How to Find a Critical Value: z, t and Chi-Square Tables

A critical value is the cut-off on a test statistic’s distribution that separates “reject the null hypothesis” from “do not reject.” The values of the test statistic beyond that cut-off form the critical region, and a null hypothesis is rejected when the test statistic lies within that region. Finding one takes three inputs: the distribution your test uses (z, t or chi-square), the significance level (alpha), and, for t and chi-square, the degrees of freedom. This guide gives the lookup tables, the formula, two worked examples, and the software calls that replace the table.

How to find a critical value in four steps

1. Pick the distribution. Use z (the standard normal) when the test statistic is a z statistic, t when it is a t statistic, and chi-square for chi-square tests.
2. Set alpha and the tail. A one-tailed test puts all of alpha in one tail. A two-tailed test splits it, so the upper-tail probability is alpha/2. For example, a two-tailed test at alpha = 0.05 uses the 0.025 upper-tail column. See one-tailed vs two-tailed tests for how to choose.
3. Get the degrees of freedom (not needed for z). For a one-sample t test, df = n – 1. For a chi-square test of independence on an r by c table, df = (r – 1)(c – 1). The guide on degrees of freedom covers other designs.
4. Read the table or call the function at that row and column, then compare the critical value with your test statistic.

How to calculate a critical value: the formula

A table is a printed copy of an inverse cumulative distribution function. In symbols, for a distribution with cumulative distribution function F, the upper-tail critical value at tail probability p is:

critical value = F^-1(1 – p)

For a two-tailed z or t test at significance level alpha, p = alpha/2, so the critical values are plus and minus F^-1(1 – alpha/2). For a one-tailed test, p = alpha. There is no closed-form arithmetic shortcut for the t and chi-square cases, which is why people use a table or software. The two-tailed alpha in the tables below is simply two times the upper-tail column heading.

Critical value calculator: z (standard normal) table

Source: NIST/SEMATECH e-Handbook of Statistical Methods, critical values of the normal distribution (itl.nist.gov/div898/handbook), values to three decimal places as published.

Upper-tail probability	Two-tailed alpha	z critical value
0.100	0.20	1.282
0.050	0.10	1.645
0.025	0.05	1.960
0.010	0.02	2.326
0.005	0.01	2.576
0.001	0.002	3.090

The most-used values are 1.645 (one-tailed 0.05), 1.960 (two-tailed 0.05) and 2.576 (two-tailed 0.01). Their role in intervals is explained in the z-score guide and the confidence interval guide.

Student’s t critical values table

Source: NIST/SEMATECH e-Handbook of Statistical Methods, critical values of the Student’s t distribution, values to three decimal places as published. Columns are upper-tail probabilities; the last row (infinity) equals the z values above.

df	0.10	0.05	0.025	0.01	0.005
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.440	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.860	2.306	2.896	3.355
9	1.383	1.833	2.262	2.821	3.250
10	1.372	1.812	2.228	2.764	3.169
12	1.356	1.782	2.179	2.681	3.055
14	1.345	1.761	2.145	2.624	2.977
15	1.341	1.753	2.131	2.602	2.947
20	1.325	1.725	2.086	2.528	2.845
25	1.316	1.708	2.060	2.485	2.787
30	1.310	1.697	2.042	2.457	2.750
40	1.303	1.684	2.021	2.423	2.704
60	1.296	1.671	2.000	2.390	2.660
Infinity	1.282	1.645	1.960	2.326	2.576

For a two-tailed test at alpha = 0.05, use the 0.025 column; for alpha = 0.01, use 0.005. Degrees of freedom not listed in this excerpt (for example 11 or 35) are not interpolated here; use software for an exact value.

