Power vs Type II error (power = 1 - beta)

Power vs Type II error: the short answer

Both describe what happens when the null hypothesis $H_0$ is false. A Type II error is failing to reject it anyway, and its probability is $\beta$ ("beta"). Power is the other outcome, correctly rejecting a false $H_0$, with probability $1 - \beta$. The two probabilities add to 1 because the two outcomes are the only two available. There is one quantity here, not two.

So the interesting part is not the difference. It is the qualifier both of them carry, and it is the one students drop. Neither $\beta$ nor power is a property of a test on its own. Each is computed by naming one specific value the parameter might really have, and each changes the instant you name a different one. The same test has one power against $\mu = 105$ and a different power against $\mu = 108$. A power quoted with no alternative value attached is not a number anyone can check.

Neither one is a single number for a test

Take the z test the site's power visualizer runs, chosen because the population standard deviation is known, so nothing is estimated on the way to the answer and every number below can be checked against a z table.

Test $H_0: \mu = 100$ against $H_a: \mu > 100$, with $\sigma$ ("sigma") known to be 15 and $n = 25$. The standard error is $15/\sqrt{25} = 3$, and at $\alpha = 0.05$ the test rejects once the sample mean $\bar{x}$ ("x-bar") clears $100 + 1.6449(3) = 104.9347$. That $z^*$ is the four-decimal critical value; a z table prints it as 1.645, which shifts the cutoff to 104.935 and every probability below by less than 0.0001.

Nothing in that setup mentions the truth. Supply one and the two numbers appear.

The test never changed. The cutoff sat at 104.9347 in every row, $\alpha$ stayed at 0.05, and $n$ stayed at 25. Only the assumed truth moved, and power went from the significance level itself to near certainty. Read the last two columns and you have read power and $\beta$ at the same time, which is the whole relationship.

One caution about the first row. The right-hand column is $\beta$ in every row except that one, where $\mu = 100$ makes $H_0$ true. A Type II error cannot happen when the null is correct, so 0.9500 there is the chance of a correct decision, not an error rate.

The two boundaries that catch people out

Two rows of that table deserve their own note, because both contradict a rule students invent.

Power equals $\alpha$ when the truth sits at the null value. At $\mu = 100$ power reads 0.0500, exactly the significance level. With no effect to detect, the only rejections still available are Type I errors, so the rejection rate is $\alpha$ by construction. Change $n$ to 10 or to 200 and that row does not move.

Power can fall below $\alpha$. This test looks upward only, so it is blind to a true mean under 100. Against $\mu = 95$ its power is 0.0005, roughly a hundredth of $\alpha$. The two-sided version of the same test cannot do that: with $\alpha$ split across both tails the power curve bottoms out at 0.0500 when $\mu = 100$ and climbs in both directions, reaching 0.3848 against $\mu = 95$. The two-sided test pays for that coverage elsewhere, since against $\mu = 105$ its power is 0.3848 rather than 0.5087. That is one concrete price of the one-sided or two-sided decision.

The four levers, and which way each one pushes

Four things move power. Every direction below was recomputed one lever at a time, holding the other three fixed, from the same baseline: the test above aimed at a true mean of 105, where power is 0.5087 and $\beta$ is 0.4913.

The cost column is where they differ, and it is what a good exam answer names. Two of them buy power outright, and they buy it the same way: raising $n$ and lowering $\sigma$ both shrink the standard error while leaving $\alpha$ exactly where you set it. Raising $\alpha$ is a trade, not a gain: it lowers $\beta$ and raises the false-alarm rate in the same move, which is the judgment covered in how to choose a significance level. The distance from the null is not a lever at all, only a fact you discover afterward. The $\sigma$ route is the one students forget on free-response: a tighter measurement process, or a design that strips out a source of variation, raises power with no extra data and no extra false alarms; see standard error vs standard deviation for why those two are not the same quantity.

Say which lever you mean. "Increasing the sample size increases power" is true only with the other three held fixed, and dropping that qualifier is the most common way a correct sentence turns into a wrong one.

Power vs Type II error side by side

Wherever the two columns differ they are describing one event from opposite ends, and the two numeric rows differ by nothing more than a subtraction. The rest of the rows are shared outright, because these are one event counted twice. In practice the choice between the words is just whichever one the question asks you to name.

The mix-ups worth killing

Five sentences to stop writing, each with the correction attached.

• "Power is 0.5087, so the null is about 51 percent likely to be false." Power says nothing about whether $H_0$ is false. It assumes a stated alternative value is already the truth and reports how often the test would notice.
• "Alpha and beta add to 1." They do not. In the baseline test they are 0.05 and 0.4913, which sum to 0.5413 and mean nothing. Power and $\beta$ are the complements; $\alpha$ and $\beta$ are unrelated numbers about two different states of the world.
• "My p-value was 0.31, so there is a 31 percent chance I made a Type II error." A p-value is computed assuming $H_0$ is true, and under a true $H_0$ a Type II error cannot occur at all. $\beta$ is never read off your data; see what a p-value means.
• "This test has a power of 0.80." Against what? Without the alternative value the sentence carries no information, and the table above is why: the same test ran from 0.0500 to 0.9543 with nothing changed but the assumed truth.
• "We failed to reject, so there is no effect." A failure to reject means the evidence was not convincing, not that $H_0$ is true. A low-power test fails to reject constantly even when an effect is real, which is exactly why "no convincing evidence" and "no effect" are different claims. The full reasoning is in Type I vs Type II error and power.