0.12204224
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10
3.162278
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0.12204224
5
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3.162278
3
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1
The Chi-Square Distribution Calculator evaluates the probability density function (PDF) and statistical properties of the chi-square (χ²) distribution, one of the most important distributions in inferential statistics. The chi-square distribution with k degrees of freedom is defined as the distribution of the sum of squares of k independent standard normal random variables. It plays a central role in hypothesis testing, confidence interval construction, and goodness-of-fit analysis.
The chi-square distribution is ubiquitous in statistical practice. The chi-square goodness-of-fit test determines whether observed categorical data match expected frequencies. The chi-square test of independence assesses whether two categorical variables in a contingency table are related. In regression analysis, the chi-square distribution governs the distribution of deviance statistics in generalized linear models. The likelihood ratio test, arguably the most general hypothesis testing framework, relies on the asymptotic chi-square distribution of the test statistic.
The distribution is parameterized solely by its degrees of freedom k. For small k, the distribution is heavily right-skewed; as k increases, it becomes more symmetric and approaches a normal distribution (by the central limit theorem, since it is a sum of independent variables). The mean equals k and the variance equals 2k, so the coefficient of variation is √(2/k), which decreases as degrees of freedom increase. The chi-square distribution is a special case of the gamma distribution with shape α = k/2 and rate β = 1/2.
Mathematically, the chi-square distribution is also central to the construction of confidence intervals for the population variance. If a sample of size n has sample variance s² from a normal population with true variance σ², then (n−1)s²/σ² follows a χ²(n−1) distribution. This result directly yields the confidence interval for σ². Additionally, the chi-square distribution underlies both the t-distribution (as the denominator involves a chi-square variable) and the F-distribution (which is a ratio of two scaled chi-square variables).
The chi-square distribution is additive: the sum of independent chi-square variables with degrees of freedom k₁ and k₂ is chi-square with k₁ + k₂ degrees of freedom. This property is fundamental to combining test statistics from independent experiments and to understanding how complex test procedures decompose into simpler components. The moment generating function M(t) = (1 − 2t)^(−k/2) exists for t < 1/2, from which all moments can be readily derived.
The chi-square PDF with k degrees of freedom is:
$$f(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \quad x > 0$$
This is equivalent to a Gamma(k/2, 1/2) distribution. The key moments are:
$$\mu = k, \quad \sigma^2 = 2k, \quad \text{Mode} = \max(k - 2, 0)$$
For hypothesis testing, you compare your computed test statistic against critical values from the chi-square distribution. The p-value is 1 − CDF(test statistic), representing the probability of observing a value at least as extreme under the null hypothesis.
The PDF value gives the density at point x. If your test statistic falls in a region of very low density in the upper tail, this suggests the null hypothesis may be false. The mean equals the degrees of freedom and represents the expected value of the chi-square statistic under the null hypothesis. The mode at k − 2 (for k ≥ 2) shows the most probable value. The standard deviation √(2k) helps gauge whether an observed statistic is unusually large. As a rule of thumb, values beyond the mean plus 2 standard deviations warrant attention.
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With df=5, the critical value at α=0.05 is 11.07. The PDF at this point is low (0.028), confirming this is in the tail region. Values exceeding 11.07 lead to rejecting the null hypothesis.
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At df=20, x=20 equals the mean, near the center of the distribution. The PDF of 0.060 confirms this is in the high-density region. The mode at 18 shows slight right-skew even at df=20.
Degrees of freedom (df) represent the number of independent standard normal variables being squared and summed. In a goodness-of-fit test, df = (number of categories − 1 − number of estimated parameters). In a test of independence on an r×c contingency table, df = (r−1)(c−1). For a variance confidence interval from a sample of size n, df = n − 1. The degrees of freedom determine the shape, mean, and spread of the distribution.
Calculate the test statistic χ² = Σ(O − E)²/E, where O and E are observed and expected frequencies for each category. Determine df = k − 1 (k categories) minus the number of parameters estimated from data. Compare χ² to the critical value from the chi-square table at your significance level. If χ² exceeds the critical value, reject the null hypothesis that the data follow the expected distribution. Ensure all expected frequencies are at least 5 for the approximation to be reliable.
If Z is a standard normal variable, then Z² follows a χ²(1) distribution. More generally, if Z₁, ..., Zₖ are independent standard normals, then Z₁² + ... + Zₖ² follows χ²(k). This is the defining relationship. As k increases, by the central limit theorem, the chi-square distribution approaches a normal distribution with mean k and variance 2k. The approximation N(k, 2k) is reasonable for k > 30.
Use the chi-square test for large samples where all expected cell frequencies are at least 5. Use Fisher's exact test when sample sizes are small, expected frequencies fall below 5, or you need an exact p-value rather than an approximation. For 2×2 tables with moderate samples, Yates' continuity correction can improve the chi-square approximation. In general, Fisher's exact test is always valid but computationally intensive for large tables, while the chi-square test is an efficient approximation for large samples.
To test H₀: σ² = σ₀² against alternatives, compute (n−1)s²/σ₀², where s² is the sample variance. Under H₀, this statistic follows χ²(n−1). For a confidence interval, the 100(1−α)% CI for σ² is [(n−1)s²/χ²_{α/2}, (n−1)s²/χ²_{1−α/2}] where the chi-square quantiles have n−1 degrees of freedom. This test requires the population to be normally distributed; it is sensitive to departures from normality.
No. The chi-square distribution is defined only for non-negative values (x ≥ 0). This makes sense because it is the sum of squared standard normal variables, and squares are always non-negative. The PDF is zero for x ≤ 0. For df = 1 or 2, the density is highest near zero; for df ≥ 3, the density starts at zero, rises to a peak at the mode (df − 2), and then decays exponentially in the right tail.
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