For more than 2 observations:
chi square =
= (summation from 1 to number of observations ( (((observer A value) - (observer B value)) ^ 2) / ((observer A value) + (observer B value)) ) )
From this value, the probability that the differences between the 2 observers is due to chance can be calculated. The equation can be simplified from an integral depending on whether there is an even or odd degree of freedom.
Even Degrees of Freedom
For even degrees of freedom, this is relatively simple.
probability due to chance (chisquare, degree of freedom) =
= ((e) ^ ((-1) * (chisquare) / 2)) * (summation of i from 0 to I ( (((chisquare) / 2) ^ (i)) / (factorial (i)))
where:
• I = (1/2 * ((degree of freedom) - 2))
2 degrees of freedom
probability = e^((-1) * (chisquare) / 2)
4 degrees of freedom
probability = (e^((-1) * (chisquare) / 2)) * (1 + ((chisquare) / 2) )
6 degrees of freedom
probability = (e^((-1) * (chisquare) / 2)) * (1 + ((chisquare) / 2))+ (((chisquare) ^2) / 8))
8 degrees of freedom
probability = (e^((-1) * (chisquare) /2 )) * (1 + ((chisquare) / 2))+ (((chisquare) ^2) / 8) + + (((chisquare) ^3) / 48))
10 degrees of freedom
probability = (e^((-1) * (chisquare) / 2)) * (1 + ((chisquare) / 2))+ (((chisquare) ^2) / 8)+ (((chisquare) ^3) / 48) + (((chisquare) ^4) / 384))
Odd Degrees of Freedom
For odd degrees of freedom, this is quite complex, and it is easier to use the Excel function CHIDIST.
probability due to chance (chisquare, degree of freedom) =
= 1 - ( (1 / (gamma function (I + 1))) * (summation from 0 to infinity (((-1)^i) * (((chisquare) / 2) ^ (I + i + 1)) / ((factorial (i)) * (I + i + 1)))
where:
• I = (1/2 * ((degree of freedom) - 2))
