What is the chi-square critical value at df=1 and alpha=0.05?

The chi-square critical value at df=1 and α=0.05 is 3.841. This is the most commonly used chi-square critical value in statistics. If your test statistic exceeds 3.841 with 1 degree of freedom at the 95% confidence level (α=0.05), you reject the null hypothesis. This value is confirmed by the NIST Engineering Statistics Handbook. At 99% confidence (α=0.01), the critical value is 6.635.

What is the chi-square critical value at df=3 and alpha=0.05?

The chi-square critical value at df=3 and α=0.05 is 7.815. This means if you have 3 degrees of freedom and your chi-square test statistic exceeds 7.815, you reject the null hypothesis at the 5% significance level. For a 4-category goodness-of-fit test (df=4−1=3), the critical value is 7.815. This value matches the NIST Engineering Statistics Handbook.

What is the chi-square critical value at df=5 and alpha=0.05?

The chi-square critical value at df=5 and α=0.05 is 11.071. This is used in 6-category goodness-of-fit tests (df=6−1=5). If your chi-square statistic exceeds 11.071, reject the null hypothesis at α=0.05. The same value appears in the NIST chi-square critical values table.

What is the chi-square critical value at df=4 and alpha=0.05?

The chi-square critical value at df=4 and α=0.05 is 9.488. This applies to 5-category goodness-of-fit tests (df=5−1=4). At α=0.01 with df=4, the critical value is 13.277.

What is the chi-square critical value at df=2 and alpha=0.05?

The chi-square critical value at df=2 and α=0.05 is 5.991. This is commonly used in 2×2 contingency table tests of independence where df=(2−1)×(2−1)=1 gives 3.841, and df=2 (from a 2×3 or 3×2 table) gives 5.991.

What is a chi-square table used for?

A chi-square table lists critical values of the chi-square (χ²) distribution for different degrees of freedom and significance levels. It is used to determine whether to reject the null hypothesis in goodness-of-fit tests, tests of independence in contingency tables, and tests of homogeneity. It can also be used to find p-values by locating where your test statistic falls between columns.

How do you read a chi-square table?

To read a chi-square table: (1) Calculate your degrees of freedom (df). (2) Choose your significance level α (e.g., 0.05). (3) Find the row for your df and the column for your α. (4) The intersection is the critical value. If your test statistic exceeds this value, reject the null hypothesis.

Where can I download a chi-square table PDF?

You can download a free printable chi-square table PDF from this page. Three versions are available: (1) Standard table (df 1–30), (2) Extended table (df 1–100), and (3) Annotated reference card with step-by-step reading guide. All include critical values for significance levels 0.005, 0.01, 0.025, 0.05, and 0.10.

How do you calculate degrees of freedom for a chi-square test?

For a goodness-of-fit test: df = number of categories − 1. For a test of independence in a contingency table: df = (rows − 1) × (columns − 1). For a variance test: df = n − 1. Example: a 3×4 contingency table has df = (3−1) × (4−1) = 6.

Do the values in this table match the NIST chi-square critical values?

Yes. All critical values in this chi-square table match the NIST Engineering Statistics Handbook (Section 1.3.6.7.4). For example: df=1, α=0.05 → 3.841; df=5, α=0.05 → 11.070 (we show 11.071 due to rounding); df=10, α=0.05 → 18.307. The NIST table is the authoritative government reference for chi-square critical values.

What is the difference between left-tail and right-tail chi-square critical values?

Right-tail critical values (the standard table) show the chi-square score where P(χ² > x) = α. They are used in goodness-of-fit tests, independence tests, and homogeneity tests. Left-tail critical values show P(χ² < x) = α. They are needed for two-sided variance tests where you must compare against both an upper and lower critical value.

What is a chi-square distribution table?

A chi-square distribution table (also written chi-squared distribution table or χ² distribution table) is a reference table listing critical values of the chi-square distribution. Each row represents a different degrees of freedom value and each column represents a different significance level α. The table values are the chi-square scores that cut off the right-tail probability equal to α. The chi-square distribution is always right-skewed and non-negative.

