Distribution of difference between sample means

In the previous section, we saw that the difference between two independent quantities X1 and X2 has a distribution with mean and standard deviation

We can apply this to obtain the distribution of the difference between the means of two random samples.

Shape of the distribution

If the distributions are normal in each group, ...
... the two sample means have normal distributions, and their difference is also normally distributed.
If the distributions are not normal in the two groups, ...
... the Central Limit Theorem states that the distributions of the two sample means approach a normal shape when the sample size increases, so their difference also approaches a normal shape.

In general therefore,

Irrespective of the distributions within the two groups,


Simulation: Manipulative skills of job applicants

The diagram below models an experiment in which a sample of male college students and a sample of experienced female industrial workers were asked to perform a manipulative task involving insertion of pins into a board. The number of pins inserted in one minute was recorded from each subject.

(The means and standard deviations are those obtained from an actual experiment that was described earlier.)

Click Accumulate then take several samples. Observe that the empirical distribution of the difference between the means matches the theoretical pink normal distribution.