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# Chi-Square Test Calculator

The chi-square distribution with *d* degrees of freedom gives distribution of the sum of the squares of *d* normally distributed random variables. Its probability density function is

The chi-square distribution is used in hypothesis testing to determine the likelihood of the null hypothesis--that the observations are consistent with a theoretically assumed distribution. The test statistic is χ² is

where O_{i} is the number of observations in group *i*, and E_{i} is the expected number in group *i*. The degrees of freedom is usually 1 less than the number of observations. In practical applications, researchers often reject the null hypothesis if the probability is less than a given threshold, such as 0.05 or 0.01.

### Chi-Square Test Calculator

Use the calculator above to compute the chi-square test statistic and probability for a set of observations. (You can also use a chi-square table.)

### Example

A researcher operates under the assumption that 40% of people in a given area (between the ages of 18 and 65) are employed full-time outside the home, 35% are employed part-time outside the home, and 25% fall into other categories. This assumption is the*null hypothesis*. He samples 320 people and comes up with the following observed data:

Full-time outside the home: 113

Part-time outside the home: 117

Other: 90

The expected data is

Full-time outside the home: 320(0.40) = 128

Part-time outside the home: 320(0.35) = 112

Other: 320(0.25) = 80

The degrees of freedom is 2 and the chi-square value is

(113-128)²/128 + (117-112)²/112 + (90-80)²/80 = 3.23102

The probability of obtaining these observations, given a theoretical distribution of 40%-35%-25%, is 0.198789. This is not low enough to reject the hypothesis; he has no reason to assume that the underlying distribution is

*not*40%-35%-25%.

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