Type II Error Rate Calculator
Estimate beta = P(fail to reject H0 when the alternative is true), power = 1 - beta, critical boundaries from alpha, and planning sample sizes for common z-test and t-test mean designs.
🎯Real Study Planning Presets
📝Hypothesis Test Inputs
Z rows use normal formulas. T rows use a clearly labeled noncentral/normal planning approximation.
One-sided tests lose power if the chosen direction conflicts with the alternative.
Common values are 0.10, 0.05, 0.025, and 0.01.
For one-sample tests this is the total sample size.
This is the H0 value or the control mean baseline.
The smallest real difference you want to detect should be entered here.
Use known sigma for z-tests; use a planning SD estimate for t-tests.
Used only by two-sample tests. A balanced design has ratio 1.00.
🔢Current Test Snapshot
📐Critical Values From Alpha and Tails
| Alpha | One-Sided z Boundary | Two-Sided z Boundaries | Meaning for Beta | Common Use |
|---|---|---|---|---|
| 0.10 | 1.282 | ±1.645 | Wider rejection region, lower beta | Screening studies |
| 0.05 | 1.645 | ±1.960 | Standard false-positive control | General testing |
| 0.025 | 1.960 | ±2.241 | Stricter rejection, higher beta | Interim checks |
| 0.01 | 2.326 | ±2.576 | Much harder to reject H0 | High-stakes claims |
| 0.005 | 2.576 | ±2.807 | Power needs larger n | Multiple testing |
| 0.001 | 3.090 | ±3.291 | Beta can become very high | Genome-wide style screens |
⚙Test Design Reference
| Design | Standard Error | Critical Value | Noncentrality Used | Beta Formula Used Here |
|---|---|---|---|---|
| One-sample z | sigma / sqrt(n) | z from alpha | delta / SE | Exact normal shifted by ncp |
| One-sample t | s / sqrt(n) | t with df = n - 1 | delta / SE | Noncentral-t planning, normal approximation |
| Two-sample z | sigma sqrt(1/n1 + 1/n2) | z from alpha | delta / SE | Exact normal shifted by ncp |
| Two-sample t | sp sqrt(1/n1 + 1/n2) | t with df = n1 + n2 - 2 | delta / SE | Noncentral-t planning, normal approximation |
| Two-sided rejection | Same design SE | -c and +c | Signed delta / SE | beta = Phi(c - ncp) - Phi(-c - ncp) |
| One-sided upper | Same design SE | +c | Signed delta / SE | beta = Phi(c - ncp) |
| One-sided lower | Same design SE | -c | Signed delta / SE | beta = 1 - Phi(-c - ncp) |
📊Preset Comparison Grid
| Scenario | Design | Effect d | n / Group | Alpha | Approx Beta | Approx Power |
|---|---|---|---|---|---|---|
| Blood pressure drug | Two-sample t | 0.33 | 80 | 0.05 | 0.44 | 56% |
| A/B revenue lift | Two-sample z | 0.25 | 500 | 0.05 | 0.02 | 98% |
| Battery degradation | One-sample t | -0.45 | 32 | 0.05 | 0.29 | 71% |
| Lab assay shift | One-sample z | 0.50 | 36 | 0.01 | 0.18 | 82% |
| Fill weight check | One-sample z | -0.40 | 45 | 0.05 | 0.24 | 76% |
| Tutoring score gain | Two-sample t | 0.40 | 60 | 0.05 | 0.41 | 59% |
| Call time reduction | Two-sample z | -0.30 | 180 | 0.05 | 0.20 | 80% |
| Process drift | One-sample t | 0.35 | 50 | 0.05 | 0.31 | 69% |
🧮Effect Size and Sample Size Lookup
| Standardized Effect d | Typical Label | One-Sample n for 80% | Two-Sample n / Group for 80% | Planning Note |
|---|---|---|---|---|
| 0.20 | Small | About 199 | About 393 | Easy to miss without large n |
| 0.30 | Small-medium | About 90 | About 175 | Beta remains high in pilots |
| 0.40 | Moderate | About 52 | About 99 | Often feasible in operations |
| 0.50 | Medium | About 34 | About 64 | Classic benchmark effect |
| 0.65 | Medium-large | About 21 | About 39 | Good power with modest n |
| 0.80 | Large | About 15 | About 26 | Usually visible in small studies |
| 1.00 | Very large | About 10 | About 17 | Beta is low unless alpha is strict |
📘Full Formula Breakdown
💡Actionable Power Planning Tips
ℹMethod Notes
a p-value of 0.07, so you didn’t reject the null hypothesis. 05, so you didn’t reject the null hypothesis. But then you start to wonder, maybe my treatment had no effect at all? Maybe I just didn’t design a sensitive enough study to pick up on it.
That’s called a Type II error (or beta), and it’s your uncertainty about whether the result is truly meaningful or not: the probability that you missed a real signal due to an underpowered study. While most people are concerned with alpha (the chance you’ll get a false positive); they tend to overlook the fact that beta. The chance you’ll get a false negative, can also be quite costly.
Why Power Analysis Matters
If you’re in clinical trials, missing a life saving drug is a different type of tragedy than falsely approving a placebo. And in business, wasting market share by launching a product that really does work but flunked a weak test isn’t any better.
The calculator above will crunch the numbers for you, but it’s when you know what they mean that real decision-making comes into play. One minus beta is simply power. Wanting 80% power means taking on a 20% probability of having missed true effect size.
Sounds good, until you remember that one study in every ten of a truly effective intervention is going to appear as noise. The sample size could of been too low; variability could have drowned out any effect. That’s part of the reason we care so much about effect size.
Effect size isn’t just about magnitude of difference but about the magnitude of the difference relative to the noise present in the data. That tiny change in average blood pressure could be clinically important, but if there’s a lot of variability from reading to reading then you’ll miss the change because of the static. More people was needed to drown out the static.
How much do you want to change? This is a question that needs an honest answer before you choose your inputs. The alternative mean represent the smallest difference you care about detecting. Set it too high and you may easily detect significance, but at the expense of finding smaller (but still important) effects. Better to plan for a modest, yet useful, change rather than a miraculuos one.
Similarly, the standard deviation input is critical. A rough guess here can greatly skew your sample size calculation. Underestimate the variance and your study will be underpowered; your beta will be higher then you think. The page’s reference table makes this clear as it shows how your sample size must jump if your effect sizes drops from large to small.
There is such a thing as alpha versus beta too. Yes, it’s cool that you tightened up alpha to 0.01 (making your test harder to trick), but this narrows down the region where effects will lead to rejection, making it less likely to detect a genuine effect. You raised the bar; now you’ll miss more jumps.
So the need for power analysis isn’t simply a token gesture for ethics boards. It is a juggling act between feasibility and rigor.
If you’re pretty sure how an effect should go (i.e., in one direction rather than another), then you can increase power with one-sided tests. However, if you guessed wrong and the effect goes the other way, then you’re screwed. Most researchers sticks to two-sided tests because caution is typically cheaper than embarrassment.
Preparing for beta allows you to respect the fact that randomness exists. Variability will happen. That’s okay, that’s not your job; it’s to plan for it. If you run through these scenarios without collecting any data at all, then you’re choosing how much evidence you want to see before believing something.
This prevents you from embarrassing yourself by missing an opportunity because your test came back with no result. This helps you create a study that’ll realy answer the question you’re curious about. This is a little upfront work that saves you months of spinning your wheels.
There’s nothing complicated about the numbers. What sets apart good research from guessing? It’s the discipline to apply this correctly. Set the power. Define the effect. Control the sample size. Everything else is probability working as designed.
