One of the assumptions of the paired t test is that the underlying data is normally distributed. I've read either the paired differences should be normally distributed, or, that the residuals should be normally distributed (which I thought was the case for ANOVAs / regressions).

I've also read that the residuals in this case ARE the paired differences (link), or that the residuals for a paired t test are the paired differences for each pair minus the mean difference overall (link).

So, is there one simple answer to my question: when checking the assumption of normally distributed data for a paired t test, should you be checking the paired differences, or residuals?


1 Answer 1


The paired differences.

A paired t-test is the same as a one-sample t-test on the differences, so the assumption is that the differences are normally distributed. The residuals are simply the differences minus the mean difference.

n = 10
x = rnorm(n)
y = x + 1 + rnorm(n, 0, 1)
t.test(x, y, paired=T)
# Paired t-test
# data:  x and y
# t = -3.0863, df = 9, p-value = 0.01301
# alternative hypothesis: true difference in means is not equal to 0
# 95 percent confidence interval:
#   -1.9138020 -0.2948852
# sample estimates:
#   mean of the differences 
# -1.104344 

d = y - x
# One Sample t-test
# data:  d
# t = 3.0863, df = 9, p-value = 0.01301
# alternative hypothesis: true mean is not equal to 0
# 95 percent confidence interval:
#   0.2948852 1.9138020
# sample estimates:
#   mean of x 
# 1.104344 
  • 1
    $\begingroup$ +1 and let me emphasize that a paired t-test is a one-sample test. $\endgroup$
    – Dave
    Commented Nov 10, 2020 at 17:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.