Chi-Square Distribution

Chi-square PDF, CDF, and critical values

X ~ χ²(k = 3) · P(X ≥ 7.815)
P(X ≥ x)
0.049994
Density f(x)
0.022406
P(X ≤ x)
0.950006
P(X ≥ x)
0.049994
Mean (k)
3
Variance (2k)
6
Chi-square density curve with the selected area shaded. P(X ≥ 7.815)Chi-square density curve with the selected area shaded. P(X ≥ 7.815)xf(x)2468101214160.050.10.150.20.25

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The Chi-Square Distribution Calculator draws the χ²(k) density curve for any degrees of freedom and shades the area you choose — the left tail P(X ≤ x), the right tail P(X ≥ x), or the critical value for an upper-tail area α. It reports the density f(x), both tail probabilities, and the distribution's mean and variance. Useful for chi-square goodness-of-fit and independence tests, p-values, and confidence intervals for a variance.

أمثلة شائعة

  • χ²(3): the 95th percentile is 7.815, so P(X ≥ 7.815) = 0.05
  • χ²(1): P(X ≤ 3.841) = 0.95 (the one-degree critical value)
  • χ²(10): mean = 10, variance = 20, P(X ≤ 10) ≈ 0.560
  • Critical value for α = 0.05 with k = 4 is 9.488
  • χ²(2) is an exponential curve: P(X ≥ 5.991) = 0.05