What significance level should I use for a chi-square test?

The most common significance level is α = 0.05 (5%), corresponding to 95% confidence. Use α = 0.01 for more rigorous research requiring 99% confidence. Use α = 0.10 for exploratory studies where missing a real effect (Type II error) is more costly than a false positive. In most academic and research contexts, α = 0.05 is the standard choice.

How do I use a chi-square table to find a p-value?

To find a p-value from a chi-square table: (1) Calculate your test statistic χ². (2) Find the row for your degrees of freedom. (3) Scan across that row to find where your χ² statistic falls between two column values. (4) The p-value is between the two α values for those columns. For example, if df=3 and χ²=8.5, it falls between 7.815 (α=0.05) and 9.348 (α=0.025), so 0.025 < p < 0.05. Use the calculator above for an exact p-value.

Chi-Square Table: Critical Values PDF | df 1

Q: What is chi-square value 3.841 and what does it mean?

3.841 is the chi-square critical value for df=1 at α=0.05 (95% confidence level). It means: if you perform a chi-square test with 1 degree of freedom and your test statistic exceeds 3.841, the result is statistically significant at the 5% significance level and you reject the null hypothesis. This value comes from the right tail of the chi-square distribution where 5% of the area lies above 3.841 when df=1.

Chi-Square Critical Values: Quick Reference (α = 0.05)

Most-looked-up values at α = 0.05 (95% confidence). Right-tail critical values — reject H₀ if χ² exceeds these.

df = 1, α = 0.05

95% confidence

3.841

Most common critical value

df = 2, α = 0.05

95% confidence

5.991

3×2 contingency table

df = 3, α = 0.05

95% confidence

7.815

4-category goodness-of-fit

df = 4, α = 0.05

95% confidence

9.488

5-category goodness-of-fit

df = 5, α = 0.05

95% confidence

11.071

6-category goodness-of-fit

df = 6, α = 0.05

95% confidence

12.592

7-category test

df = 1, α = 0.01

99% confidence

6.635

Strict significance level

df = 3, α = 0.01

99% confidence

11.345

Strict 4-category test

Chi-Square Critical Value: df=1, α=0.05 = 3.841 (95% Confidence)

3.841 is the chi-square critical value at 1 degree of freedom and α = 0.05 (95% confidence level). It means: if you run a chi-square test with df=1 and your test statistic exceeds 3.841, the result is statistically significant — you reject the null hypothesis. This value comes from the right tail of the χ² distribution where 5% of the probability lies above 3.841 when df=1. Confirmed by NIST (value: 3.841). At 99% confidence (α=0.01): 6.635. At 90% confidence (α=0.10): 2.706.

Chi-Square Critical Value: df=3, α=0.05 = 7.815

At 3 degrees of freedom and α = 0.05, the chi-square critical value is 7.815. If your test statistic exceeds 7.815, reject H₀ at the 5% significance level. Common uses: 4-category goodness-of-fit test (df = 4−1 = 3); a 2×4 contingency table (df = (2−1)(4−1) = 3). At α=0.01: 11.345. At α=0.10: 6.251. Source: NIST Engineering Statistics Handbook.

Chi-Square Critical Value: df=5, α=0.05 = 11.071

At 5 degrees of freedom and α = 0.05, the critical value is 11.071 (NIST: 11.070). Reject H₀ if χ²_calc > 11.071. Common use: 6-category goodness-of-fit test (df = 6−1 = 5); a 2×6 or 3×4 contingency table with df=5. The NIST chi-square critical values table for df=5, α=0.05 shows 11.070 (our value rounds to 11.071). At α=0.01: 15.086.

Chi-Square Critical Value: df=4, α=0.05 = 9.488

At 4 degrees of freedom and α = 0.05, the critical value is 9.488. Used in 5-category goodness-of-fit tests (df = 5−1 = 4) and contingency tables with df=4. At α=0.025: 11.143. At α=0.01: 13.277.

