A coefficient is statistically significant if its p-value is less than 0.05, the conventional threshold used across most fields. This means there’s less than a 5% probability that the result you’re seeing happened by random chance alone. Most statistical software displays this p-value directly in the regression output, making it straightforward to check. But the p-value isn’t the only way to evaluate significance, and understanding what’s happening behind that number will help you interpret your results with more confidence.
What the P-Value Actually Tells You
When you run a regression, each coefficient comes with a p-value that tests a specific question: is this coefficient meaningfully different from zero? The starting assumption, called the null hypothesis, is that the true coefficient equals zero, meaning the variable has no real effect. The p-value tells you how likely you’d be to see your result (or something more extreme) if that assumption were true.
A p-value of 0.03, for example, means there’s only a 3% chance of getting a coefficient this large if the variable truly had no effect. Since 3% is below the 0.05 cutoff, you’d call it statistically significant. A p-value of 0.14, on the other hand, means you can’t rule out that the result is due to chance, so you would not call that coefficient significant.
The 0.05 threshold isn’t arbitrary. It roughly corresponds to values that fall more than two standard deviations from the mean in a normal distribution, which captures about 95% of expected outcomes. Some researchers advocate for a stricter cutoff of 0.005 to reduce false positives, but 0.05 remains the standard in most disciplines.
Reading a Regression Output Table
Statistical software like SPSS, R, Stata, and Excel all produce a coefficient table when you run a regression. The key columns to look at are the coefficient estimate (sometimes labeled “B” or “Coef”), the standard error, the t-statistic, and the p-value (often labeled “Sig.” or “P>|t|”). Some programs also add asterisks next to significant coefficients as a visual shortcut.
The process is simple: compare each coefficient’s p-value to your chosen significance level, typically 0.05. If the p-value is smaller, the coefficient is statistically significant, and you can interpret it as being reliably different from zero. If the p-value is larger, you should not interpret that coefficient as having a real effect. As one widely cited statistics resource from the University of Texas puts it: if a coefficient’s test statistic is not significant, don’t interpret it at all, because you can’t be sure the true value isn’t zero.
For example, if a variable called “female” has a p-value of 0.051, it just barely misses the 0.05 threshold, and you would not reject the null hypothesis that its coefficient equals zero.
How the T-Statistic Works
The p-value you see in the output is calculated from the t-statistic, which is simply the coefficient divided by its standard error. This ratio measures how many standard errors the coefficient sits away from zero. A larger t-statistic means the coefficient is more precisely estimated and further from zero, which produces a smaller p-value.
As a quick rule of thumb, a t-statistic with an absolute value greater than about 2 is typically significant at the 0.05 level. The exact threshold depends on your sample size (specifically, the degrees of freedom), but for any reasonably sized dataset, 2 is a reliable benchmark. You can look up precise critical values in a t-distribution table if needed, but most people simply read the p-value from their software output.
Why Standard Error Matters
The standard error of a coefficient reflects how much uncertainty surrounds your estimate due to the fact that you’re working with a sample, not the entire population. A smaller standard error means your estimate is more precise, which pushes the t-statistic higher and the p-value lower.
This is why sample size has such a large influence on significance. With a large enough sample, even a tiny coefficient can become statistically significant because the standard error shrinks. Conversely, a large and potentially meaningful coefficient can fail to reach significance if your sample is small and the standard error is wide. The same estimated effect will produce different p-values depending on how precisely it’s measured.
Using Confidence Intervals Instead
A 95% confidence interval gives you the same information as a p-value test at the 0.05 level, just presented differently. If the 95% confidence interval for a coefficient does not contain zero, the coefficient is statistically significant. If the interval includes zero, it’s not.
Confidence intervals have an advantage: they show you the plausible range of the true effect, not just whether it’s different from zero. A coefficient of 2.18 with a 95% confidence interval of 1.80 to 2.56 tells you the effect is significant (zero isn’t in the range) and gives you a sense of how large the effect probably is. This is more informative than a p-value alone, which tells you nothing about the size of the effect.
Significant Does Not Mean Important
One of the most common mistakes is treating “statistically significant” as a synonym for “important” or “large.” It is not. Statistical significance only tells you that the observed effect is unlikely to be zero. It says nothing about whether the effect is large enough to matter in practice.
A study with thousands of participants might find that a variable increases an outcome by 0.001 units with a p-value of 0.001. That’s highly significant statistically, but the effect is so small it may be completely irrelevant in the real world. Meanwhile, a study with 30 participants might find an effect of 15 units with a p-value of 0.08. That’s not statistically significant, but the effect size could be practically meaningful if confirmed with more data.
The American Statistical Association has emphasized that a p-value without context provides limited information. A p-value near 0.05, taken by itself, offers only weak evidence against the null hypothesis. And a large p-value doesn’t prove the null hypothesis is true; it just means you don’t have enough evidence to reject it. The primary product of any analysis should be the effect size itself, not the p-value.
A Step-by-Step Checklist
- Find the p-value column in your regression output table. It may be labeled “Sig.,” “P>|t|,” “Pr(>|t|),” or simply “p-value.”
- Compare it to 0.05 (or whatever significance level you chose before running the analysis). If the p-value is less than 0.05, the coefficient is statistically significant.
- Check the t-statistic as a quick confirmation. An absolute value above roughly 2 aligns with significance at the 0.05 level.
- Look at the confidence interval. If the 95% interval does not include zero, that confirms significance.
- Evaluate the coefficient’s size. Ask whether the magnitude of the effect is large enough to be meaningful in your specific context, regardless of the p-value.
Following these steps together gives you a more complete picture than any single number can. A coefficient that passes the significance test, has a tight confidence interval that stays well away from zero, and represents a meaningfully large effect is one you can interpret with real confidence.