Z Critical Value Calculator

Find the z critical value for a confidence level or significance level, with one-tailed and two-tailed rejection regions.

Enter lookup settings

Fixed to the standard normal distribution. Results update as you change the tail type or input value.

Tail type

Input mode

For a two-tailed lookup, the confidence level is the central area and alpha is split equally across both tails.

Common confidence levels

\mathrm{CL}

Confidence level

Enter a percent greater than 0 and less than 100.

Critical value result

$z_1=z_{\alpha/2}$ Lower critical value

-1.959964

$z_2=z_{1-\alpha/2}$ Upper critical value

1.959964

\alpha

0.05

\alpha/2

0.025

2.5%

1-\alpha

95%

Confidence level

Distribution

Z\sim N(0,1)

standard normal

\Phi^{-1}(p)

0.975

p=1-\alpha/2

Rejection region

Reject H0 when z is less than or equal to the lower critical value or greater than or equal to the upper critical value.

Normal curve with shaded alpha

Values are rounded for display; use the full precision in the formulas when comparing borderline test statistics.

Step-by-Step Lookup

Convert the input to alpha

\alpha = 1-\frac{95}{100} = 0.05

Assign alpha to the tail or tails

\frac{\alpha}{2}=\frac{0.05}{2}=0.025

Identify the required quantile

z^{*}=\Phi^{-1}\left(1-\frac{\alpha}{2}\right)=\Phi^{-1}\left(0.975\right)=1.95996399

Read the critical value

z_1=\left(-1.95996399\right),\ z_2=1.95996399

What a z critical value means

A critical value is a cutoff on the standard normal curve. In hypothesis testing, it defines the rejection region: if the test statistic falls beyond the cutoff, the result is unusual enough under the null hypothesis to reject $H_0$ . In confidence intervals, the critical value sets how many standard errors are needed on each side of the estimate.

For a z critical value, the calculator uses the standard normal distribution $Z \sim N(0,1)$ and performs an inverse lookup. Instead of asking for the probability to the left of a known z-score, it starts with an area such as $\alpha=0.05$ or a confidence level such as $95\%$ and finds the z-score that creates that area.

Alpha versus confidence level

The significance level $\alpha$ is the probability assigned to the rejection region. A $5\%$ significance level means $\alpha=0.05$ .

The confidence level is the central coverage used for confidence intervals, usually written as $1-\alpha$ . A $95\%$ confidence level corresponds to:

\alpha = 1 - 0.95 = 0.05

For two-tailed critical values, that total alpha is split equally:

\alpha_{\text{per tail}} = \frac{\alpha}{2}

For one-tailed critical values, the full alpha stays in the selected tail.

One-tailed and two-tailed critical values

In a right-tailed test, the rejection region is in the high end of the curve, so the calculator finds:

z_{\alpha} = \Phi^{-1}(1-\alpha)

In a left-tailed test, the rejection region is in the low end of the curve:

z_{\alpha} = \Phi^{-1}(\alpha)

In a two-tailed test, the rejection region is split between both ends:

\pm z^{*} = \pm \Phi^{-1}\left(1-\frac{\alpha}{2}\right)

That is why a $95\%$ two-tailed z critical value is about $\pm 1.96$ , while a $95\%$ one-tailed lookup uses about $1.645$ for the right tail or $-1.645$ for the left tail.

How to use this calculator

Choose the tail type: two-tailed, right-tailed, or left-tailed.
Choose whether your input is a confidence level or significance level $\alpha$ .
Use a preset confidence level or enter a custom value.
Read the critical value and rejection-region statement.
Compare your test statistic with the critical value or bounds.

For example, with a two-tailed test and $\alpha=0.05$ , the rejection rule is $z \le -1.96$ or $z \ge 1.96$ . With a right-tailed test and $\alpha=0.05$ , the rejection rule is $z \ge 1.645$ .

How to read the shaded alpha region

The shaded part of the curve is the rejection area. In a two-tailed test, the calculator shades both tails because extreme values in either direction count against the null hypothesis. In a right-tailed test, only the right tail is shaded. In a left-tailed test, only the left tail is shaded.

The vertical marker is the boundary between the non-rejection area and the rejection area. A test statistic beyond the marker is in the shaded alpha region.

Critical values in confidence intervals

For a two-sided z confidence interval, the same two-tailed lookup is used. A $95\%$ interval leaves $\alpha=0.05$ outside the interval, with $0.025$ in each tail, so the critical value is $z^* \approx 1.96$ . The margin of error is then:

\text{margin of error} = z^* \times \text{standard error}

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