What is the best method of performing a chi-squared test with only one set of data? I am trying to perform a chi-squared test on the following data:
Achieved: 31
Expected: 19.87179
From a total of: 182
Is it possible to perform such a test on just one item of data like this? 
All the examples I have encountered use multiple sets of data.
Ultimately I would like to be able to quantify the % chance that the above result occurred at random.
 A: The basic chi-square statistic for a test of a proportion being from a population with an expected of 19.8 is (O-E)^2/E =  6.52. (We do need to ask you here whether that was a numeric expected of 19.87 of a proportion or a percentage. If it's a percentage then you need to compare 31 to N*E or 36.)   The statistic given is the normal approximation to an exact test, but it's going to be pretty accurate with an expected of 19. Assuming that was not an expected percentage, then you do the test by comparing it to the 95th percentile of the chi-square distribution which is 3.84. Since it exceeds that critical value (by a wide margin, being very close to the 99th percentile) you would say that there is a significant difference of that proportion from the expected. If you are into p-values (and I'm not) this would have a p-value of (1-0.989333) so let's round it to the nearest hundredth and say p= 0.01. If it were a percentage number, your chi-square statistic would be a very chance-like value of less than 1 and a p-value of 1-pchisq(0.7380947,1) = 0.390.
A: This sounds to me like a standard problem in faint disguise. 
Observed frequencies are total frequency 182, achieved 31, so not achieved 151; expected frequencies are achieved 19.87179, so not achieved is 182 $-$ 19.87179. 
Your chi-square statistic must be calculated from both pairs of observed and expected, with 1 d.f. I get Pearson chi-square statistic of 6.9956 with a P-value of 0.008. 
