Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test compares observed frequencies with frequencies expected under an assumed distribution. The notation χ²=Σ(O_i−E_i)²/E_i shows which quantities are connected. Here, O_i: Observed frequency in category i (count); E_i: Expected frequency under H₀ (count); k: Number of categories (dimensionless). Dimensionless; observed and expected counts use the same category units. An application depends on whether the supplied quantities match the definitions and reference directions. Common uses include Test whether categorical data fit a specified probability distribution.. Read the variables and their dimensions before substituting numbers; this often reveals whether the result should be a length, rate, probability, or dimensionless statistic.

Start with a diagram or a list of known quantities, assign each one a symbol, and convert units to a consistent form. Then choose the appropriate equation and substitute values only afterward. Example: For O=(18,32), E=(25,25): χ²=49/25+49/25=3.92. With two categories and no estimated parameters, df=1. Check: square each difference, then divide by E_i. Verification is part of the calculation: Under H₀, the statistic is approximated by a chi-square distribution Also compare the magnitude and unit with the context. Round only at the end, since early rounding can affect multi-step calculations, especially when small differences or sums are involved.

Under H₀, the statistic is approximated by a chi-square distribution. If no model parameters are estimated from data, degrees of freedom are usually k−1; estimated parameters reduce them. Expected cell counts should be sufficiently large, classically at least 5. Categories must be mutually exclusive and cover the same observation unit. These are not minor footnotes: when assumptions fail, the formula can produce a precise but inappropriate result. Before calculating, identify which values are observed, estimated, or modeled, and whether special cases change the equation. Test whether categorical data fit a specified probability distribution. For numerical methods, also check step size and stability; for statistics, sampling design and distribution; for calculus, differentiability, boundary, and orientation. Use another method when these conditions do not hold.

Both relations belong to the same topic, but answer different questions. “Chi-Square Goodness-of-Fit Test”: The chi-square goodness-of-fit test compares observed frequencies with frequencies expected under an assumed distribution. The companion relation “Binomial Distribution” The binomial distribution gives the probability of exactly k successes in n independent trials with success probability p. Choose based on the quantity sought and the model: Test whether categorical data fit a specified probability distribution. The worked example For O=(18,32), E=(25,25): χ²=49/25+49/25=3.92. With two categories and no estimated parameters, df=1. Check: square each difference, then divide by E_i. shows which quantities actually enter this relation. Compare both formulas using the same data and identify the companion’s extra assumption. This clarifies which relation fits.

A statistic alone proves neither causation nor a hypothesis. Check sampling, independence, denominator and distribution assumptions before interpreting a test or interval. The symbols must also match their definitions exactly: O_i: Observed frequency in category i (count); E_i: Expected frequency under H₀ (count); k: Number of categories (dimensionless). Do not calculate mechanically; write the assumption beside the setup, then check dimensions and sign. Also note: Under H₀, the statistic is approximated by a chi-square distribution The example For O=(18,32), E=(25,25): χ²=49/25+49/25=3.92. With two categories and no estimated parameters, df=1. Check: square each difference, then divide by E_i. lets you follow these checks directly. If the assumptions do not fit, choose another relation from the chapter instead of forcing values into an unsuitable formula.