The One Sample or Matched Pairs Case This is a rank test. We are dealing with the single random sample and the random sample of matched pairs that is reduced to a single sample by considering differences. Basically this means that we are doing another rank-based test and dealing with a single sample, that is assumed to have been selected at random from a population. We intend to reduce a sample of matched pairs to a single sample by comparing differences. We make the matched pair (X,Y) into a single observation on a bivariate random variable. This means that this is a single observation on a variable that is duo-fold. In Section 3.4 we learned about the sign test. We analyzed matched pairs of data similarly, but by reducing each pair to a plus or a minus, or a tie. Once the problem has been reduced, we can apply the binomial test (a distribution that requires there be only two possible outcomes) to the resultant single sample [of +s and -s]. Similarly, the test of this section reduces a matched pair (X,Y) to a single observation by considering the difference between the values in the matched pair. We subtract the Y value by X. (Y-X). We then can perform an analysis on the Differences as a sample of single observations. Whereas the sign test simply noted whether the difference was postive, negative, or zero, the test of this section notes the sizes of the differences relative to the negative differences. The model resembles that of the sign test. The important difference is an additional assumption of symmetry of the distribution of differences. We should clarify the meaning of symmetric as it applies to a distribution and discuss the influence of such symmetry on the scale of measurement. Symmetry is easy to define if the distribution is discrete (countable). It is symmetric if the left half of the graph of the probability function is the mirror image of the right half. For example, the binomial distribution is symmetric if p = 1/2. The discrete uniform distribution is always symmetric. For other than discrete distributions, there is a more abstract definition of symmetry. The distribution of a random variable X is symmetric about a line x=c for some constant c, if the probability of X <= (c-x) equals the probability of X>= (c+x) for each possible value of x. If a distribution is symmetric, the mean coincides with the median because both are located exactly in the middle of the distribution, at the line of symmetry. Also, the required scale of measurement is changed from ordinal to interval. With an ordinal scale of measurement, two observations of the random variable need only to be distinguished on the basis of which is larger and which is smaller. It is not necessary to know which one is farthest from the median, such as when two observations are on either side of the median. If the assumption of symmetry is a meaningful measurement, the distance between two observations is a meaningful one. The scale of measurement is therefore more than just ordinal, it is interval. A test presented by Wilcoxon in 1945 is designed to test whether a particular sample came from a population with a specified mean or median. It may also be used in situations where observations are paired, such as "before" and "after" observations on each of the several subjects, to see if the second random variable in the pair has the same mean as the first. Note than in symmetric distribution the mean equals the median, so the two terms can be used interchangably.