# How do I compare the distributions and magnitudes of a set of positive and negative values?

Suppose we have a set of positive and negative values and wish to test two hypotheses: firstly that these values are evenly distributed about zero and, secondly, the medians of the absolute values of the positive and negative values are equal.

The former can be done using the sign test and the latter using the Wilcoxon (also known as the Mann-Whitney) rank sum test with either an asymptotic, bootstrap or exact p-value. This can be done in SPSS using *analyze > nonparametric tests>* choosing *binomial* to perform the sign test or *2 independent samples* and using the default Mann-Whitney U option. For the Mann-Whitney test we are asking if the rank sums (adjusted for the number of negative and positive scores) are equal whereas for the sign test we are asking if the numbers of negative and positive values are equal.

An example

Seven scores equal to -30, -20, -10, -5, -4, -3 and 2 have ranks (ignoring sign): of 7, 6, 5, 4, 3, 2 and 1 respectively.

Rank sum of abs(positive value) = 1 Rank sum of abs(negative values) = 2+3+4+5+6+7 = 27

The rank sum of either of the above can then be compared to the rank sum expected assuming the absolute values of the scores are evenly distributed between those which are positive and those which are negative. We would expect the rank sums for 1 and 6 observations to be 4 and 24 respectively if the median of the absolute values of positive and negative scores are equal. It turns out that the probability of observing such a distribution by chance is equal to the P(Picking the lowest score out of seven by chance) = 1/7 as you have an equal chance of picking any of the seven scores or 2/7ths if you allow the randomly chosen score to be either the lowest or the highest (two-tailed p).

Notice for the comparison of the medians of the absolute values you will probably need to specify two groups (corresponding to negative and positive) values. Since we are only interested in the ranks of the observations one can construct a dummy data set to test equality of positive and negative median absolute values.

The example scores above: -30, -20, -10, -5, -4, -3 and 2 can be entered into SPSS as any seven distinct scores with the smallest value (corresponding to the one observed positive value which takes the lowest absolute value) as 'group 1' and the remaining six values as coded as 'group 2'. SPSS then outputs the result described above with a one-tailed p-value of one seventh and a two-tailed p-value of two-sevenths of observing such a distribution of absolute values of positive and negative scores.