8.6: Interpreting p-values and Significance
- Page ID
- 58929
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Once we’ve completed a hypothesis test and found a p-value, what does that number really tell us?
A p-value is more than a number — it represents a decision point, but also carries the weight of uncertainty.
What is a p-value, really?
Let's review our working definition of a p-value:
Definition: p-value
The p-value is the probability of observing a sample statistic as extreme or more extreme than what we saw — assuming the null hypothesis is true.
It reflects how surprising our result would be if the null hypothesis were actually correct.
A small p-value means our observed result would be very unlikely under the null. A large p-value means the result is quite consistent with the null.
Important: A small p-value does not prove that the null is false. It just suggests the data are inconsistent with the null — and we may want to reject it.
Drawing Conclusions from Your p-value
We compare the p-value to our significance level \( \alpha \) to make a formal decision:
- If \( p \leq \alpha \): reject \( H_0 \) → there's statistically significant evidence for \( H_A \)
- If \( p > \alpha \): fail to reject \( H_0 \) → there's not enough evidence against it
For example, if \( \alpha = 0.05 \) and our p-value is 0.03, we reject the null.
Statistical Significance
If the p-value is less than or equal to our \( \alpha \) level, we say the result is statistically significant.
This means the result is unlikely to be due to chance alone — according to the assumptions of our model.
Limits of p-values: p-hacking and Misuse
P-values are useful — but not magical. Let’s talk about the risks of misusing them.
What is p-hacking?
P-hacking is when someone tries many comparisons or statistical tweaks, only reporting the p-values that look significant — even if those results occurred just by chance.
- Trying dozens of statistical tests until “something sticks”
- Selectively reporting only results with p < 0.05
- Stopping a data collection early because p < 0.05
This creates false positives — appearing to have evidence when there isn’t any. It’s why replicable studies and good design matter.
What can we do?
- Plan your test before collecting data
- Report all results — not just significant ones
- Understand that one significant p-value doesn’t “prove” your claim
Thinking Beyond the P-value
While significance testing is useful, it should never be your only consideration. Always ask:
- Is this result practically significant? (Does the effect matter in the real world?)
- What’s the effect size? (Is the difference meaningful?)
- What are the risks of a Type I or Type II error in context?
- Would this result hold with a larger or different sample?
P-values are just one tool — use with care, context, and critical thinking.
If you find a statistically significant result (p = 0.04), but the difference is only 0.2 points on a 100-point scale, what should you conclude? What other info would you want before acting on that result?