A true p-value cannot be negative. P-values are probabilities, and probabilities always fall between 0 and 1. If you’ve encountered what looks like a negative p-value, you’re almost certainly looking at one of a few common situations: a negative log transformation, a negative test statistic being confused with the p-value itself, or a software error. Understanding which one applies to your case clears up the confusion quickly.
Why P-Values Are Always 0 or Above
A p-value represents the probability of seeing results as extreme as yours (or more extreme) if there were truly no effect or no difference in what you’re studying. It answers the question: “How likely is this outcome under pure chance?” Since it’s a probability, it’s mathematically bounded between 0 and 1. A p-value of 0.03 means there’s a 3% chance of observing your result if the null hypothesis were true. A p-value of 0.50 means a coin-flip level of likelihood. There is no scenario in standard statistics where a probability dips below zero.
The American Psychological Association’s reporting guidelines reinforce this: researchers report exact p-values to two or three decimal places (like p = .006 or p = .03), and when values get extremely small, they simply write p < .001 rather than reporting tinier decimals. The scale always stays positive.
The Negative Log Transformation
The most common reason people see negative numbers associated with p-values is the negative log transformation, written as -log10(p). This is a standard technique in genomics, bioinformatics, and other fields that deal with thousands of statistical tests at once.
Here’s what it does: it flips the p-value scale so that smaller (more significant) p-values become larger numbers. A p-value of 0.05 becomes about 1.3 on the -log10 scale. A p-value of 0.001 becomes 3. A p-value of 0.0001 becomes 4. This makes it much easier to spot significant results in large datasets because the most important data points rise to the top of a chart instead of clustering near zero.
Volcano plots, which are widely used in gene expression studies, put -log10(p) on the vertical axis and the size of the effect on the horizontal axis. As one review in Briefings in Bioinformatics describes it, the higher a data point sits on the y-axis, the smaller (more significant) its original p-value. If you’re reading a paper or looking at software output that shows these transformed values, what you’re seeing isn’t a negative p-value. It’s a mathematical conversion designed to make visualization easier. The underlying p-value is still positive.
Negative Test Statistics vs. P-Values
Another common source of confusion is mixing up a test statistic with the p-value. Many statistical tests produce a test statistic (like a t-value, z-score, or correlation coefficient) that can absolutely be negative, and then separately produce a p-value that cannot.
Correlation coefficients, for example, range from -1 to +1. A negative correlation means that as one variable increases, the other decreases. But the p-value attached to that correlation only tells you how likely that relationship is to have appeared by chance. It says nothing about direction. A correlation of -0.75 with a p-value of 0.002 means a strong inverse relationship that is very unlikely to be a fluke. The negative sign belongs to the correlation, not the p-value.
The same applies to t-tests. A negative t-statistic simply indicates the direction of the difference between groups (Group A scored lower than Group B, for instance). The p-value derived from that t-statistic remains between 0 and 1. If you’re scanning a results table and see a negative number next to a very small positive number, the negative number is probably the test statistic and the positive number is the p-value.
Software Output That Looks Negative
Some statistical software packages display p-values in scientific notation, which can look confusing at first glance. A value like 3.2E-05 means 0.000032, not negative 5. The “E-05” part indicates the decimal point has moved five places to the left. This is still a valid, positive p-value.
In rare cases, software bugs or floating-point rounding errors in programming can produce a p-value that displays as a tiny negative number, like -2.2e-16. This is a computational artifact, not a real probability. It effectively means the p-value is so close to zero that the computer’s rounding produced a slightly negative number. Treat it as approximately zero.
What to Do When You See One
If you encounter an apparently negative p-value, check three things. First, look at the axis label or column header. If it says “-log10(p)” or “log p-value,” you’re looking at a transformed scale, not a raw p-value. Second, check whether the negative number is actually the test statistic (t, z, r) rather than the p-value itself. These are often reported side by side in tables, and it’s easy to read the wrong column. Third, if the value is something like -1.5e-16, it’s a rounding glitch in the software and functionally equals zero, meaning the result is highly significant.
In every case, the conclusion is the same: the actual p-value is not negative. What you’re seeing is either a transformation, a neighboring statistic, or a computational quirk. Once you identify which one, the interpretation of your results stays straightforward.