There is quite a bit of overlap and the most common form of the permutation test is a form of a randomization test.

Some purists consider the true permutation test to be based on every possible permutation of the data. But in practice we sample from the set of all possible permutations and so that is a randomization test.

There are also bootstrap tests, if we don't find every possible bootstrap sample but rather sample from the possible set (what is usually done) then this is also a randomization test (but not a permutation test).

Think of the difference like any other statistic that you are collecting. These differences are just some values that you have recorded. You calculate their mean and standard deviation to understand how they are spread (for example, in relation to 0) in a unit-independent fashion.

The usefulness of the SD is in its popularity -- if you tell me your mean and SD, I have a better understanding of the data than if you tell me the results of a TOST that I would have to look up first.

Also, I'm not sure how the difference and its SD relate to a correlation coefficient (I assume that you refer to the correlation between two variables for which you also calculate the pairwise differences). These are two very different things. You can have no correlation but a significant MD, or vice versa, or both, or none.

By the way, do you mean the standard deviation of the mean difference or standard deviation of the difference?

**Update**

OK, so what is the difference between SD of the difference and SD of the mean?

The former tells you something about how the measurements are spread; it is an estimator of the SD in the population. That is, when you do a single measurement in A and in B, how much will the difference A-B vary around its mean?

The latter tells us something about how well you were able to estimate the mean difference between the machines. This is why "standard difference of the mean" is sometimes referred to as "standard error of the mean". It depends on how many measurements you have performed: Since you divide by $\sqrt{n}$, the more measurements you have, the smaller the value of the SD of the mean difference will be.

SD of the difference will answer the question "how much does the discrepancy between A and B vary (in reality) between measurements"?

SD of the mean difference will answer the question "how confident are you about the mean difference you have measured"? (Then again, I think confidence intervals would be more appropriate.)

So depending on the context of your work, the latter might be more relevant for the reader. "Oh" - so the reviewer thinks - "they found that the difference between A and B is x. Are they sure about that? What is the SD of the mean difference?"

There is also a second reason to include this value. You see, if reporting a certain statistic in a certain field is common, it is a dumb thing to *not* report it, because not reporting it raises questions in the reviewer's mind whether you are not hiding something. But you are free to comment on the usefulness of this value.

## Best Answer

It is hard to comment without context. Many terms may be ambiguous or there may be different procedures and methodologies of doing things that at first sight may be the same, but because of the "technical details" are not. When reading terms like this, they should always be accompanied by the definitions of the terms and the actual methodology that was used. If they are not, it is a guessing game.

Referring to the quotes, the usual meaning of the three phrases would be different.

## 1. "De-meaning the equation gives..."

De-meaning usually means subtracting the mean from all the values. If the mean is

$$ \bar x = \frac{1}{N} \sum_{i=1}^N x_i $$

then de-meaning is the operation that produces $x_i' = x_i - \bar x$ for all observations $i=1 \dots N$.

## 2. "Differencing the mean of the outcome eliminates...."

This one might mean different things. For example, similar language was used here to describe the difference-in-differences method:

Here it is about the difference

betweenmeans.## 3. "The mean-difference provides..."

"Mean-difference" would usually mean that you calculate the mean of differences. For example, you have the $z_i = x_i - y_i$ datapoints for $i = 1 \dots N$ and calculate mean-difference, i.e. the mean of $z_i$'s.

So the terms

are notinterchangeable.