Derivation

signal=noiseless underlying signal

n1=random noise present in measurement 1, average 
 value = zero

n2=random noise present in measurement 2 (Separate
 sample, same population)

sdn=standard deviation of n1 or n2 (same population, on average equal)

measurement 1: m1=signal+n1 
 
measurement 2: m2=signal+n2 (signal is exactly the 
 same in both)

difference=m1-m2, so that the signal cancels out 
 exactly, leaving n1-n2, the standard deviation of
 which is LARGER than that of n1 or n2 separately,
 because it's composed of TWO independent noises.  

To figure this out, we must use the rules for error
 propagation. When two random variables are added
 or subtracted, the standard deviation of the
 result is the square root of the sum of the
 squares of the individual standard deviations. 
 In other words:

std(m1-m2) = std(n1-n2) = 
  sqrt(std(sdn)^2 + std(sdn)^2)

Square both sides:
 std(m1-m2).^2 = std(sdn)^2 + std(sdn)^2

Combine terms on right side:
 std(m1-m2).^2 = 2*(std(sdn)^2)

Divide both sides by 2:
 (std(m1-m2).^2)/2 = std(sdn)^2)

Take square root of both sides
 sqrt((std(m1-m2).^2)/2) = std(sdn)

QED

Reference: See the first line of https:// 
terpconnect.umd.edu/~toh/spectrum/ErrorPropagation.pdf)