The chi-square curve is a family of curves that depend on a parameter called degrees of freedom (d.f.). The chi-square curve is an approximation to the probability histogram of the chi-square statistic for multinomial model if the expected number of outcomes in each category is large. The chi-square curve is positive, and its total area is 100%, so we can think of it as the probability histogram of a random variable. The balance point of the curve is d.f., so the expected value of the corresponding random variable would equal d.f.. The standard error of the corresponding random variable would be (2×d.f.)^½. As d.f. grows, the shape of the chi-square curve approaches the shape of the normal curve. This page shows the chi-square curve.