Critical Value Calculator - T, Z & Chi-Square Critical Value Solver

Statistics Tool

Critical Value Calculator

Find the t critical value, z critical value, or chi-square critical value for any confidence level, significance level, and degrees of freedom — with step-by-step working and printable reference tables.

Confidence Level:

Custom confidence level (%):

Tail Type:

Degrees of Freedom (df):

Confidence Level:

Custom confidence level (%):

Tail Type:

Significance level (α):

Degrees of Freedom (df):

Tail Type:

T Critical Value: Use when the population standard deviation is unknown and estimated from a sample — the standard case for t-tests. Requires degrees of freedom (df).

Z Critical Value: Use when the population standard deviation is known, or the sample size is large (typically n≥30), so the t-distribution converges to the standard normal distribution.

Chi-Square Critical Value: Use for tests of variance, goodness-of-fit tests, and tests of independence (contingency tables) — anywhere the test statistic follows a chi-square distribution.

95% two-tail, df=24

90% one-tail, df=15

99% two-tail, df=9

95% one-tail, df=29

95% two-tail → ±1.96

90% two-tail → ±1.645

99% two-tail → ±2.576

95% one-tail → 1.645

α=0.05, df=10, right

α=0.01, df=5, right

α=0.05, df=20, two-tail CI

α=0.10, df=15, left

Error

Critical Value

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What to Do With This Critical Value

Compare your test statistic (t, z, or χ²) directly against this critical value to decide reject vs. fail-to-reject — or use it as the first input to a margin of error / confidence interval calculation.

→ Evaluate your t-test result with this critical value → Use this as input to the Margin of Error Calculator

Visual: Critical Value & Rejection Region

0——

Rejection region Critical value boundary

Step-by-Step Working

Adjust Display Precision

Decimal places:

Student's t Table — Critical Values by Degrees of Freedom

Standard t-distribution critical value table. Columns show one-tail probabilities (the two-tail probability is double each column header). Click any row to load that df into the calculator above.

df	One-Tail (α)
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
11	1.363	1.796	2.201	2.718	3.106
12	1.356	1.782	2.179	2.681	3.055
13	1.350	1.771	2.160	2.650	3.012
14	1.345	1.761	2.145	2.624	2.977
15	1.341	1.753	2.131	2.602	2.947
16	1.337	1.746	2.120	2.583	2.921
17	1.333	1.740	2.110	2.567	2.898
18	1.330	1.734	2.101	2.552	2.878
19	1.328	1.729	2.093	2.539	2.861
20	1.325	1.725	2.086	2.528	2.845
21	1.323	1.721	2.080	2.518	2.831
22	1.321	1.717	2.074	2.508	2.819
23	1.319	1.714	2.069	2.500	2.807
24	1.318	1.711	2.064	2.492	2.797
25	1.316	1.708	2.060	2.485	2.787
26	1.315	1.706	2.056	2.479	2.779
27	1.314	1.703	2.052	2.473	2.771
28	1.313	1.701	2.048	2.467	2.763
29	1.311	1.699	2.045	2.462	2.756
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
120	1.289	1.658	1.980	2.358	2.617
∞ (z)	1.282	1.645	1.960	2.326	2.576

Note: as df → ∞, the t-distribution converges to the standard normal distribution — the bottom row (∞) matches the standard z critical values exactly.

Critical Value Formulas — Interactive Reference

1T Critical Valuet* : P(T>t*)=α/2

Found by inverting the t-distribution CDF for a given degrees of freedom — the value where the tail area equals α (one-tailed) or α/2 (two-tailed).

Use directly to evaluate a result from the T-Test Calculator.

2Z Critical Valuez* = Φ⁻¹(1−α/2)

The inverse of the standard normal CDF Φ. Common values: 90%→1.645, 95%→1.96, 99%→2.576 (all two-tailed).

3Chi-Square Critical Valueχ²: P(X>χ²)=α

Found by inverting the chi-square CDF (regularized lower incomplete gamma function) for a given df — asymmetric, always positive.

4One-Tailed vs Two-Tailedα vs α/2 per tail

Two-tailed critical values split α across both tails (α/2 each), producing a ± pair. One-tailed critical values place all of α in one tail, producing a single boundary value.

5Degrees of Freedom Effectdf↑ → t* → z*

As degrees of freedom increase, the t-distribution's tails thin out and its critical values converge toward the corresponding z critical values. By df≈120, t* and z* are nearly identical.

6Critical Value ↔ p-Value|stat|>crit ⇔ p<α

Comparing your test statistic to the critical value and comparing the p-value to α are mathematically equivalent decision rules — they always agree.

