Z Alpha (zα and zα/2): Meaning, Values and How to Find It

Z alpha (written zα, or z_α) is the critical value of the standard normal distribution that has an area of α in its upper tail. For the common two-sided case you use zα/2, which cuts off α/2 in each tail. At α = 0.05 the one-sided value zα is 1.645 and the two-sided value zα/2 is 1.960. The two are easy to confuse, and different books index them differently, so this guide gives the values, the notation clash, and a worked example with the arithmetic shown.

What is z alpha? Definition

Z alpha is a point on the z axis of the standard normal distribution (mean 0, standard deviation 1). Formally, zα is the number for which P(Z > zα) = α. Because the standard normal curve is symmetric, the matching lower-tail point is −zα. Statisticians call it a critical value because a test statistic beyond it falls in the rejection region, and because it sets the width of a confidence interval.

The α here is the significance level, the probability you accept of a Type I error. If you need a refresher on the symbol itself, see what alpha means in statistics.

Z alpha vs z alpha/2: one-tailed vs two-tailed

Use zα when all of α sits in one tail (a one-tailed test). Use zα/2 when α is split between two tails (a two-tailed test or a two-sided confidence interval). Splitting α is why the two-sided cut-off is larger: at α = 0.05, each tail holds 0.025, so the cut-off moves out from 1.645 to 1.960. The one-tailed vs two-tailed comparison explains when each design is justified.

Notation clash: zα/2 versus z1−α/2

Many textbooks index z by the upper-tail area, so zα/2 is positive and about 1.96 at α = 0.05. The NIST/SEMATECH e-Handbook of Statistical Methods instead indexes by the cumulative (lower-tail) probability: its 100(1−α)% confidence interval for a mean with known σ is Ȳ ± z_1−α/2 · σ/√N, where z_1−α/2 is “the 1−α/2 critical value of the standard normal distribution”. Both notations name the same number.

Textbook (upper-tail) notation	Lower-tail (NIST-style) notation	Value at α = 0.05
zα	z1−α	1.645
zα/2	z1−α/2	1.960

Formula and what each symbol means

The two-sided confidence interval for a population mean with known standard deviation is:

x̄ ± zα/2 × σ/√n

• x̄: the sample mean.
• zα/2: the critical value that leaves α/2 in the upper tail. The confidence level is 100(1 − α)%.
• σ: the known population standard deviation.
• n: the sample size, so σ/√n is the standard error of the mean.

For an upper-tailed z test, reject H₀ when the test statistic z = (x̄ − μ₀) / (σ/√n) is greater than zα. For a two-sided test, reject when |z| > zα/2.

Common z alpha values (one-tailed and two-tailed)

α	One-tailed zα	Two-tailed zα/2	Two-sided confidence level
0.10	1.282	1.645	90%
0.05	1.645	1.960	95%
0.025	1.960	2.241	97.5%
0.01	2.326	2.576	99%
0.005	2.576	2.807	99.5%

The values 1.645, 1.960 and 2.576 match the standard normal critical values tabulated in the NIST/SEMATECH e-Handbook (section 1.3.6.7.1); the others come from the same standard normal distribution function. Note the diagonal: the one-tailed zα at α = 0.025 equals the two-tailed zα/2 at α = 0.05, because both leave 0.025 in one tail. To find other values, see how to find a critical value.

Worked example with the arithmetic

A sample of n = 64 measurements has mean x̄ = 52.4. The population standard deviation is known to be σ = 8.

1. Standard error: σ/√n = 8/√64 = 8/8 = 1.
2. 95% interval (α = 0.05, zα/2 = 1.960): margin = 1.960 × 1 = 1.96, so 52.4 − 1.96 = 50.44 and 52.4 + 1.96 = 54.36. The interval is [50.44, 54.36].
3. 99% interval (α = 0.01, zα/2 = 2.576): margin = 2.576 × 1 = 2.576, so the interval is 52.4 ± 2.576 = [49.82, 54.98] (rounded to two decimals).

Now test H₀: μ = 50 against an upper-tailed alternative. The statistic is z = (52.4 − 50)/1 = 2.40.

