# Is there a specific name for this formula?

I was wondering if there was a specific name for this formula and if it is involved in an established statistical process (like how chi-squared tests are for comparing observation classes or t tests are for distributions without a known population stddev), or if this is an original development from the referenced article author. I wish to understand its usage a bit more is why.

How it is referenced in the article:

A further test, based on the work of Ellegard and Thomson, was devised to determine "Distinctive Numbers" ... The "distinctive numbers" which result are therefore given the symbol D. This test is in fact finding where x-epsilon is the expected occurrence, x-0 the observed occurrence and x the average occurrence in the grand total. The effect of this is to increase the denominator when the Stowe manuscript uses a word either more often or less often than the average, giving a lower D value and weighting the result so that increased notice is taken of occurrences when abnormalities in the texts coincide.

The formula in question: $$D=\sum\frac{[(x_\epsilon-\bar{x}) - (x_0-\bar{x})]^2}{|x_\epsilon-\bar{x}|}$$

The article where I found it: Harvey, P. (1970). Stylistic Analysis

## 1 Answer

It looks something like a $\chi^2$ test, though the denominator is slightly different.

If you run through the (trivial) algebra, the numerator reduces to ($x_\epsilon - x_o)^2$, which is just like the $\chi^2$ test's numerator of $(N_{\textrm{expected}} - N_{\textrm{observed}})^2$. The denominator for a $\chi^2$ test is normally just the expected value ($x_\epsilon$ in your notation). Here, they appear to be adjusting the expected value for each candidate according to its prevalence in their entire corpus. I can see why you'd want to do this pragmatically, but it does mean that you can no longer treat the result as an actual $\chi^2$ value (e.g., for looking it up on a table).

If you didn't want to call it a "distictive number", I suspect you might get away with calling it a "modified $\chi^2$ value" or something like that--I'm not sure it has another name.