What question does Chi-Square Independence Calculator answer?

Analyze an observed contingency table with expected counts, chi-square, degrees of freedom and an approximate p-value. The implemented relationship is R[chi-square-calculator]=Eᵢⱼ = rowᵢ × columnⱼ / N; χ² = Σ(Oᵢⱼ−Eᵢⱼ)²/Eᵢⱼ; df = (r−1)(c−1); x₁=Observed counts (one table row per line).

Which inputs does Chi-Square Independence Calculator use?

It uses the visible fields Observed counts (one table row per line). No hidden value is substituted for an omitted required input.

What happens when Chi-Square Independence Calculator receives an invalid or extreme value?

The page rejects missing, ambiguous, out-of-range, or non-finite values and refuses to display a non-finite result.

Why can Chi-Square Independence Calculator differ from another result?

Different unit conventions, endpoint policies, fee or rate assumptions, formula domains, and premature rounding can produce a different answer.

Can I use Chi-Square Independence Calculator as professional advice?

No. It answers the displayed mathematical question and cannot account for omitted real-world facts.

How can I verify the Chi-Square Independence Calculator result?

Recalculate the two evaluated examples without rounding intermediate values and compare the declared formula or convention with NIST/SEMATECH — Contingency tables.

Chi-Square Independence Calculator

Resultp = 0.003873388838

Model and interpretation

Use an at-least 2×2 table of nonnegative integer counts from independent observations. Empty row or column totals are invalid. This chi-square reference approximation is intended for expected counts of at least 5 in every cell; smaller expectations trigger a warning. It is not Fisher's exact test or proof of causation. 2 ≤ r,c ≤ 20; N ≤ 10⁹.

χ²: 19.178
df: 6
p: 0.00387339
N: 309
#(E < 5): 0
#(E < 1): 0

Data table (12)

i	j	Oᵢⱼ	Eᵢⱼ
1	1	15	22.5113
1	2	21	20.9903
1	3	45	38.9385
1	4	13	11.5599
2	1	26	22.9903
2	2	31	21.4369
2	3	34	39.767
2	4	5	11.8058
3	1	33	28.4984
3	2	17	26.5728
3	3	49	49.2945
3	4	20	14.6343

Method source: NIST/SEMATECH — Contingency tables Report a correction

Formula

Chi-Square Independence Calculator evaluates the registered expression “R[chi-square-calculator]=Eᵢⱼ = rowᵢ × columnⱼ / N; χ² = Σ(Oᵢⱼ−Eᵢⱼ)²/Eᵢⱼ; df = (r−1)(c−1); x₁=Observed counts (one table row per line)” with the validated inputs. Full calculation precision is retained until the result is formatted for display.

R[chi-square-calculator]=Eᵢⱼ = rowᵢ × columnⱼ / N; χ² = Σ(Oᵢⱼ−Eᵢⱼ)²/Eᵢⱼ; df = (r−1)(c−1); x₁=Observed counts (one table row per line)

NIST/SEMATECH — Contingency tables

What this calculator answers

Chi-Square Independence Calculator applies the registered method “Eᵢⱼ = rowᵢ × columnⱼ / N; χ² = Σ(Oᵢⱼ−Eᵢⱼ)²/Eᵢⱼ; df = (r−1)(c−1)” to the displayed Observed counts (one table row per line) inputs. Both worked examples come from the production calculation engine, and NIST/SEMATECH — Contingency tables defines the cited method or convention; no hidden inputs or current external data are inferred.

Variables, defaults, and limits

Observed counts (one table row per line): Observed counts (one table row per line) is a structured text input. The displayed starter value is 15; 21; 45; 13 26; 31; 34; 5 33; 17; 49; 20; it is an example, not a hidden assumption.; Default: 15; 21; 45; 13 26; 31; 34; 5 33; 17; 49; 20; Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.

Formula and calculation rule

R[chi-square-calculator]=Eᵢⱼ = rowᵢ × columnⱼ / N; χ² = Σ(Oᵢⱼ−Eᵢⱼ)²/Eᵢⱼ; df = (r−1)(c−1); x₁=Observed counts (one table row per line)

Worked examples