# Are there alternative names for the Pearson chi-squared test?

I have two books I am using to try to get a feel for statistics and neither mentions the Pearson's chi-squared test, although it has a Wikipedia article to itself. Are there any alternative names or terms that I can look up to find information about it? I would like to be able to tie in the Wikipedia article with what is in the books.

The two books are "Statistics for Dummies" by Deborah J Rumsey, and "A Primer in Data Reduction" by A.S.C. Ehrenberg.

Following from the Martin Westerings comment, there is no entry for "chi-squared" in the index, so what should I look for there?

• In relation to the Ehrenberg book a quick peek at the contents suggests that if the chi-squared tests are to be found anywhere they'll be under "tests for proportions", in chapter 10 (specifically, p145-149). If they're not there, I would guess they're not all that likely to be anywhere else. Nov 20, 2017 at 23:08
• "Contingency table" and even "table" or "2x2 table" are often good targets for searches. You can also consider looking up information about "count data."
– whuber
Nov 20, 2017 at 23:29

Q1: I can't find the $\chi^2$-test (chi squared test) in my book. Is there an alternative name?

There aren't really alternative names. The chi-squared test is described in "statistics II for dummies".

The wikipedia article is already very fine though. I do not expect that the 'dummies' series does much better (disclaimer: I am not a fan of any "... for dummies" book... they teach you to remain a dummy).

Q2: What should I look for? Part I

statistics for dummies

The first book from the series 'statistics for dummies' seems to speak about 2-way tables and 'checking independence' (chapter 19).

In this simple case (low number of categories) a test for equality two proportions is done (chapter 15).

A primer for data reduction sired l I can not find much online about this book. But I can find the questions in this book.

Question 10.5 in the chapter on 'tests of significance' is about the chi-squared test to determine independence. Also the term 'chi-squared distribution' occurs in the index of this book, which points to chapter 10.

Chapter 3 'structured tables' helps you to write and interpret tables. Making the inference more qualitative, e.g. the answer to question 3.3 speaks about 'interactions' without giving any calculated measure, number, or test, that supports it and basing it on an intuitive look at the numbers in the table.

Chapter 4 'observed' distributions' helps you to write and interpret frequency tables of (observational) sample distributions.

Q2: What should I look for? Part II

The chi-squared test is a test that can be applied to multiple types of problems. If you are strict then you could say that Pearson's chi-squared test is a bit less general and only applies to test in which the chi-squared statistic is calculated by the observed and expected frequencies or counts.

(there can be other statistics which are chi-squared distributed, e.g. the log of a likelihood ratio or the test statistic in the Cochran–Mantel–Haenszel test which is based on odds-ratios).

While I referred to the two-way table as an application that you may be looking for. You should note that the (Pearson's) chi-square test is more general than the application. So you should tell us a bit more information and clarify your question. Why, are you looking for the chi-squared test?

(historically the chi-squared test started as a goodness of fit developed by Pearson around 1900, with some creativity a contingency table and test of independence can be seen as a fit as well, fitting a model of Independence and then testing it's goodness of fit)

• I have edited my question to take account of Martin Westerings's comment, Nov 20, 2017 at 22:44

The way I understand it, Pearson's chi-square refers to the test statistic.

$$\chi^2= \sum_{i=1}^n\frac{(O_i - E_i)}{E_i}$$

There are two commonly encountered forms of the test: 1) a goodness-of-fit test (of observed counts) to expected counts under some model (e.g., a proportional model) and 2) the chi-square contingency test (where the expected counts are inferred from the product of the row sum and column sum, divided by the total count...in other words, the multiplication rule is used to infer expected probabilities based on the assumption that the categories are independent..and is then converted to an expected count by multiplying it by the total count).

The degrees of freedom for the first test is the number of categories minus one minus the number of parameters inferred from the data.

The degrees of freedom for the second test is equal to the product of the number of row minus one and the number of columns minus one.

If I recall, the Pearson chi-square statistic is an approximation to the G-statistic.

• Chi-square statistics can arise in other ways too. These two forms are, in my experience, the forms that arise most frequently at elementary level. Picky point: $n$ in your formula refers to the number of categories, whereas at both elementary and more advanced levels it is often the number of observations in a sample. Some agnostic notation such as $\sum_{\rm categories}$ might suit. Nov 21, 2017 at 10:11
• Good point Nick. It would great if there was a variable that was known to represent categories....because spelling out categories in a summation would be a space killer. Nov 22, 2017 at 2:01
• I like the notation $\sum_{i=1}^I$ or $\sum_{i=1}^I \sum_{j=1}^J$ but a notation such as $\sum_{\rm bins}$ is attractively simplified compared with a small snowstorm of summation signs and subscripts, especially for multiway tables. Nov 22, 2017 at 8:24