Chi-square critical values table

Source: NIST/SEMATECH e-Handbook of Statistical Methods, critical values of the chi-square distribution (upper-tail values), three decimal places as published. Chi-square tests are upper-tail tests, so you read the column that equals your alpha. See the chi-square test entry for the test itself.

df	0.10	0.05	0.025	0.01
1	2.706	3.841	5.024	6.635
2	4.605	5.991	7.378	9.210
3	6.251	7.815	9.348	11.345
4	7.779	9.488	11.143	13.277
5	9.236	11.070	12.833	15.086
6	10.645	12.592	14.449	16.812
7	12.017	14.067	16.013	18.475
8	13.362	15.507	17.535	20.090
9	14.684	16.919	19.023	21.666
10	15.987	18.307	20.483	23.209
15	22.307	24.996	27.488	30.578
20	28.412	31.410	34.170	37.566
30	40.256	43.773	46.979	50.892

Worked example 1: a two-tailed t critical value

A researcher runs a one-sample t test on n = 15 measurements at alpha = 0.05, two-tailed. The numbers here are illustrative.

1. Degrees of freedom: df = 15 – 1 = 14.
2. Tail probability: 0.05 / 2 = 0.025, so use the 0.025 column.
3. Table lookup: row 14, column 0.025 gives 2.145.
4. The critical values are -2.145 and +2.145. Reject the null hypothesis if the t statistic is below -2.145 or above +2.145. A statistic of 2.40 would be rejected; a statistic of 1.90 would not.

Worked example 2: a chi-square critical value

A chi-square test of independence is run on a 2 by 3 table at alpha = 0.05. The numbers here are illustrative.

1. Degrees of freedom: (2 – 1)(3 – 1) = 2.
2. Table lookup: row 2, column 0.05 gives 5.991.
3. Reject independence if the chi-square statistic exceeds 5.991. A statistic of 7.20 exceeds it; a statistic of 4.10 does not.

How to compute a critical value in Excel, R and Python

Software returns the same values with more digits than a printed table. Function names below are standard; confirm argument order in your software’s documentation. Outputs rounded to three decimals match the tables above.

Goal	Excel	R	Python (scipy.stats)
z, two-tailed 0.05 (1.960)	=NORM.S.INV(0.975)	qnorm(0.975)	norm.ppf(0.975)
t, df 14, two-tailed 0.05 (2.145)	=T.INV.2T(0.05, 14)	qt(0.975, 14)	t.ppf(0.975, 14)
Chi-square, df 2, alpha 0.05 (5.991)	=CHISQ.INV.RT(0.05, 2)	qchisq(0.95, 2)	chi2.ppf(0.95, 2)

The R and Python functions take the cumulative probability (1 minus the upper-tail probability), which is why the t call uses 0.975 for a two-tailed 0.05 test. Excel’s T.INV.2T takes the two-tailed alpha directly.

Frequently asked questions

How do you find the critical value?

Choose the distribution, set alpha (split in half for a two-tailed test), find the degrees of freedom if the distribution needs them, and read the matching table cell or call the inverse-CDF function in software.

How do you calculate a critical value without a table?

Evaluate the inverse cumulative distribution function at 1 minus the tail probability: qt(1 - alpha/2, df) in R, =T.INV.2T(alpha, df) in Excel, or t.ppf(1 - alpha/2, df) in Python.

What is the critical value for a 95% confidence level?

For the normal distribution it is 1.960 (two-tailed alpha = 0.05). For the t distribution it depends on df, for example 2.145 at df = 14 and 2.228 at df = 10.

Why does the t critical value change with sample size?

The t distribution has heavier tails when df is small, so the cut-off is larger. As df grows, the t values approach the z values, as the table’s infinity row shows.

Is the critical value the same as the p-value?

No. The critical value is a fixed cut-off set by alpha; the p-value is computed from your data. The two approaches lead to the same decision.

Sources

All sources checked 28 October 2026.

• NIST/SEMATECH e-Handbook of Statistical Methods, critical values of the normal distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm
• NIST/SEMATECH e-Handbook of Statistical Methods, critical values of the Student’s t distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm
• NIST/SEMATECH e-Handbook of Statistical Methods, critical values of the chi-square distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm
• NIST/SEMATECH e-Handbook of Statistical Methods, hypothesis testing (definition of critical values and the critical region): https://www.itl.nist.gov/div898/handbook/prc/section1/prc131.htm