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I ran a textbook example in an Excel sheet - and replicated the answer. However when I retain the ratio, but double the sample size my result is not significant.

This strikes me as illogical. Please can someone explain where I have gone wrong, or why a fixed ratio does not give same chi-square result.

Test used X2 = (Abs(O-E)-0.5)squared / E
df= 1 (2 categories)
Test 1 (text book example- significant diff)
Females 4
Males 12
Total 16
Expected Females 8
Expected Males 8
Yates corrected X2 = 3.06

Test 2 - same ratio, diff sample size
Females 8
Males 24
Total 32
Expected Females 16 
Expected Males 16
Yates corrected X2 = 7.03
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  • $\begingroup$ degree of freedom is not 1.It is 16 -1= 15 and 32 -1 = 31for test -2. Please correct it. $\endgroup$ – Subhash C. Davar Nov 14 '17 at 10:33
  • $\begingroup$ @subhashc.davar Sorry, but that is just confused at a very elementary level. $\endgroup$ – Nick Cox Nov 14 '17 at 11:58
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There are two questions here, specific and general.

In general, the logic of significance tests certainly does include consideration of sample size.

In my favourite software I get (without Yates' correction)

observed frequencies from keyboard; expected frequencies equal

         Pearson chi2(1) =   4.0000   Pr =  0.046
likelihood-ratio chi2(1) =   4.1860   Pr =  0.041

  +-------------------------------------------+
  | observed   expected   obs - exp   Pearson |
  |-------------------------------------------|
  |        4      8.000      -4.000    -1.414 |
  |       12      8.000       4.000     1.414 |
  +-------------------------------------------+

observed frequencies from keyboard; expected frequencies equal

         Pearson chi2(1) =   8.0000   Pr =  0.005
likelihood-ratio chi2(1) =   8.3720   Pr =  0.004

  +-------------------------------------------+
  | observed   expected   obs - exp   Pearson |
  |-------------------------------------------|
  |        8     16.000      -8.000    -2.000 |
  |       24     16.000       8.000     2.000 |
  +-------------------------------------------+

Specifically, it is hard to see why you describe the results as you do. Your output doesn't include observed significance levels at all, but either test yields significance at conventional level (all P-values are $\le$ 0.046, compared with the conventional level of 0.05 = 5%). The second has higher chi-square and thus for the same number of degrees of freedom a lower P-value (in ordinary language, a more significant result).

On the general point, take a different example: You're tossing a coin and get 6/10 heads. Is the coin fair (with probability 0.5 for heads, 0.5 for tails)? Do you regard possible proportions 60/100, 600/1000, ... 6 billion/10 billion, ... in exactly the same way? 6/10 could easily be a fluke, in ordinary language, but at some point as you get higher chi-square and lower P-values, you should change your mind.

The use of Yates' correction here is a side-issue except insofar as it affects threshold significance levels.

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  • $\begingroup$ Excel tells me that the X2 value is greater and the probability lower- as per Nick Cox answer. HOWEVER the book had me look up an Appendix and informed me that my X2 value had to be LESS than a threshold value for p < 0.5 OR p < 0.01 and that leaves me most confused. I am going to trust Excel (!) and ignore the black box sensation that gives me! $\endgroup$ – m4sterbunny May 26 '17 at 13:03
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    $\begingroup$ Unnamed and so inaccessible book, so I can't possibly comment reliably. I agree that chi-square should be larger than stated threshold values to qualify as more significant. $\endgroup$ – Nick Cox May 26 '17 at 13:07
  • $\begingroup$ @m4sterbunny It might be worth quoting the book, with some context and asking about it (in another question) $\endgroup$ – Glen_b May 26 '17 at 17:13
  • $\begingroup$ Degree of freedom here should be calculated as n-1 and not as number of categories - 1. $\endgroup$ – Subhash C. Davar Nov 14 '17 at 10:38
  • $\begingroup$ @subhashc.davar Your suggestion is extraordinary. Care to flesh it out? $\endgroup$ – Nick Cox Nov 14 '17 at 10:42

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