How to find the standard deviation of the difference between two means?

Statistical analyses are very often concerned with the difference between means.

Answer: The expression for calculating the standard deviation of the difference between two means is given by z = [(x1 - x2) - (µ1 - µ2)] / sqrt (  σ1² / n1 +  σ2² / n2)

The sampling distribution of the difference between means can be thought of as the distribution that would result if we repeated the following three steps over and over again.

Explanation:

A confidence interval for the difference between two means tells a specific range of values within which the difference between the means of the two populations may exist.

These intervals can be easily calculated by, for example, a producer who wishes to estimate the difference in mean daily output from two machines; a medical researcher who wishes to estimate the difference in mean response by patients who are receiving two different drugs; etc.

The confidence interval for the difference between two means contains all the values of (µ1 - µ2) (the difference between the two population means), which would not be rejected in the two-sided hypothesis test 

Given samples from two normal populations of size n1 and n2 with unknown means µ1 and µ2  and known standard deviations σ1 and σ2, the test statistic comparing the means is known as the two-sample z statistic

z = [(x1 - x2) - (µ1 - µ2)] / sqrt (  σ1² / n1 +  σ2² / n2)

Hence, the expression for calculating the standard deviation of the difference between two means is given by z = [(x1 - x2) - (µ1 - µ2)] / sqrt (  σ1² / n1 +  σ2² / n2)