**n+ , n- ,
p <= **

Crudest and most insensitive test. It is also the most convincing and easiest to apply. The level of significance can almost be estimated without the help of a calculator or table. If the Sign-test indicates a significant difference, and another test does not, you should seriously rethink whether the other test is valid.

*H0:*

The median value of the distribution is *m* (generally *m* = 0),
values larger (+) and smaller (-) than the median are equally likely.

When matched pairs are used, the probability of observing (A, B) is
equal to that of observing (B, A) and the value of A-B has median value 0.

*Assumptions:*

None other than H0.

*Scale:*

Ordinal, but sometimes nominal when it is used as a binomial test

*Procedure:*

Label observations as +/-, positive/negative, greater/smaller, left/right,
black/white, male/female, or whatever dichotomy you want to use and count the
number of times each label is observed. n+ is the number of times a "+" is
observed and n- the number of times a "-" is observed.
Ignore all *zero* or *equal* observations.

*Level of Significance:*

n+ and n- are
binomial distributed with *p* = *q* = 1/2
and *N* = (n+) + (n-).

If k is the smaller of (n+) and (n-) then:

p <= 2 * Sum (i=0 to k) {*N*!/(i!*(*N*-i)!)}/4

(with k! = k*(k-1)*(k-2)*...*1 is the factorial of k and 0! = 1)

*Approximation:*

If (n+) + (n-) = *N* > 25, then
*Z* = (| n+ - n- | - 1)/sqrt( *N* )
can be approximated with a
Standard Normal distribution. In our example, we calculate the exact
probabilities upto *N* = 100.

For *N* > 30, the
Student t-test can be used. However, the
Wilcoxon Matched-Pairs Signed-Ranks test is a better alternative.

*Remarks:*

The Sign-test answers the question: *How often?*, whereas other
tests answer the question *How much?*. It must be kept in mind that
these two questions might have different answers.

When the problem concerns the *sizes* of the *differences*,
the
Wilcoxon Matched-Pairs Signed-Ranks Test should be preferred. This test
is also distribution free, but it is much more sensitive than the Sign-test.

Finally, the Sign-test is an almost ideal *quick-and-dirty* test that
can be used to browse datasets or to check the results of other tests
(e.g., as in: "All six subjects show an increase, so why does your test
insist that p > 0.05?").

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