula gives r = .234, and correlating the scores with the 0/1 group code gives exactly the same .234.
• r squared. r² = .055, identical to eta squared from a one-way ANOVA on the same data: group membership accounts for about 5.5% of the variance in scores.
• Cohen's d. The pooled-SD d, and d = t × √(1/n₁ + 1/n₂), both give 0.475. The shortcut 2r / √(1 − r²) gives 0.481, slightly high because it assumes a large sample.

Converting between r and d

Both numbers describe the same difference, on different scales. d is a mean difference in standard deviation units; r is a correlation. The conversions for two independent groups are:

• From t to d (exact): d = t × √(1/n₁ + 1/n₂). Use this whenever the paper gives t and the group sizes.
• From r to d (equal groups): d ≈ 2r / √(1 − r²). It is close when the groups are similar in size and the sample is not tiny.
• From d to r (equal groups): r ≈ d / √(d² + 4).

The benchmarks are not interchangeable. Cohen's labels of .10, .30 and .50 for r correspond to d values of 0.20, 0.63 and 1.15, so a "medium" r is a bigger effect than a "medium" d of 0.5. For more on what d values mean, see How do you interpret Cohen's d?.

Cautions before you compare r values

• r depends on how the sample is split; d does not. In part 4, the true d is 0.5 in every row, yet r falls from .243 with a 50/50 split to .223 at 30/70, .196 at 20/80 and .148 at 10/90. Two studies with the same mean difference can report quite different r values just because one recruited unequal groups.
• Paired designs give a different quantity. Applying the same formula to a paired t test (with df = n − 1) gives a number tied to the spread of the difference scores. It is often larger than the independent-groups r for the same mean difference and should not be compared with it directly.
• Use Student's t for the conversion. The exact match with the point-biserial correlation holds for the equal-variance test. Plugging in a Welch t and its fractional df gives a close but not identical value.
• Not the same as r from a rank test. Some papers report r = Z / √N after a Mann–Whitney or Wilcoxon test. That is a related effect size built from the test's Z statistic, not this one. See t test or Mann–Whitney in small samples?.

If you are choosing which test and effect size to report, the test chooser can help, and the DASS blog has more plain-language guides to reporting results.

How to report effect size r for a t test in APA style (7th edition)

Report r right after the test, with two decimals and no leading zero, because a correlation cannot exceed 1. Using the example above:

"Participants in the treatment group scored higher (M = 55.06, SD = 9.58, n = 40) than those in the control group (M = 49.81, SD = 12.33, n = 40), t(78) = 2.13, p = .037, r = .23, mean difference = 5.25, 95% CI [0.33, 10.17]."

If your readers expect d, give it instead or as well (d = 0.48, with a leading zero because d can exceed 1). In the method section, say how the effect size was computed, for example: "Effect sizes are point-biserial correlations computed as r = √(t² / (t² + df))."

Related tools and guides

• t-test power and sample size calculator
• APA 7 formatter for t tests
• Correlation power and sample size calculator
• How do you interpret Cohen's d?
• Does a t test need a minimum sample size?
• Paired vs independent t test: which should you use?
• Is R-squared useful or misleading?
• Point-biserial correlation (Wikipedia)

More answered questions

• How do you interpret Cohen's d, and are 0.2, 0.5 and 0.8 a real standard?
• t test or Mann-Whitney in small samples: how do you choose?
• Paired vs independent t test: which should you use?

Written by AskStats with AI assistance. This is general information, not advice for your specific data or study. When the results matter, check your approach with a qualified statistician.