When the p-value is greater than alpha, the result is not statistically significant, and you fail to reject the null hypothesis. In practical terms, this means the data you collected did not provide strong enough evidence to conclude that an effect or difference exists. If you set alpha at the common threshold of 0.05 and your p-value comes back as 0.12, for example, you cannot claim your result is statistically significant.
That sounds straightforward, but the interpretation carries important nuances that trip up students, researchers, and even published scientists. What you can and cannot conclude from a high p-value matters more than the binary significant/not-significant label.
What the Decision Actually Means
Alpha is the threshold you choose before running your test. It represents how much risk of a false positive you’re willing to tolerate. The most widely used value across medicine, psychology, economics, and most empirical sciences is 0.05, meaning you accept a 5% chance of incorrectly declaring a result significant when nothing is really going on.
The p-value is what your data produce. It tells you the probability of seeing results at least as extreme as yours if the null hypothesis were true. When the p-value lands above alpha, the data are consistent with what you’d expect under the null hypothesis. The standard move is to “fail to reject” the null hypothesis rather than “accept” it, and that phrasing is deliberate.
Why “Fail to Reject” Instead of “Accept”
This is the single most misunderstood point in introductory statistics. Failing to reject the null hypothesis is not the same as proving the null hypothesis is true. You haven’t shown that there’s no effect. You’ve shown that your data weren’t strong enough to rule the null hypothesis out.
The reason is mathematical: a strict null hypothesis states that the true effect equals exactly zero (or some other single value). To prove that, you would need an estimate with zero bias and infinite precision. That’s impossible with real data. A 2025 paper in a methods journal put it bluntly: proving the null hypothesis would require “divine knowledge” about the true value. Statistical tests simply aren’t built to confirm that nothing is happening. They’re built to detect when something is.
A classic BMJ editorial captured this with a phrase worth remembering: “absence of evidence is not evidence of absence.” The authors argued it is never reasonable to claim a study has proved no effect exists, because some uncertainty will always remain. The distinction matters in real decisions. Imagine a trial testing whether an intervention reduces HIV transmission produces a non-significant result, but the confidence interval is compatible with anything from a 40% reduction to a 50% increase. Saying “the intervention doesn’t work” would be misleading. The study simply didn’t have enough information to tell.
The Role of Statistical Power
A p-value above alpha doesn’t always mean the effect isn’t real. It can also mean your study wasn’t powerful enough to detect it. Statistical power is the probability that your test will correctly identify a true effect, calculated as 1 minus the probability of a Type II error (a false negative). A power of 0.80 is the conventional minimum, meaning an 80% chance of catching a real effect if one exists.
Two things most commonly drag power down: small sample sizes and small effect sizes. For the same observed effect and the same variability in your data, the p-value decreases as sample size increases. So a study with 30 participants might produce a p-value of 0.09 for the exact same effect that a study with 300 participants would flag at 0.001. The underlying reality hasn’t changed. Only the study’s ability to detect it has.
Research on underpowered studies reveals a troubling pattern. As power drops, results tend to fall into two buckets: accurate but non-significant, or significant but inaccurate (exaggerated or even pointing in the wrong direction). Neither outcome is useful. If your p-value is greater than alpha and you suspect low power might be the reason, the honest conclusion is that your study was inconclusive, not that the effect doesn’t exist.
Type II Errors: The False Negative Risk
Every time you fail to reject the null hypothesis, there’s a chance you’re making a Type II error: concluding there’s no significant effect when one actually exists. The probability of this error is called beta. While alpha (the Type I error rate) is something you control by choosing your significance threshold, beta depends on your sample size, the true size of the effect, and the variability in your data.
Type I and Type II errors sit on opposite ends of a seesaw. Lowering alpha to be more cautious about false positives (say, from 0.05 to 0.01) makes it harder to reach significance, which raises the risk of a false negative unless you compensate with a larger sample. This tradeoff is worth keeping in mind: a stricter alpha doesn’t make your conclusions more reliable if it just shifts the errors from one type to another.
Non-Significant Does Not Mean Unimportant
Statistical significance and practical importance are different things. Statistical significance is a function of sample size, effect size, and variability. Practical or clinical significance asks a different question: does this result matter in the real world?
A treatment might lower blood pressure by a meaningful amount, but if the study was too small, the p-value won’t cross the alpha threshold. Conversely, an enormous study can produce a tiny, meaningless difference that clears statistical significance easily. The p-value alone can’t tell you whether a finding is worth acting on.
The American Statistical Association released a formal statement warning against over-reliance on the significance threshold. One of its core principles: scientific conclusions and policy decisions should not be based only on whether a p-value passes a specific cutoff. A large p-value is not evidence that your alternative hypothesis is wrong, because many different hypotheses could be consistent with the observed data. The ASA recommends looking at effect sizes and confidence intervals alongside the p-value to get the full picture.
What to Do With a Non-Significant Result
If your p-value is greater than alpha, here’s how to think through it rather than just stamping “not significant” on the result and moving on:
- Check the confidence interval. A wide confidence interval that includes both meaningful positive and negative effects means your study was inconclusive. A narrow interval hovering around zero is much stronger evidence that any true effect is small.
- Look at the effect size. Even without statistical significance, the estimated size of the effect tells you something. A large estimated effect paired with a wide confidence interval suggests the study needed more participants, not that the effect is absent.
- Consider sample size and power. If your study had fewer than the number of participants a power analysis would have recommended, a non-significant result is expected regardless of whether the effect is real.
- Report the actual p-value. Writing “p = 0.07” gives readers far more information than “not significant.” A p-value of 0.06 tells a very different story than a p-value of 0.85, even though both sit above the 0.05 cutoff.
The core takeaway is that a p-value above alpha closes one door (you can’t claim statistical significance) but doesn’t open another (you can’t claim the null hypothesis is true). The result is a statement about what your data can support, not a statement about reality. Treating non-significance as proof of no effect has led to useful interventions being abandoned and harmful ones being dismissed as harmless, which is why careful interpretation matters far more than the binary label.