Statistics/Distributions/Chisquare
Chisquare Distribution[edit]
Probability density function 

Cumulative distribution function 325px 

Notation  or 

Parameters  (known as "degrees of freedom") 
Support  x ∈ [0, +∞) 
CDF  
Mean  k 
Median  
Mode  max{ k − 2, 0 } 
Variance  2k 
Skewness  
Ex. kurtosis  12 / k 
Entropy  
MGF  (1 − 2 t)^{−k/2} for t < ½ 
CF  (1 − 2 i t)^{−k/2} ^{[1]} 
Chisquare distribution is related to normal distribution. A chisquare statistic is the sum of a number of independent and standard normal random variables.
Assume that we have n number of random variables Z, that are normally distributed. Therefore, we can write . If we square Z such that , then we get the chisquare distribution . If we sum n number of , we can write
.
One example could be that we want to know whether the weight of a set of eight apples is normally distributed. Chisquare distribution can be used to test for this. Assume that the apples weigh 88, 93, 110, 76, 78, 121, 92 and 86 grams, and we have knowledge of the mean and the standard deviation weight of all apples. We obtain the normally distributed Z values by subtracting the mean weight (93) and divide by the standard deviation (15.41). For example, the first apple has got using four decimal points. Square all the Z values, then taking the sum yields a Chisquared distributed random variable with mean 8 and variance 16.
Now when we have the value of the chisquare statistic Y, we compare it to the critical value of the chisquare distribution at n = 8 degrees of freedom and 95% level of significance which can found in a Chisquare statistical table. The null hypothesis is that the sample of apples is normally distributed. It is rejected if the value of the test statistic is higher than the critical value.
The chisquare distribution is a special case of the gamma distribution, where a=2 and p=k/2. The probability density function is:
Summary statistics[edit]
The mean of a chisquared is
The variance of a chisquared is
For the proof of these, see the gamma distribution.