Z Critical Value Table

The most common mistake in choosing a z critical value is repeating the rule that a sample size of 30 or more lets you use the normal distribution. That rule is wrong. The correct choice between a z critical value and a t critical value depends on whether the population standard deviation (sigma) is known, not on sample size. If sigma is known, use Z. If sigma is estimated from the sample standard deviation (s), use t. Even with n = 1000, if you estimate sigma from the sample, you need a t critical value, not a z critical value. The z critical value is the threshold in the standard normal distribution that defines the rejection region for a null hypothesis. For a two-sided test at a 95% confidence level, that threshold is 1.96. For a one-sided test at the same level, it is 1.645.

The table below gives the z critical value (z*) for one-sided and two-sided tests at the most common alpha levels. These values come from the inverse normal distribution (invNorm on a TI-84, NORM.S.INV in Excel or Google Sheets). Use the two-sided column when your alternative hypothesis is non-directional (e.g., μ ≠ μ₀). Use the one-sided column when your alternative is directional (e.g., μ > μ₀ or μ < μ₀).

The number 1.96 comes from the standard normal distribution. For a two-sided test at α = 0.05, you split the alpha equally between both tails: 0.025 in each tail. The z critical value is the quantile that leaves 0.025 in the right tail, which means 0.975 of the area is to the left. The inverse normal function, NORM.S.INV(0.975) in Excel or invNorm(0.975) on a TI-84, returns 1.95996. Most textbooks, including Moore, McCabe & Craig's Introduction to the Practice of Statistics (9th ed.) and OpenIntro Statistics (4th ed.), round this to 1.96. The unrounded value is 1.95996, but 1.96 is the standard for table lookup and reporting. The same logic applies to a one-sided test: α = 0.05 goes entirely into one tail, so the left-tail area is 0.95, and NORM.S.INV(0.95) returns 1.645.

Use a z critical value only when the population standard deviation (sigma) is known. This happens in two main scenarios. First, when you are working with a population proportion (p) rather than a mean. For a one-sample proportion test, the standard deviation of the sampling distribution is sqrt(p₀(1-p₀)/n), which uses the null proportion p₀, not an estimate from the sample. This is a known value, so the critical value comes from Z. Second, when sigma is given by prior research, a manufacturing tolerance, or a theoretical model. For example, if a factory knows its fill weights have sigma = 2 grams, a test on a sample mean uses a z critical value.

Avoid using Z when sigma is unknown. This is the failure mode of the n ≥ 30 rule. If you estimate sigma from the sample, use a t critical value instead. The difference is not small. For a two-sided test at α = 0.05 with a sample size of 10, the t critical value (df = 9) is 2.262, while the z critical value is 1.96. Using Z here would incorrectly make your rejection threshold 14% smaller, inflating your Type I error above the stated alpha.

You do not need a printed table. Any level of alpha can be turned into a z critical value using an inverse normal function. The steps are the same for both software and calculator.

Using Excel or Google Sheets

For a two-sided test at alpha α, the left-tail probability is 1 - α/2. Use the function NORM.S.INV(1 - α/2). For α = 0.10, this is NORM.S.INV(0.95) = 1.645. For a one-sided test, the left-tail probability is 1 - α. Use NORM.S.INV(1 - α). For α = 0.01, this is NORM.S.INV(0.99) = 2.326.

Using a TI-84 Plus CE

Press 2nd then VARS (DISTR). Choose invNorm. For a two-sided test, enter invNorm(1 - α/2, 0, 1). For α = 0.05, this is invNorm(0.975, 0, 1). The default mean and standard deviation are 0 and 1, so invNorm(0.975) alone works. For a one-sided test, use invNorm(1 - α).

Using the Z-Table (Standard Normal Table)

If you are using a printed Z-table from a source like the NIST/SEMATECH e-Handbook of Statistical Methods (section 1.3.6.7), locate the probability (1 - α/2 for two-sided, 1 - α for one-sided) in the body of the table. Read the z-score from the margins. For 0.975, the z-score is 1.96. For 0.95, it is 1.645. For 0.99, it is 2.326. Tables typically give three or four decimal places, which is enough for decision-making.

A z critical value is the boundary of the rejection region for a hypothesis test or the multiplier for a confidence interval. For a two-sided test, calculate your test statistic (e.g., z = (x̄ - μ₀) / (σ / √n)). If the absolute value of the test statistic exceeds the z critical value, reject the null hypothesis. For a one-sided test, reject if the test statistic exceeds the critical value in the direction of the alternative. For a confidence interval, the formula is point estimate ± (z* × standard error). The z critical value is the same number whether you are constructing an interval or testing a hypothesis, provided the distribution and alpha are the same.

The ASA Statement on Statistical Significance and P-Values (Wasserstein & Lazar, 2016) reminds readers that a critical-value-based decision (statistical significance) does not imply practical significance. A test statistic that just crosses the threshold at α = 0.05 is not evidence that the effect is large or important. It is only evidence that the observed data are unlikely under the null hypothesis, assuming the model is correct.

Four errors account for nearly all mistakes with z critical values. First, using Z when sigma is unknown: this happens when a reader follows the n ≥ 30 rule instead of checking whether sigma is known. Second, using a one-sided critical value for a two-sided test: this happens when alpha is not halved for the two-sided case. For α = 0.05, the one-sided z* is 1.645, but the two-sided z* is 1.96. Using 1.645 in a two-sided test inflates Type I error. Third, using a two-sided critical value for a one-sided test: this makes the threshold too large and deflates power. Fourth, misreading the tail for asymmetric distributions: chi-square and F critical values require right-tail inverse functions (CHISQ.INV.RT and F.INV.RT in Excel), not left-tail ones. Using the left-tail value for chi-square at α = 0.05, df = 1, returns approximately 0.0039 instead of the correct 3.841.

The single most practical step you can take is to stop relying exclusively on printed tables and verify your z critical value with software. Open a spreadsheet or a calculator and run NORM.S.INV with the correct left-tail probability. For a two-sided test at α = 0.05, that is NORM.S.INV(0.975). Compare the result to the value from your textbook or table. The software value will be 1.95996. The table value will be 1.96.The habit of checking with software protects you against the one thing that most often goes wrong: using the wrong tail probability. If you mistakenly entered NORM.S.INV(0.95) for a two-sided test, you would get 1.645, not 1.96, and you would reject the null hypothesis too often. That is the failure case. Build the check into your workflow.