A false alarm and a missed detection.
They sound alike. They aren't the same.

// α, β, and power tend to tangle, don't they

Type I errors. Type II errors. Power. The formulas are not the problem; the moment a real problem shows up, α and β and "reject / fail to reject" and "true / false" all start dancing at once, and which one is which gets hard to keep track of. That kind of moment, surely most readers know it.

StatPlay also has a section that handles these errors. But "α is here, β is here, watch them move" only labels the parts; it doesn't touch the wobble itself. A page like that is easy to scroll past.

This column is an attempt to revisit that wobble in the direction of "sort it into cases." Crossing two axes — "H₀ is true / false" and "reject / fail to reject" — gives four cases. That single move alone may make the whole picture much easier to keep straight. Worth a try, perhaps.

// One axis isn't enough

The reason error questions stay slippery is simple: an error gets talked about as a single probability on one axis.

For instance, "the Type II error is the probability of failing to reject" is a tempting line. Not exactly wrong, but a critical piece is missing — whether the underlying world has H₀ true, or H₁ true.

Work through an actual problem and the answer turns out to come in pairs:

• Type I: in the world where H₀ is true, the probability of rejecting anyway.
• Type II: in the world where H₁ is true, the probability of failing to reject.

An error is therefore determined by two coordinates: the world (H₀ true / H₁ true) and the decision (reject / fail to reject). Drop the world axis, and the description is guaranteed to wobble somewhere.

The next section pins both axes down inside one table.

// The 2×2 matrix — the spine of this column

The whole story collapses into this single table.

The vertical axis is which world is actually true. The horizontal axis is the decision being made. Two of the four cells are correct outcomes; the other two are errors.

Pause for a moment and study the table. The point to notice is that the four probabilities are not summed across the table. Reading along a row gives "in the H₀-true world, fail to reject + reject = 1"; the other row says "in the H₀-false world, fail to reject + reject = 1." In other words, each row sums to 1, and α and β live in different rows.

That "different rows" fact is the next twist.

// Try it — where does each cell sit on the distributions?

The frame is in place: errors are 2×2. But the table alone doesn't show where on the distributions each cell lives. In the mini-visualization below, clicking any cell of the 2×2 lights up the matching region of the distribution. Predict first, then click. Calling the shot in advance — "α is probably here, β over there" — is what makes the rest of the picture readable.

Once the clicks have run, two facts should be visible on the picture:

• α lives in the tails of the blue mountain (the H₀-true world). The purple mountain isn't involved.
• β lives near the center of the purple mountain (the H₁-true world). It's measured on a different mountain.

That picture — "α and β sit on different mountains" — is the topic of the next section.

// α and β are probabilities of different worlds

This is the key turn for the whole column — the point where the case-by-case sort really starts to pay off.

α and β are probabilities defined under different assumptions:

• α is the probability of rejecting in the world where H₀ is true.
• β is the probability of failing to reject in the world where H₁ is true.

So they don't live in the same probability space. Written out it sounds slight, but this carries serious consequences. For instance, the following question now sorts itself:

"What does α + β represent?"

Answer: nothing. The 2×2 table is not built to be summed down its columns. It's built so that each row sums to 1: in the H₀-true world, (1−α) + α = 1; in the H₁-true world, β + (1−β) = 1. Adding numbers across two different worlds doesn't yield an interpretable quantity. Within each world, fail to reject + reject = 1, and that's the relation the table is showing.

The familiar asymmetry between α and β also sorts naturally from here:

• α can be drawn the moment H₀ is fixed (the distribution shape is determined). That's why "α = 0.05" can be committed to before any data arrive.
• β can't be drawn until the position of H₁ is specified — that is, until an effect size δ is assumed.

α is set up front; β only becomes drawable once a δ has been committed to. The asymmetry follows directly from having only one of the two worlds fully specified — the same case-by-case point as before.

// Try it — β requires a δ to even exist

The claim above — "β can't be drawn until H₁ is placed" — is the kind of thing that's faster to feel than to argue. The control below moves α and the effect size δ independently. First move only α and watch what happens to β. Then freeze α and move only δ. The contrast between the two behaviors is what makes which quantity belongs to which world visible.

Pin down what just appeared on the canvas:

• Moving α alone changes the shaded area on the blue mountain. α is the protagonist; β can be treated as essentially unchanged here.
• Moving δ alone slides the purple mountain. β (the unshaded area between the critical values, under purple) changes a lot. α doesn't budge.

So α is a tool of the H₀ world; β is a tool of the H₁ world. They share a screen, but they live in different worlds. The case-by-case sort is showing up directly on the canvas.

// Power is just β read backward — not a new concept

At first, "power" tends to look like a third character alongside α and β. Once the cases are sorted, it isn't.

This is the probability of rejecting, still inside the H₁ world — the other side of β. It's the bottom-right cell of the 2×2. The same fact, just read as "detection rate" instead of "miss rate."

So why does power get its own name? It plugs straight into sample-size planning:

• "Keep β at most 0.2" feels less actionable than "secure power of at least 0.8." Power is a forward-facing number that drops into the formula solving for the required n.
• Sliding α and n inside StatPlay shows directly how power 1−β responds — the next interactive does this.

The takeaway: there is no need to memorize three concepts. The 2×2 is still just four cells. Power is β phrased the other way around.

// Try it — what shifts when you raise n?

One more quantity worth touching: the sample size n. "α and β trade off — push one down, the other goes up" is a common saying. That's only half the story. The full story is that increasing n loosens the trade-off itself. Below, vary only n. α stays fixed (0.05).

At n=30, the same effect size carries a high risk (β) of being judged "no effect." At n=300, that risk drops sharply. That is what "a larger n is needed" is really pointing at. It is the only legitimate way to reduce misses (β) without sacrificing α.

Pin down what showed up on the canvas:

• α doesn't move. The blue area stays at 5%.
• β drops as the mountains thin out and the overlap shrinks.
• Therefore power 1−β rises.

The "α and β are a seesaw" metaphor is half right; the complete picture is that the seesaw's pivot itself moves with n, once everything is sorted by case.

// When you want to feel one more layer

By this point, the 2×2, the world distinction between α and β, the relationship to power, and the role of n should all be lined up.

If you'd like to see how α, β, power, effect size, and n move simultaneously during an actual test, the testing topic has a heavier-duty canvas in its "Two Errors" section.

• Testing topic / Two Errors section →
• Preset effect-size links:
• Open with a small effect (δ=0.5) →
• Open with a moderate effect (δ=2.0) →

This column is the "putting it in order" room; the other visualization is the "playing with all knobs at once" room. Order first, play second — that sequence tends to leave the whole picture in order.

// Frequently asked questions