Normal Distribution

Z Critical Values Table (1.645, 1.96, 2.576 and More)

The z-values that cut off a given area in one or both tails of the standard normal distribution: the multipliers behind confidence intervals and z-tests.

  • 50% to 99.9% confidence
  • One- and two-tailed
  • Calculator included
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Z Critical Values

Each row is a confidence level. Two-tailed columns split α between both tails; one-tailed columns put all of α in one tail.

Z Critical Values Table
Two-tailed testOne-tailed test
Confidence levelα (both tails)Critical z (±)α (one tail)Critical z
50%0.50.67450.50.0000
60%0.40.84160.40.2533
70%0.31.03640.30.5244
75%0.251.15030.250.6745
80%0.21.28160.20.8416
85%0.151.43950.151.0364
90%0.11.64490.11.2816
95%0.051.96000.051.6449
96%0.042.05370.041.7507
98%0.022.32630.022.0537
99%0.012.57580.012.3263
99.5%0.0052.80700.0052.5758
99.8%0.0023.09020.0022.8782
99.9%0.0013.29050.0013.0902

Tip: click any value to highlight its row and column.

How to read this table

  1. Find the row for your confidence level (for example 95%).
  2. For a two-sided confidence interval or a two-tailed test, use the two-tailed critical value. Reject H₀ if |z| is larger.
  3. For a one-sided test, use the one-tailed column. Reject H₀ if z is beyond it in the direction of the alternative.

Worked example

A 95% confidence interval for a mean with known σ is x̄ ± z × σ/√n. The 95% row gives the two-tailed critical value 1.9600, the familiar 1.96. With x̄ = 120, σ = 15 and n = 36, the interval is 120 ± 1.9600 × 15/6 = 115.10 to 124.90.

A one-sided test at α = 0.05 uses the one-tailed value in the same row, 1.6449.

Click the underlined links to highlight the value in the table.

Calculator

For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

Two-tailed: z = Φ⁻¹(1 − α/2)  ·  One-tailed: z = Φ⁻¹(1 − α)

Critical values are Φ⁻¹(1 − α/2) for two-tailed use and Φ⁻¹(1 − α) for one-tailed use, computed in double precision and rounded to four decimals.

Frequently asked questions

Why is the 95% critical value 1.96 and not 1.645?

A 95% two-sided interval leaves 2.5% in each tail, so the cut-off is the 97.5th percentile, 1.96. The value 1.645 leaves 5% in a single tail and is used for one-sided 95% bounds and one-tailed tests at α = 0.05.

When should I use t instead of z?

Use a z critical value when the population standard deviation is known or the sample is large. When σ is estimated from a small sample, use the Student t table with the appropriate degrees of freedom.

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