7Critical Value for Margin of ErrorMOE = crit × SE

Every confidence interval formula multiplies a critical value (t* or z*) by the standard error to get the margin of error. Feed your critical value directly into the Margin of Error Calculator.

8Chi-Square for Variance CITwo values: χ²_(1−α/2), χ²_(α/2)

A confidence interval for a population variance needs two chi-square critical values — a lower one (upper-tail area = 1−α/2) and an upper one (upper-tail area = α/2) — because the chi-square distribution is not symmetric.

9Common Confidence Levels90%, 95%, 99%

90%, 95%, and 99% are the most commonly used confidence levels in practice, corresponding to α = 0.10, 0.05, and 0.01 respectively.

10Rejection RegionBeyond critical value(s)

The rejection region is the range of test statistic values beyond the critical value(s) — landing there means you reject the null hypothesis at your chosen significance level.

What Is a Critical Value?

This critical value calculator finds the exact boundary point on the t-distribution, standard normal (z) distribution, or chi-square distribution that separates "statistically significant" results from "not significant" results at your chosen confidence level. A critical value is the threshold your test statistic must exceed for you to reject the null hypothesis.

Every hypothesis test ultimately reduces to one comparison: is your calculated statistic (t, z, or χ²) more extreme than the critical value? If yes, you reject H₀. This calculator computes that threshold directly using genuine inverse-distribution mathematics — not a lookup table — so it works for any confidence level and any degrees of freedom, not just the common textbook values.

Critical Value = F⁻¹(1 − α) or F⁻¹(1 − α/2) The inverse CDF of the relevant distribution (t, z, or χ²), evaluated at the tail probability

T Critical Value vs Z Critical Value

Both t and z critical values answer the same question — "how many standard errors away from the center marks statistical significance?" — but they come from different distributions:

Z critical value: From the standard normal distribution. Used when the population standard deviation is known, or the sample size is large enough (n≥30) that the sampling distribution of the mean is essentially normal regardless of the population's shape.
T critical value: From the t-distribution, which has heavier tails than the normal distribution to account for the extra uncertainty introduced by estimating the population standard deviation from a small sample. Requires degrees of freedom (df).

As degrees of freedom increase, the t-distribution's tails shrink toward the normal distribution's tails. By df=120, the t critical value and z critical value are nearly indistinguishable (e.g., 95% two-tailed: t*≈1.980 vs z*=1.960).

How to Find a Critical Value — Step-by-Step

Step 1 — Choose your confidence level or α: Common choices are 90% (α=0.10), 95% (α=0.05), or 99% (α=0.01).
Step 2 — Determine one-tailed or two-tailed: Two-tailed tests split α across both tails (α/2 each); one-tailed tests place all of α in one tail.
Step 3 — Identify the correct distribution: Use t if the population standard deviation is unknown (need df); use z if it's known or n is large; use chi-square for variance/goodness-of-fit/independence tests.
Step 4 — Compute the inverse CDF: Find the value where the tail probability equals your target (α or α/2).
Step 5 — Apply the sign: Two-tailed critical values are reported as ± the computed magnitude; one-tailed right-tests are positive, one-tailed left-tests are negative.

Common Critical Values Table

A quick-reference summary combining z and t critical values across the three most common confidence levels (all two-tailed unless noted):

Confidence Level	α (two-tail)	z Critical Value	t Critical Value (df=10)	t Critical Value (df=30)
90%	0.10	±1.645	±1.812	±1.697
95%	0.05	±1.960	±2.228	±2.042
99%	0.01	±2.576	±3.169	±2.750

Notice how the t critical values are always larger in magnitude than the corresponding z critical value for the same confidence level — this "penalty" reflects the extra uncertainty from estimating the standard deviation with a small sample, and it shrinks as df increases (compare the df=10 column to the df=30 column above).

Worked Examples

1. T Critical Value — 95% Confidence, df=24

Confidence = 95% → α = 0.05, two-tailed → each tail = 0.025
Find t such that P(T > t) = 0.025 with df=24
t* ≈ ±2.064

2. Z Critical Value — 99% Confidence, Two-Tailed

Confidence = 99% → α = 0.01, two-tailed → each tail = 0.005
Find z such that Φ(z) = 0.995
z* = ±2.576

3. Z Critical Value — 90% Confidence, One-Tailed

Confidence = 90% → α = 0.10, one-tailed (all in one tail)
Find z such that Φ(z) = 0.90
z* = 1.282

4. Chi-Square Critical Value — α=0.05, df=10, Right-Tail

Find χ² such that P(X > χ²) = 0.05 with df=10
χ² ≈ 18.307

5. Chi-Square for a Variance Confidence Interval — α=0.05, df=20

Need two values: lower (upper-tail area 0.975) and upper (upper-tail area 0.025)
χ²_lower ≈ 9.591 | χ²_upper ≈ 34.170
The 95% CI for variance uses both boundaries: [(n−1)s²/χ²_upper, (n−1)s²/χ²_lower]

6. T Critical Value — 90% Confidence, df=15, One-Tailed

Confidence = 90% → α = 0.10, one-tailed → find t such that P(T>t)=0.10, df=15
t* ≈ 1.341

Frequently Asked Questions

What is a critical value in statistics?