Chi-Square Critical Value: df=2, α=0.05 = 5.991

At 2 degrees of freedom and α = 0.05, the critical value is 5.991. This applies to 3-category goodness-of-fit tests (df = 3−1 = 2) and 2×3 or 3×2 contingency tables (df = (2−1)(3−1) = 2). At α=0.01: 9.210. At α=0.10: 4.605.

Chi-Square Critical Value Calculator

Degrees of Freedom (df)

Significance Level (α)

χ² critical value

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Chi-Square Critical Values Table — Right-Tail (Upper-Tail)

Values are P(χ² > x) = α. Click any cell to fill the calculator above. NIST Verified

df \ α	0.10	0.05	0.025	0.01	0.005	0.001
1	2.706	3.841	5.024	6.635	7.879	10.828
2	4.605	5.991	7.378	9.210	10.597	13.816
3	6.251	7.815	9.348	11.345	12.838	16.266
4	7.779	9.488	11.143	13.277	14.860	18.467
5	9.236	11.071	12.833	15.086	16.750	20.515
6	10.645	12.592	14.449	16.812	18.548	22.458
7	12.017	14.067	16.013	18.475	20.278	24.322
8	13.362	15.507	17.535	20.090	21.955	26.124
9	14.684	16.919	19.023	21.666	23.589	27.877
10	15.987	18.307	20.483	23.209	25.188	29.588
11	17.275	19.675	21.920	24.725	26.757	31.264
12	18.549	21.026	23.337	26.217	28.300	32.909
13	19.812	22.362	24.736	27.688	29.819	34.528
14	21.064	23.685	26.119	29.141	31.319	36.123
15	22.307	24.996	27.488	30.578	32.801	37.697
16	23.542	26.296	28.845	32.000	34.267	39.252
17	24.769	27.587	30.191	33.409	35.718	40.790
18	25.989	28.869	31.526	34.805	37.156	42.312
19	27.204	30.144	32.852	36.191	38.582	43.820
20	28.412	31.410	34.170	37.566	39.997	45.315
21	29.615	32.671	35.479	38.932	41.401	46.797
22	30.813	33.924	36.781	40.289	42.796	48.268
23	32.007	35.172	38.076	41.638	44.181	49.728
24	33.196	36.415	39.364	42.980	45.559	51.179
25	34.382	37.652	40.646	44.314	46.928	52.620
26	35.563	38.885	41.923	45.642	48.290	54.052
27	36.741	40.113	43.195	46.963	49.645	55.476
28	37.916	41.337	44.461	48.278	50.993	56.892
29	39.087	42.557	45.722	49.588	52.336	58.301
30	40.256	43.773	46.979	50.892	53.672	59.703

All values are right-tail critical values χ²_α,df where P(χ² > x) = α. Click any cell to fill the calculator. Values match NIST Engineering Statistics Handbook.

NIST Chi-Square Critical Values

The NIST Engineering Statistics Handbook (Section 1.3.6.7.4) is the authoritative government reference for chi-square critical values. All values in the table above match the NIST chi-square critical values table. Key reference points confirmed by NIST:

df	α = 0.10	α = 0.05	α = 0.025	α = 0.01	α = 0.001
1	2.706	3.841	5.024	6.635	10.828
2	4.605	5.991	7.378	9.210	13.816
3	6.251	7.815	9.348	11.345	16.266
5	9.236	11.070	12.833	15.086	20.515
10	15.987	18.307	20.483	23.209	29.588

Source: NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.7.4. Note: NIST shows 11.070 for df=5, α=0.05; our table shows 11.071 (rounding difference only).

How to Read a Chi-Square Table (Step by Step)

Reading a chi-square table — also called a chi-squared table, χ² table, chi distribution table, or chi square test table — takes four steps:

1 Calculate degrees of freedom (df). Goodness-of-fit: df = categories − 1. Contingency table: df = (rows − 1) × (columns − 1). Variance test: df = n − 1.

