Found this after a quick google: "R^2 is used to quantify the amount of variability in the data that is explained by your model. It's useful for comparing the fits of different models.

The Chi-square goodness of fit test is used to test if your data follows a particular distribution. It's more useful for testing model assumptions rather than comparing models."

sounds like Chi-square is more useful if you have a function you are trying to test (or a model you are trying to fit to your data) as opposed to the R^2 which tells you how much variability there is in your data, and therefore how much the best model fits.

*Univariate analysis* stands for the statistical analysis of data collected from a single variable. See https://en.m.wikipedia.org/wiki/Univariate_analysis, for instance.

Such data are typically collected from a sample intended to be representative of some underlying population. For example, a random sample of 100 users are asked to rate their satisfaction with the Cross Validated forum on a numerical scale from 0 (not satisfied) to 3 (very satisfied). (The intermediate ratings are 1 = satisfied, 2 = moderately satisfied.) In this example, the underlying population consists of *all* current users of the Cross Validated forum.

If the collected data are analyzed with the sole purpose of describing what happens in the sample itself, then the corresponding analysis is a *descriptive univariate analysis*. In the example provided, a descriptive univariate analysis would entail simply reporting the proportion of users among the 100 considered in the sample who selected each of the satisfaction ratings.

If the collected data are analyzed with the purpose of learning something about the underlying population, then the corresponding analysis is an *inferential univariate analysis*. For our example, you might want to test if the proportion of users in the underlying population who are *not* satisfied with the Cross Validated forum is less than 0.15.

There is a univariate Chi-square test one could use as part of the inferential univariate analysis - the so-called *Chi-square goodness of fit test* (see https://stattrek.com/chi-square-test/goodness-of-fit.aspx). For the example provided, let's say that the people running the Cross Validated forum believe that 50% of all of their users are statisticians, 20% of all of their users are data scientists and the remaining 30% are scientists from other domains. They will examine their sample of 100 users to see what percentage of users in the sample fall in each of these professional categories. Using the sample-provided data, the people running the Cross Validated forum will test the following competing hypotheses via the *Chi-square goodness of fit test*:

*Null hypothesis*: The percentage of statisticians, data scientists, and scientists from other domains among *all* Cross Validated forum users is 50%, 20% and 30%, respectively.

*Alternative hypothesis*: At least one of the percentages stated in the null hypothesis is false.

P.S. If you do a Google search, you will see that many people incorrectly refer to a *Chi-square test of association" as being a univariate procedure. However, this particular testing procedure is a bivariate procedure, as it is concerned with testing the association between two categorical variables in an underlying population using the data collected on those variables from a sample representative of the population.

## Best Answer

As @whuber explained, the "problem" with use the chi-squared test to assess conformity to Benford's law is that it lacks statistical power. The discrete version of the Kolmogorov-Smirnov goodness-of-fit test (GOF) has better power. There are, however, other alternatives designed especially to test conformity to Benford's law See here for a wider discussion of the matter.