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Formula used
This calculator converts a significance level α into a critical value z for the standard normal distribution. Let Φ(z) be the normal CDF and Φ⁻¹(p) its inverse.
- Two-tailed: split alpha into both tails, so p = 1 − α/2, then z = Φ⁻¹(p) and the critical region is ±z.
- One-tailed (right): p = 1 − α, so z = Φ⁻¹(1 − α) (positive).
- One-tailed (left): p = α, so z = Φ⁻¹(α) (negative).
- Confidence level: 1 − α.
How to use this calculator
- Enter your alpha (α) as a proportion or percent.
- Select one-tailed or two-tailed based on your test.
- For one-tailed, choose left or right direction.
- Pick decimal places for rounding the reported z value.
- Optionally paste several alphas to compute a batch table.
- Click Calculate to show results above the form.
- Use Download CSV or Download PDF for exporting.
Example data table
| Alpha (α) | Two-tailed critical z (±) | One-tailed right z | One-tailed left z |
|---|---|---|---|
| 0.10 | ±1.6449 | 1.2816 | -1.2816 |
| 0.05 | ±1.9600 | 1.6449 | -1.6449 |
| 0.01 | ±2.5758 | 2.3263 | -2.3263 |
1) Alpha and Type I error
Alpha (α) is the probability of a Type I error: rejecting a true null hypothesis. In practice, α controls how extreme a standardized statistic must be before you call the result statistically significant. Smaller α means stricter evidence and larger critical values.
2) Tail choices and meaning
A tail describes where the rejection region sits on the normal curve. A two-tailed test splits α into both ends, because departures in either direction matter. A one-tailed test places all α in a single end, used when only unusually large or unusually small outcomes are meaningful.
3) Converting alpha into a probability p
To convert α into a z cutoff, first convert α into a cumulative probability p for the standard normal CDF Φ(z). For two-tailed tests, p = 1 − α/2. For a right-tailed test, p = 1 − α. For a left-tailed test, p = α.
4) The inverse normal quantile
The critical z value is the inverse normal quantile: z = Φ⁻¹(p). This is the point where the area to the left equals p. Because Φ(z) increases steadily, higher p produces higher z, so stricter α yields larger cutoffs.
5) Common reference values
With α = 0.05, the two-tailed cutoff is ±1.9600, while the one-tailed right cutoff is 1.6449. With α = 0.10, the two-tailed cutoff is ±1.6449 and one-tailed right is 1.2816. With α = 0.01, the two-tailed cutoff is ±2.5758 and one-tailed right is 2.3263.
6) Confidence level links
Confidence level is 1 − α and appears in interval estimation. For example, α = 0.05 corresponds to 95% confidence and α = 0.01 corresponds to 99% confidence. The same z values show up in margins of error where margin = z · (standard error).
7) Direction and sign
Sign matters only for one-tailed direction. A left-tail cutoff is negative (for example, α = 0.05 gives z ≈ −1.6449), while a right-tail cutoff is positive. Two-tailed reporting usually uses ±z because both ends are symmetric.
8) Practical notes for use
If you already have a p-value, you may compare it directly to α. Critical values are useful for reporting rejection rules, drawing rejection regions, and converting confidence levels into cutoffs. In batch mode, enter several α values (such as 0.20, 0.15, 0.025) to build a quick reference table. Tiny differences versus printed tables can occur.
FAQs
1) What is a z critical value?
Z critical value is the cutoff on a standard normal curve that defines a rejection region. If your z statistic goes beyond this cutoff, the result is significant at the chosen alpha level.
2) What alpha should I use for 95% confidence?
Use alpha = 0.05 for a 95% confidence level. For a two-tailed interval, the common critical value is about ±1.9600. For a one-tailed cutoff at alpha 0.05, it is about 1.6449.
3) Why does two-tailed use alpha/2?
Because significance must cover both extremes. A two-tailed test rejects for unusually large or unusually small values, so the total alpha is split evenly: alpha/2 in each tail.
4) Can I use this tool for p-values?
Not directly. This tool converts alpha into a cutoff. If you already have a p-value, compare the p-value to alpha. If you have a z statistic, compare it to the critical z.
5) When should I use t instead of z?
If your population standard deviation is unknown and the sample is small, use a t distribution and a t critical value with degrees of freedom. For large samples, z and t values are often very close.
6) Why is the left-tail z negative?
The left tail corresponds to small cumulative probabilities p. Since the normal curve is centered at zero, probabilities below 0.5 occur for negative z, so a left-tail cutoff is negative.
7) How many decimals should I keep?
Four decimals are usually enough for reporting. If you will reuse the z value inside other calculations, keep more precision during computation and round only in the final displayed result.