Chi-square or chi-squared?

Question

The $\chi^2$ test/distribution is referred to as either "chi-square" (more frequently) or else "chi-squared" (less frequently).

What is the history behind the name?

Footnote 2 in this paper by Peter Scott makes the following claim (without any corroboration):

The notation of $\chi^2$ is traditional and possibly misleading. It is a single statistical variable, and not the square of some quantity. It is therefore not chi squared, but chi-square. The notation is merely suggestive of its construction as the sum of squares of terms. Perhaps it would have been better, historically, to have called it $\xi$ or $\zeta$.

The Wikipedia article "Pearson's chi-squared test" states that it was first investigated by Karl Pearson in this paper from 1900, in which he apparently just uses the notation $\chi^2$. More historical information can be found in Plackett's 1983 article, "Karl Pearson and the Chi-squared Test".

Some evidence that suggests that it was first called "chi-square", and only later was "chi-squared" used, is the fact in MathSciNet the first paper with "chi-squared" in the title was published in 1958, whereas "chi-square" is used in the title of articles from 1940 onwards.

• Who first used the term "chi-square" in print?

• And why was "chi-square" not "chi-squared" used?

• And when was "chi-squared" first used in print?

Answer 1

It is a single statistical variable, and not the square of some quantity.

I'm not sure on the history, or the proper name (either variation seems fine to me), but this statement is just not true. First of all, for any non-negative random variable, you can always define another random variable by its square-root. More specifically, for any random variable $\chi^2 \sim \text{ChiSq}(n)$ it is also true that $\chi \equiv \sqrt{\chi^2} \sim \text{Chi}(n)$, and you clearly have $\chi^2 = (\chi)^2$, so any chi-squared random variable is the square of a corresponding chi random variable.

Moreover, the chi-squared distribution is usually derived as the distribution of the sum of squares of independent standard normal random variables $Z_1,...,Z_n$ so that:

$$\chi^2 \equiv ||\mathbf{Z}||^2 = Z_1^2 + \cdots + Z_n^2 \sim \text{ChiSq}(n).$$

Obviously it is trivial to define the corresponding quantity:

$$\chi \equiv ||\mathbf{Z}|| = \sqrt{Z_1^2 + \cdots + Z_n^2} \sim \text{Chi}(n),$$

and then you again have $\chi^2 = (\chi)^2$. So the idea that the chi-squared random variable is not the square of any other relevant quantity is clearly not true. Any chi-squared random variable can be conceived as the square of a corresponding chi-random variable, and it can also be derived as the norm of a vector of independent standard normal random variables.

Answer 2

As a matter of fact Pearson's original paper uses $\chi$ alone (without the exponent) frequently. Equation $(i)$ in the paper defines $\chi^2$ but the text speaks of using the equation to find $\chi$. Given this background, it seems absurd to say that $\chi^2$ is not a square.