Here is the original article describing Kincaid’s Automated Readability Index (ARI) . Apparently, there are several tools online that do not calculate ARI grade levels the right way.
The Instructions for Automated Readability Index section explains how this is done.
See also Appendix E of this PDF.
Historical Typo: Note that in the Document Resume section of the paper, the author’s name was mispelled as Kinkaid. This has caused a spurious record across some journals and reference works as no one known as Kinkaid wrote the article. This reminds me of The most influential paper that Gerard Salton never wrote.