2 Choose your significance level (α). Most common: α = 0.05 (95% confidence). Stricter research: α = 0.01.

3 Find the row for your df and move across to the column for your α value.

4 Compare your test statistic. If χ²_calc > χ²_critical, reject H₀. Example: df=3, α=0.05 → critical value = 7.815.

Chi-Square Distribution Table (χ²)

The chi-square distribution table — also written as chi-squared distribution table, chi-squared table, χ² distribution table, chi distribution table, or chi square test table — is a reference giving critical values of the chi-square distribution. The chi-square distribution is a family of right-skewed curves defined entirely by degrees of freedom (df). As df increases, the distribution shifts right and becomes more bell-shaped.

Distribution Properties

χ² is always ≥ 0. Mean = df. Variance = 2×df. Sum of k squared standard normals: if Z₁…Z_k ~ N(0,1), then Z₁²+…+Z_k² ~ χ²(k).

Finding p-values

Scan your df row across the table. If your χ² falls between two column values, the p-value is between those two α values. For an exact p-value, use the calculator above or Excel: =CHISQ.DIST.RT(x, df).

Test Statistic Formula

χ² = Σ(O−E)²/E, where O = observed frequency, E = expected frequency. Always compare this calculated value against the right-tail critical value from the chi-square distribution table.

Chi-Square Test Table: Hypothesis Testing Guide

A chi-square test table provides the critical values needed to complete a chi-square hypothesis test. Below are the three main uses of the chi-square test table:

📊

Goodness-of-Fit Test

Tests whether observed frequencies match expected distribution. df = k − 1. Example: fair six-sided die? df = 6 − 1 = 5 → critical value at α=0.05 is 11.071.

🔗

Test of Independence

Tests if two categorical variables are related. df = (r−1)(c−1). Example: gender vs preference in 2×3 table: df = 1×2 = 2 → critical value = 5.991.

📐

Variance Test

Tests if population variance equals a specified value. df = n − 1. Uses both upper and lower critical values (two-tailed). See the left-tail table below.

Chi-Square Table: Left-Tail (Lower-Tail) Critical Values

Left-tail critical values where P(χ² < x) = α. Used in two-tailed variance tests and lower-tail rejection regions. For a two-sided test at α=0.05: use the right-tail table for χ²_0.025 (upper bound) and this table for χ²_0.025 (lower bound). Values match NIST lower-tail table.

df \ α (left)	0.005	0.010	0.025	0.050	0.100
1	—	—	0.001	0.004	0.016
2	0.010	0.020	0.051	0.103	0.211
3	0.072	0.115	0.216	0.352	0.584
4	0.207	0.297	0.484	0.711	1.064
5	0.412	0.554	0.831	1.145	1.610
6	0.676	0.872	1.237	1.635	2.204
7	0.989	1.239	1.690	2.167	2.833
8	1.344	1.646	2.180	2.733	3.490
9	1.735	2.088	2.700	3.325	4.168
10	2.156	2.558	3.247	3.940	4.865
11	2.603	3.053	3.816	4.575	5.578
12	3.074	3.571	4.404	5.226	6.304
13	3.565	4.107	5.009	5.892	7.042
14	4.075	4.660	5.629	6.571	7.790
15	4.601	5.229	6.262	7.261	8.547
16	5.142	5.812	6.908	7.962	9.312
17	5.697	6.408	7.564	8.672	10.085
18	6.265	7.015	8.231	9.390	10.865
19	6.844	7.633	8.907	10.117	11.651
20	7.434	8.260	9.591	10.851	12.443
25	10.520	11.524	13.120	14.611	16.473
30	13.787	14.953	16.791	18.493	20.599

Left-tail values: P(χ² < x) = α. For a two-tailed variance test at α=0.05 with df=10: lower CV = 3.247 (left-tail at 0.025), upper CV = 20.483 (right-tail at 0.025). Source: NIST lower-tail chi-square table.

Chi-Square Table PDF — Free Download

Download a free printable chi-square table PDF. All three versions include critical values for significance levels 0.005, 0.01, 0.025, 0.05, and 0.10. Suitable for exams, coursework, and research. These are the same values as the chi-square distribution table PDF and chi square distribution table PDF free download.

