Back-calculate the pooled SD from means, group sizes, and the p-value

backcalculate_sd.r

# means, SDs, and group sizes
m1 <- 52
m2 <- 40
sd1 <- 6
sd2 <- 4
n1 <- 20
n2 <- 40

# pooled SD
sdp <- ((n1-1)*sd1^2 + (n2-1)*sd2^2) / (n1 + n2 - 2)
sdp

# independent samples t-test and corresponding p-value
tval <- (m1 - m2) / (sdp * sqrt(1/n1 + 1/n2))
pval <- 2*pt(abs(tval), df=n1+n2-2, lower.tail=FALSE)
pval

# back-calculate the pooled SD from the p-value, means, and group sizes
tval <- qt(pval/2, df=n1+n2-2, lower.tail=FALSE)
(m1 - m2) / (tval * sqrt(1/n1 + 1/n2))