<\/span><\/h2>\nThe p-value is the probability of observing a result as extreme as, or more extreme than, what you actually observed, assuming the null hypothesis is true<\/strong>.<\/p>\nBreaking that down:<\/p>\n
\n- Null hypothesis (H\u2080)<\/strong> \u2014 the default assumption of “no effect” or “no difference” (e.g., “this coin is fair,” “this new drug has no effect compared to placebo”)<\/li>\n
- Extreme result<\/strong> \u2014 a result as unusual or more unusual than what you observed<\/li>\n
- Assuming H\u2080 is true<\/strong> \u2014 this is the critical, often-missed condition: the p-value is calculated under the assumption<\/em> that there’s nothing going on, not as a measure of whether that assumption is actually correct<\/li>\n<\/ul>\n
<\/span>What a P-value Is NOT<\/span><\/h2>\nThis is where most confusion happens. A p-value is not<\/strong>:<\/p>\n\n- The probability that the null hypothesis is true<\/strong> \u2014 a p-value of 0.03 does not mean “there’s a 3% chance there’s no real effect.” It means “if there were no real effect, you’d see a result this extreme only 3% of the time.”<\/li>\n
- The probability that your results happened by chance<\/strong> \u2014 similar misconception, same underlying error<\/li>\n
- A measure of effect size<\/strong> \u2014 a very small p-value doesn’t necessarily mean a large or important effect; with a large enough sample size, even a tiny, practically meaningless difference can produce a very small p-value<\/li>\n
- The probability that you’d get the same result if you repeated the study<\/strong> \u2014 that’s a different concept (related to statistical power and replication), not what the p-value measures<\/li>\n<\/ul>\n
<\/span>The Significance Threshold (Alpha)<\/span><\/h2>\nResearchers typically set a threshold, called alpha (\u03b1)<\/strong>, before running a study \u2014 commonly 0.05 \u2014 to decide how surprising a result needs to be before rejecting the null hypothesis.<\/p>\n\n- If p \u2264 \u03b1<\/strong> (commonly p \u2264 0.05) \u2014 the result is considered “statistically significant,” meaning the observed data would be quite unlikely if the null hypothesis were true<\/li>\n
- If p > \u03b1<\/strong> \u2014 the result is not statistically significant; this does not prove the null hypothesis is true, only that this particular study didn’t find strong enough evidence against it<\/li>\n<\/ul>\n
The 0.05 threshold is a convention, not a law of nature \u2014 some fields use stricter thresholds (like 0.01 or even 0.001), especially when false positives are particularly costly (e.g., certain areas of medical research or physics).<\/p>\n
<\/span>Worked Example<\/span><\/h2>\nSuppose a company claims a new studying technique improves test scores. You run a study comparing students using the new technique against students using a standard method, and find the new-technique group scored 4 points higher on average.<\/p>\n
\n- Null hypothesis (H\u2080):<\/strong> There’s no real difference between the two techniques; any observed difference is due to random variation between students<\/li>\n
- Run the statistical test<\/strong> (e.g., a t-test) comparing the two groups<\/li>\n
- Result:<\/strong> the test produces p = 0.02<\/li>\n<\/ol>\n
Interpretation:<\/strong> If the new technique truly had no effect, you’d see a difference this large (or larger) only 2% of the time just from random variation between students. Since 0.02 < 0.05, this result is considered statistically significant \u2014 evidence against the null hypothesis, suggesting the technique likely has some real effect.<\/p>\nWhat this does not<\/strong> tell you: exactly how much better the technique is in practice, whether it’s worth the cost\/effort to implement, or that there’s a 98% chance the technique works. It only tells you the observed difference would be unlikely under the “no real effect” assumption.<\/p>\n<\/span>Statistical Significance vs Practical Significance<\/span><\/h2>\nA result can be statistically significant without being practically meaningful. With a large enough sample size, even a trivial difference (say, an average test score improvement of 0.1 points) can produce a very small p-value. This is why researchers increasingly report effect sizes<\/strong> alongside p-values \u2014 a measure of how large the difference actually is, not just how unlikely it is to be due to chance.<\/p>\n\n
\n\n\n| <\/th>\n | Statistical significance<\/th>\n | Practical significance<\/th>\n<\/tr>\n<\/thead>\n |
\n\n| Question answered<\/td>\n | Is this result unlikely to be pure chance?<\/td>\n | Is this result actually large enough to matter?<\/td>\n<\/tr>\n |
\n| Affected by sample size<\/td>\n | Yes, heavily<\/td>\n | No (effect size is independent of sample size)<\/td>\n<\/tr>\n |
\n| Example<\/td>\n | p = 0.001 with a tiny, trivial effect<\/td>\n | A large, meaningful effect regardless of p-value<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/span>Common Student Mistakes<\/span><\/h2>\n\n- Interpreting p-value as “probability the null hypothesis is true”<\/strong> \u2014 the most common error, and technically backwards from what the p-value actually measures<\/li>\n
- Treating p = 0.05 as a hard scientific cutoff<\/strong> \u2014 it’s a conventional threshold, and results just above or below it (p = 0.048 vs p = 0.052) aren’t meaningfully different in practice, despite one being “significant” and the other not<\/li>\n
- Assuming statistical significance means practical importance<\/strong> \u2014 always check effect size alongside the p-value<\/li>\n
- “P-hacking”<\/strong> \u2014 running many tests or repeatedly checking data until a p-value below 0.05 appears by chance, which inflates the risk of false positives; this is considered poor research practice<\/li>\n<\/ul>\n
<\/span>Frequently Asked Questions<\/span><\/h2>\nWhat does it mean if a p-value is very small, like 0.001?<\/strong> It means that, assuming the null hypothesis were true, a result this extreme would be very rare \u2014 happening only about 0.1% of the time by chance. This is typically strong evidence against the null hypothesis, though it still doesn’t tell you the size or practical importance of the effect.<\/p>\nIs a p-value of 0.05 the same as a 95% chance the result is real?<\/strong> No \u2014 this is one of the most common misinterpretations. A p-value of 0.05 means that if the null hypothesis were true, you’d see a result this extreme 5% of the time by chance. It says nothing directly about the probability that the null hypothesis itself is true or false.<\/p>\nWhy is 0.05 the standard threshold?<\/strong> It’s a widely adopted convention, originally popularized by statistician Ronald Fisher, rather than a mathematically derived “correct” cutoff. Different fields and different types of research use stricter or more lenient thresholds depending on the cost of false positives.<\/p>\nCan a study have a high p-value but still show a meaningful effect?<\/strong> Yes \u2014 a high p-value might simply reflect a small sample size that doesn’t provide enough statistical power to detect a real effect, rather than proving there’s no effect at all. This is why non-significant results shouldn’t automatically be treated as proof of “no effect.”<\/p>\n","protected":false},"excerpt":{"rendered":"A p-value tells you how surprising your data would be if there were actually no real effect happening \u2014 that […]<\/p>\n","protected":false},"author":2,"featured_media":2614,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_seopress_robots_primary_cat":"none","_seopress_titles_title":"P-value Explained Simply (With a Worked Example)","_seopress_titles_desc":"Understand what a p-value actually means, common misconceptions, the 0.05 threshold, and statistical vs practical significance, with a worked 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