Standard Chi-Square Table PDF df 1–30 · All α levels · Printable Extended Chi-Square Table PDF df 1–100 · All α levels · Printable Annotated Reference Card PDF With how-to-read guide · Printable

One-Tailed vs Two-Tailed Chi-Square Test

One-Tailed (Right-Tail) — Standard

Used in goodness-of-fit, independence, and homogeneity tests. Reject H₀ when χ²_calc > χ²_α,df from the right-tail table above.

df=5, α=0.05 → χ² = 11.071

Two-Tailed — Variance Tests

Used when testing H₀: σ² = σ₀². Split α between both tails. Compare against both the upper CV (right-tail table at α/2) and lower CV (left-tail table at α/2).

df=5, α=0.05 → lower: 0.831 | upper: 12.833

Chi-Square Table for Contingency Tables

In a test of independence, df = (rows − 1) × (columns − 1). Use the chi-square table to find the critical value for your specific table size.

Worked Example: 3×2 Contingency Table

Survey of 200 people: 3 age groups (rows) × 2 preference categories (columns). df = (3−1)×(2−1) = 2. At α = 0.05, look up df=2 in the chi-square critical value table → 5.991. If your calculated χ² > 5.991, the variables are not independent.

df = (r−1)(c−1)

Degrees of Freedom

E = (Row × Col) / n

Expected Frequency

χ² = Σ(O−E)²/E

Test Statistic

Frequently Asked Questions About the Chi-Square Table

What is the chi-square critical value at df=1, α=0.05?

3.841. The chi-square critical value at df=1 and α=0.05 (95% confidence) is 3.841. If your test statistic exceeds 3.841 with 1 degree of freedom, the result is statistically significant. At α=0.01: 6.635. At α=0.10: 2.706. Confirmed by NIST.

What is chi-square value 3.841 and where does it come from?

3.841 is the chi-square critical value at df=1 and α=0.05 (95% confidence level). It represents the chi-square score above which 5% of the probability falls when df=1. If your test statistic exceeds 3.841, the result is statistically significant. At α=0.01 the value is 6.635; at α=0.10 it is 2.706.

Do the values in this table match NIST chi-square critical values?

Yes. All values match the NIST Engineering Statistics Handbook (Section 1.3.6.7.4). Key verified values: df=1, α=0.05 → 3.841; df=3, α=0.05 → 7.815; df=5, α=0.05 → 11.070 (NIST) / 11.071 (rounding); df=10, α=0.05 → 18.307. The NIST table is the US government's authoritative statistical reference.

How do I find a chi-square critical value without a table?

Use: (1) The calculator above — enter df and α. (2) Excel: =CHISQ.INV.RT(alpha, df), e.g. =CHISQ.INV.RT(0.05,1) returns 3.841. (3) Python: scipy.stats.chi2.ppf(1-alpha, df). (4) R: qchisq(1-alpha, df). All return the same values as the printed table.

How do I use the chi-square table to find a p-value?

Find your df row. Scan across to find where your calculated χ² falls between two column values. The p-value lies between those two α values. Example: df=3, χ²=8.5 → falls between 7.815 (α=0.05) and 9.348 (α=0.025) → 0.025 < p < 0.05. For exact p-values use the calculator above.

How do you calculate degrees of freedom?

Goodness-of-fit: df = categories − 1. Test of independence: df = (rows − 1) × (columns − 1). Variance test: df = n − 1. Example: a 3×4 contingency table has df = (3−1) × (4−1) = 2 × 3 = 6.

What is a chi-square test table vs a chi-square distribution table?

They refer to the same table. A "chi-square test table" is the critical values table used during hypothesis testing. A "chi-square distribution table" emphasizes it shows the theoretical distribution's critical values. Other names: chi-squared table, χ² table, chi distribution table, chi square critical value table — all the same reference.

Can a chi-square value be negative?

No. Chi-square values are always ≥ 0 because the formula χ² = Σ(O−E)²/E involves squared differences. The minimum value 0 occurs when observed frequencies perfectly match expected frequencies.

What does it mean when my statistic exceeds the critical value?

Reject the null hypothesis. The result is statistically significant — observed data differs from expected by more than chance at your chosen α level. If it does not exceed the critical value, fail to reject H₀ (do not conclude the null is true).