A critical value is the boundary point on a probability distribution (t, z, or chi-square) that separates the 'reject the null hypothesis' region from the 'fail to reject' region at a chosen significance level. If your test statistic exceeds the critical value, the result is statistically significant.

What is the difference between a t critical value and a z critical value?

A z critical value comes from the standard normal distribution and is used when the population standard deviation is known or the sample size is large. A t critical value comes from the t-distribution, which has heavier tails to account for the extra uncertainty of estimating the population standard deviation from a small sample, and depends on degrees of freedom (df).

How do you find a critical value step by step?

First choose your significance level α (e.g., 0.05 for 95% confidence). Then determine whether the test is one-tailed or two-tailed — two-tailed tests split α across both tails (α/2 each). Finally, look up or compute the value on the relevant distribution (t, z, or chi-square) where the tail area equals your chosen α or α/2.

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

For a 95% confidence level (two-tailed, α=0.05), the z critical value is ±1.96. For a one-tailed 95% test, the z critical value is 1.645.

What is the t critical value and how does it depend on degrees of freedom?

The t critical value is the boundary on the t-distribution for a given significance level and degrees of freedom (df = n−1 for a one-sample test). As df increases, the t-distribution's tails become thinner and the t critical value approaches the corresponding z critical value — at df=∞ they are identical.

When do you use a chi-square critical value instead of t or z?

Chi-square critical values are used for tests involving variance, goodness-of-fit tests, and tests of independence (contingency tables) — situations where the test statistic follows a chi-square distribution rather than a t or normal distribution.

What is the difference between a one-tailed and two-tailed critical value?

A two-tailed critical value splits your significance level α across both ends of the distribution (α/2 in each tail) and produces a ± critical value, used when testing for any difference. A one-tailed critical value places the entire α in a single tail and produces one value, used when testing for a difference in one specific direction.

How is the critical value related to the p-value?

They are two equivalent ways of making the same decision. If your test statistic's magnitude exceeds the critical value, then its p-value is necessarily less than α, and you reject the null hypothesis. Comparing statistic-to-critical-value and comparing p-value-to-α always produce the same conclusion.

Why does the t critical value get smaller as sample size increases?

Larger sample sizes mean more degrees of freedom, which makes the t-distribution's tails thinner and closer to the normal distribution. With more data, there is less uncertainty in estimating the population standard deviation, so a smaller critical value is needed to mark the same significance level.

Can critical values be negative?

Yes. For a two-tailed test, critical values come in a ± pair (e.g., ±1.96). For a one-tailed left-tail test, the critical value itself is negative (e.g., −1.645), marking the boundary in the lower tail of the distribution.

Related Calculators

Quick Reference

z*(90%, two-tail) = 1.645Standard normal

z*(95%, two-tail) = 1.960Standard normal

z*(99%, two-tail) = 2.576Standard normal

t* = F⁻¹(1−α/2, df)t-distribution

χ²* : P(X>χ²*)=αChi-square (right-tail)

Reject H₀ if |stat| > critDecision rule

Most Searched

z* for 95% confidence

t* for 95%, df=24

z* for 99% confidence

χ² for α=0.05, df=10

t* one-tail, df=15

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3. Z Critical Value — 90% Confidence, One-Tailed

4. Chi-Square Critical Value — α=0.05, df=10, Right-Tail

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Critical Value Calculator – T, Z & Chi-Square Critical Value Solver

Critical Value Calculator

What Is a Critical Value?

T Critical Value vs Z Critical Value

How to Find a Critical Value — Step-by-Step

Common Critical Values Table

Worked Examples

1. T Critical Value — 95% Confidence, df=24

2. Z Critical Value — 99% Confidence, Two-Tailed

3. Z Critical Value — 90% Confidence, One-Tailed

4. Chi-Square Critical Value — α=0.05, df=10, Right-Tail

5. Chi-Square for a Variance Confidence Interval — α=0.05, df=20

6. T Critical Value — 90% Confidence, df=15, One-Tailed

Frequently Asked Questions

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