Workers × neighborhoods (3×4)
- Input
- 90 60 104 95 / 30 50 51 20 / 30 40 45 35
- Expected output
- χ² = 24,5712, gl = 6, p = 0,0004
Cramér V = 0.137.
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chi square test of independence calculator
In the "χ² independence" tab the table is editable: add rows and columns (up to 10 by 10), type the counts or paste straight from a spreadsheet. The calculator builds the expected frequencies, sums the contributions (O − E)² / E, highlights the cells that weigh most and warns when expected counts fall below 5.
Cramér V = 0.137.
Rejects H0 at 5%; V = 0.182.
Still rejects H0.
V = 0.25.
Every expected count is 2.
The p-value is the probability of observing a statistic as extreme as your sample, or more, assuming H0 is true. It is not the probability that H0 is true. With p = 0.0794 and α = 0.05, the data are not "unlikely enough" to reject H0; with p = 0.0028, they are. Very small p-values appear in scientific notation: 2.5e-7 means 0.00000025.
Goodness of fit compares one variable with an expected distribution that you define (for example 1/6 for each face). Independence uses two variables and computes the expected counts from the table own totals.
In "Paste table from a spreadsheet", paste the copied cells (tab separated) and click "Use this table". It must be rectangular, numbers only, up to 10 rows and 10 columns.
No. The χ² grows with the sample size. Strength comes from Cramér V (0 to 1), read by Cohen ranges: 0.1 small, 0.3 medium and 0.5 large for 1 df, with lower limits when the table is larger.
10 values read (separator: comma).
10 values read (separator: comma).
Interpretation
There is not enough statistical evidence, at significance level α = 0.05, that the difference between the population means (group 1 − group 2) is different from 0 (observed difference: -1.58; p-value = 0.079394). Effect size: Cohen d = -0.8322 (large). Failing to reject H0 does not prove it true: the data simply do not provide enough evidence against it, which can also reflect a small sample.
| Group | Size (n) | Mean | s |
|---|---|---|---|
| Group 1 | 10 | 0.75 | 1.789 |
| Group 2 | 10 | 2.33 | 2.0022 |
Difference between means (group 1 − group 2): -1.58. Standard error: 0.8491.
F test of variances: F = 0.7983, p = 0.742720. No sign of different variances.
Levene (Brown-Forsythe): F = 0.2482, p = 0.624374.
SE = √(s₁²/n₁ + s₂²/n₂) = √(3.2006/10 + 4.009/10) = 0.8491t = ((x̄₁ − x̄₂) − δ₀) / SE = ((0.75 − 2.33) − 0) / 0.8491 = -1.8608df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ] = 17.7765p = 2 · P(T ≥ |t|) = 2 · P(T ≥ 1.8608) = 0.079394tcrit = ±2.1028CI 95% = [-3.3655; 0.2055]Two independent samples, each approximately normal (or large). Welch does not require equal variances; the pooled t does. If the same subjects were measured twice, use the paired t.