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The following table is of UK domiciled university applicants to UK universities in $2013$ and of acceptances, grouped by self-declared disability.

A $\chi^2$ test on the numbers shows an extremely statistically significant difference between the groups (perhaps suggesting that acceptances are not completely random) even though the quoted two columns of percentages are the same when rounded to one decimal place. This curiosity is related to the large numbers involved, discussed in another question.

The differences in success rates between groups is not large: they in fact range from $72.7\%$ for those declaring a mental health condition to $80.8\%$ for those declaring an autistic spectrum disorder or Asperger's Syndrome, with a figure of $77.2\%$ both overall and for the large majority with no disability declared.

What is a good measure of how close the distributions of applications and acceptances are?

How should the fact that most individuals are in a single subgroup be taken into account? If the total number of acceptances is fixed then the success rate for those with no disability declared must be within the range $75.2\%$ to $83.9\%$ even if everybody else was accepted or everybody else was rejected. 

Disability                              Applicants    Acceptances   
                                          No.    %      No.   %
Autistic spectrum disorder or 
    Asperger's Syndrome                 2,635   0.5   2,130  0.5
Blind or partially sighted                565   0.1     440  0.1
Deaf or serious hearing impairment      1,035   0.2     805  0.2
Long standing illness (e.g. diabetes, 
          epilepsy, heart condition)    4,515   0.8   3,355  0.8
Mental health condition                 5,170   0.9   3,760  0.9
Specific learning difficulty (e.g. 
     dyslexia, dyspraxia)              23,480   4.2  18,085  4.2
Physical impairment or 
     mobility difficulties              1,325   0.2   1,020  0.2
Other disability or medical condition   4,795   0.9   3,755  0.9
Two or more of the above                1,695   0.3   1,275  0.3
No disability declared                518,105  92.0 400,065 92.0
Totals*                               563,319  100  434,688 100

Using R:

applic <- c(2635, 565, 1035, 4515, 5170, 23480, 1325, 4795, 1695, 518105) 
accept <- c(2130, 440,  805, 3355, 3760, 18085, 1020, 3755, 1275, 400065) 
chisq.test(cbind(accept, applic - accept))

gives

        Pearson's Chi-squared test
data:  cbind(accept, applic - accept)
X-squared = 107.5192, df = 9, p-value < 2.2e-16
Community
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Henry
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  • Wouldn't exploring *where* the differences exist be of more interest than measuring "how close" they are to being random? You could learn that autistic applicants have a 3% greater chance of acceptance and people with illness or mental health conditions have a 3% to 4% lesser chance. The other variations are consistent with a random mechanism. – whuber Jan 09 '15 at 21:39
  • I might use the maximum of the absolute differences between the groups and the overall average, as in, "No group's acceptance rate was more than 4.5% different from the overall acceptance rate", inadequate though this is to summarize the results. – jbowman Jan 09 '15 at 23:24
  • There are other similar tables (such as gender, age, and ethnic origin) which show different disparities as well has having a variety of distributions between the size of the largest group and of the others. I was wondering if there was some sensible summary statistic that I had missed which like the $\chi^2$ statistic gave a greater weight to disparities in the larger groups but which was not so dependent on the total number of individuals as the $\chi^2$ statistic. For example the $\chi^2$ statistic divided by the total number of individuals would be unitless. – Henry Jan 10 '15 at 00:23
  • Are you looking for something like an effect size? a confidence interval? If you just want to see how close the distributions are, what about some visualization of the data? – Glen_b Jan 10 '15 at 03:23
  • @Glen_b: I am looking for a summary measurement of "closeness" which is insensitive to scale (e.g. multiplying all the data by $10$) and can be used for meaningful comparisons with other splits of the same population by different factors and for comparisons with other cases for example when the success rates change. – Henry Jan 10 '15 at 14:35

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