chi-square distribution
E596523
The chi-square distribution is a continuous probability distribution commonly used in statistics to model the sum of squared standard normal variables and to conduct hypothesis tests such as goodness-of-fit and tests of independence.
Observed surface forms (3)
| Surface form | Occurrences |
|---|---|
| Pearson's chi-squared test | 1 |
| chi-squared distribution | 1 |
| chi-squared test | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
probability distribution
ⓘ
right-skewed distribution ⓘ |
| assumes |
independent observations in contingency tables
ⓘ
sufficiently large expected cell counts for chi-square approximation ⓘ |
| belongsTo | family of Pearson distributions ⓘ |
| cdf | regularized gamma function of x/2 and k/2 ⓘ |
| characteristicFunction | φ(t) = (1 - 2it)^{-k/2} ⓘ |
| cumulativeShape | increasing and concave for small x ⓘ |
| definedOn | nonnegative real numbers ⓘ |
| hasProperty |
approaches normal distribution as degrees of freedom increase
ⓘ
nonnegative random variable ⓘ right-skewness decreases as degrees of freedom increase ⓘ |
| hasScaleParameter | 2 in gamma representation ⓘ |
| hasShapeParameter | k/2 in gamma representation ⓘ |
| introducedBy | Karl Pearson NERFINISHED ⓘ |
| isSpecialCaseOf | gamma distribution ⓘ |
| kurtosisExcess | 12/k ⓘ |
| mean | k (degrees of freedom) ⓘ |
| mode | k - 2 for k ≥ 2 ⓘ |
| momentGeneratingFunction | M(t) = (1 - 2t)^{-k/2} for t < 1/2 ⓘ |
| origin |
distribution of quadratic forms in normal variables
ⓘ
sum of squares of independent standard normal variables ⓘ |
| parameter | degrees of freedom ⓘ |
| f(x;k) = 1/(2^{k/2} Γ(k/2)) x^{k/2-1} e^{-x/2} for x>0 ⓘ | |
| relatedTo |
F-distribution
ⓘ
Student's t-distribution NERFINISHED ⓘ gamma distribution ⓘ standard normal distribution ⓘ |
| skewness | √(8/k) ⓘ |
| specialCase | exponential distribution when k = 2 ⓘ |
| support | [0, ∞) ⓘ |
| symbol | χ² ⓘ |
| tailBehavior | heavy right tail for small degrees of freedom ⓘ |
| usedFor |
confidence interval for variance
ⓘ
goodness-of-fit test ⓘ test of homogeneity ⓘ test of independence ⓘ variance test ⓘ |
| usedIn |
analysis of variance
ⓘ
categorical data analysis ⓘ contingency table analysis ⓘ hypothesis testing ⓘ likelihood ratio tests ⓘ maximum likelihood estimation theory ⓘ |
| usedWith |
Pearson's chi-square test
NERFINISHED
ⓘ
likelihood ratio chi-square test ⓘ |
| variance | 2k (degrees of freedom) ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Pearson's chi-squared test
this entity surface form:
chi-squared test
this entity surface form:
chi-squared distribution