• One-tailed, α = 0.05: 2.40 > z0.05 = 1.645, so reject H₀. At α = 0.01 it is also rejected, because 2.40 > z0.01 = 2.326.
• Two-tailed, α = 0.05: 2.40 > z0.025 = 1.960, so reject. Two-tailed at α = 0.01: 2.40 < z0.005 = 2.576, so do not reject.

The same data give different decisions at α = 0.01 depending on whether you use zα or zα/2, which is why the tail choice must be made before looking at the data.

Interpretation: reading z against z alpha

Situation	Decision
Upper-tailed test, z > zα	Reject H0 at level α
Upper-tailed test, z ≤ zα	Do not reject H0
Two-tailed test, |z| > zα/2	Reject H0 at level α
Two-tailed test, |z| ≤ zα/2	Do not reject H0
Confidence interval	Larger zα/2 (higher confidence) gives a wider interval

Equivalently, you can compare the p-value with α; see how to find a p-value from a z-score. In the example above, the two-tailed p-value for z = 2.40 is about 0.016, below 0.05 and above 0.01, matching the decisions.

Assumptions and common mistakes

• Known σ. The z interval above assumes σ is known (NIST states the interval for known σ). When σ is estimated from a small sample, the t distribution is the usual choice; see z-test vs t-test.
• Using α instead of α/2. A 95% two-sided interval uses 1.960, not 1.645. Using 1.645 gives a 90% interval.
• Dropping the sign. Lower-tail critical values are negative: −1.645 or −1.960. A lower-tailed test rejects when z < −zα.
• Mixing notations. z_0.95 in lower-tail notation is 1.645, but a book that indexes by upper-tail area would call that z_0.05. Check which convention a source uses.
• Confusing zα with a z-score. A z-score is computed from your data; zα is a fixed cut-off that depends only on α.
• Choosing the tail after seeing the data. Decide one-sided or two-sided before analysis.

How to report it

State the confidence level, the interval and the assumption behind it. Example wording: “The 95% confidence interval for the mean was [50.44, 54.36] (n = 64, known σ = 8, critical value z = 1.96).” For a test: “The upper-tailed z test was significant, z = 2.40, p = .008.” Report the exact test statistic and p-value, and say whether the test was one- or two-tailed.

How to calculate z alpha in Excel, R or Python

Each of these returns the value that leaves α in the upper tail, so for zα/2 you pass α/2.

Excel

=NORM.S.INV(1-0.05/2)   ' z alpha/2 at alpha = 0.05, returns 1.959964
=NORM.S.INV(1-0.05)     ' z alpha at alpha = 0.05, returns 1.644854

R

qnorm(1 - 0.05/2)   # 1.959964
qnorm(1 - 0.05)     # 1.644854

Python (SciPy)

from scipy.stats import norm
norm.ppf(1 - 0.05/2)   # 1.959964
norm.ppf(1 - 0.05)     # 1.644854

Frequently asked questions

What is z alpha/2 for 95%?

1.960. A 95% two-sided interval has α = 0.05, so 0.025 sits in each tail and zα/2 = z0.025 = 1.960.

Why is z alpha/2 1.96 and not 1.645?

1.645 leaves 5% in one tail. A two-sided 95% interval leaves only 2.5% in each tail, which requires a point further out: 1.96.

What is the difference between z alpha and z alpha/2?

zα puts all of α in one tail (one-tailed test); zα/2 splits α across both tails (two-tailed test or two-sided interval). For the same α, zα/2 is larger.

What is z alpha for a 99% confidence level?

For a two-sided 99% interval, α = 0.01 and zα/2 = z0.005 = 2.576. For a one-tailed test at α = 0.01, zα = 2.326.

Is z alpha the same as a z-score?

No. A z-score measures how far an observation or statistic lies from the mean in standard-deviation units. Z alpha is the threshold you compare it with.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.1, “Cumulative distribution function of the standard normal distribution”: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (checked 2026-10-29).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.1.4, “What are confidence intervals?”: https://www.itl.nist.gov/div898/handbook/prc/section1/prc14.htm (checked 2026-10-29).