Performing experiments using FTFs

A table of critical values for chi-square

df  p = 0.05  p = 0.01 
3.841  6.635 
5.991  9.210 
7.815  11.345 
9.488  13.277 
11.070  15.086 
12.592  16.812 
14.067  18.475 
15.507  20.090 
16.919  21.666 
10  18.307  23.209 
11  19.675  24.725 
12  21.026  26.217 
13  22.362  27.688 
14  23.685  29.141 
15  24.996  30.578 
16  26.296  32.000 
17  27.587  33.409 
18  28.869  34.805 
19  30.144  36.191 
20  31.410  37.566 
21  32.671  38.932 
22  33.924  40.289 
23  35.172  41.638 
24  36.415  42.980 
25  37.652  44.314 

How to use this table

The table lists critical values of χ² for degrees of freedom, df, from 1 to 25. It quotes critical values at two levels: with a probability of error, p = 0.05, i.e., 1:20, and p = 0.01, i.e., with an error of 1:100. Usually an error rate of 1 in 20 is quite sufficient. This means that there is a chance of one in twenty of detecting a change in the sample when there is no real change in the population. If you choose the lower threshold, you tend to make the opposite error of being over-cautious.

The number of degrees of freedom for a contingency table column is the number of cells, minus one, usually written as df = r-1 where r is the number of rows in the column. The number of degrees of freedom for an entire table or set of columns, is df = (r-1) x (c-1), where r is the number of rows, and c the number of columns.

We should remind the reader that each cell in the expected distribution must have at least 5 cases in it; if not, collapse values together, as in the two-features-in-a-clause example here (see the effect of mood on ditransitive). The number of degrees of freedom will then fall.

If your value of df is greater than 25, you will need to refer to other tables. (NB. Sometimes error levels are quoted by the probability that the test is successful, i.e., p = 0.95; 0.99, as below. If you see figures like this, just subtract them from 1.)

The mathematics of chi-square

You may be interested to know that critical values of chi-square may be calculated from first principles. This is useful if you need to calculate the critical value for fractional degrees of freedom or for different error levels.

The graph plots critical threshold lines for p = 0.95 (red) through to p = 0.99 and even 1 (top line). Note that the line straightens out as df increases so you can estimate critical values for higher degrees of freedom relatively easily.

The shaded area is beyond the limit (p = 1 / probability of an error is 0). The critical value for p = 1 is finite and can be exceeded by a χ² test, especially if a large amount of data is found.

Philosophical question: would an error level of 0 mean that an experimental result was guaranteed correct? (A. No, it means that there is a limit to how measurable the reproducibility of an experiment is.)

Online resources

»   Statistical tables from the US National Institute of Standards and Technology (NIST) handbook
» Critical value calculator courtesy of David W. Stockburger
» 2 x 2 χ² (Excel)
»   Further reading: z-squared: the origin and use of χ² (PDF)

FTF home pages by Sean Wallis and Gerry Nelson.
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This page last modified 28 January, 2021 by Survey Web Administrator.