# How to interpret correlations with negative numbers in SPSS

I'm trying to establish a bivariate Pearson correlation between two groups of variables in SPSS, however one of the groups has positive decimal numbers and the other negative decimal numbers.

The results show a significant negative correlation between the two groups. If the negative numbers were positive instead this analysis would show a significant positive correlation.

What Im wondering is, how do I interpret these results? Can I accurately say that there's a negative correlation?

• Note that use of SPSS by poster is immaterial to the question. I've removed the tag. – Nick Cox May 5 '13 at 14:56

Yes, there is a negative correlation. The positive correlation means there is a positive relationship between the variables; as one variable increases or decreases, the other tends to increase or decrease with it. The negative correlation means that as one of the variables increases, the other tends to decrease, and vice versa.

If the negative numbers were positive instead this analysis would show a significant positive correlation.

Not necessarily! For example, if you add a large enough constant to all the negative numbers so that they're all positive - i.e. add something a little larger than the absolute value of the smallest (most negative) number - then the correlation would still be negative.

You interpret the negative correlation as a negative correlation between the sets of numbers you have. The fact that if you transformed some of the numbers to different numbers (such as by multiplying them by $-1$) can yield a different correlation is immaterial -- if you had wanted to know that correlation with a transformed variable, you would have transformed the numbers before computing the correlation.

Correlations are between two variables; if you have groups of variables, then you will many correlations. For example, if there are five variables in group A and 3 in group B you will have 15 correlations.

Whether the values of the variables are positive, negative or a mixture, the correlation can be positive or negative and can be anywhere from -1 to +1. Correlation is a measure of the linear relationship between 2 variables. For example,

set.seed(1929100) #Pick random seed
x <- rnorm(1000) #Standard normal, mean = 0, sd = 1
y <- x^2 # Square
cor(x,y) #Correlation
plot(x,y)


there is a perfect relationship between x and y, but the correlation with this seed is -0.10