# How do I run a statistical test for two sets of scores that have both positive and negative values?

I have a set of scores with a maximum range from -12 to +12 for a set of categories of a variable. There are two populations that each have a set of scores for the same categories of the same variable. The sets of scores look something like this:

Category Population One Population Two
Red +7 -3
Orange +1 -2
Yellow 0 +5
Purple +2 -1
Green -10 +1

The scores between the two populations are very different for each category (i.e. Green has a strong negative association for Pop One, and a positive association for Pop Two). Additionally, the categories also have clearly different effects on the population association (i.e., Green scores very negatively compared to Red in Population One).

Is there a way to statistically “prove” whether the differences are significant?

Is there then a way to "prove" which pairs are the most significantly different (aka, Green is the most different between Pop One and Two)?

Note that the scores for a given population will always sum to zero or near zero, so comparing the averages will be insignificant.

Is it justifiable to report significance based on the standard deviation of each population? i.e. "The score for Green in Population One is two standard deviations into the negative, so it is the most significant score." OR "Any score below one standard deviation away from 0 was not considered significant"

If it is justifiable to use this, should I calculate the standard deviation using the median rather than the mean, since the mean will be zero or near zero?

It is feasible to get the score for each individual in each population, and then have the "population score" be the mean of the individuals. Would this make it easier to run statistical tests on significance?

• Given that the averages are essentially the same, please tell us the sense in which the scores are "very different." And could you explain what might be special about the presence of negative scores? That seems irrelevant.
– whuber
Jun 8, 2023 at 21:02
• The scores are very different in that what is important is the relationship between the category and the population. The two populations clearly have very different relationships to the different categories. The problem (I have been told) is that negative numbers do not function well within some statistical tests, maybe that is not true? Jun 10, 2023 at 16:19
• Could you be specific about what "clearly ... different relationships" might be? I cannot see any clear differences based on this tiny sample. What statistical tests do you have in mind that cannot accommodate negative numbers?
– whuber
Jun 10, 2023 at 17:36
• Statististical test relate to whether or not samples are different relative to the probabilistic variations that may occur in sampling. Your example does not give details about how you obtained the data and what it represents. This makes it difficult to match it with a statistical test... Jun 16, 2023 at 13:10
• ... example: Say I have five coloured coins. One coin is red with +7 on one side and -3 on the other side, one coin is orange with +1 on one side and -2 on the other side, one coin is yellow with +0 on one side and +5 on the other side, one coin is purple with +2 on one side and -1 on the other side, one coin is green with -10 on one side and +1 on the other side. The table shows the results of flipping the five coins each once in population A and another time in population B. There are $2^5 = 32$ equally probably outcomes. Are the results different? Yes. Is it statistically significant? No... Jun 16, 2023 at 13:14

Is there a way to statistically “prove” whether the differences are significant?

Statistical significance can only be computed* when you have an idea about the statistical variations that occur in your samples.

You haven't given any information about your samples that allow to make such estimates.