Sometimes in a fieldwork it is only possible to collect small samples due to limitations of available individuals or of time:
the Comparison of Two Means allows you to reliably compare the two entire populations
from which the samples are taken. It can tell you if the two samples
(given as numerical nonranked data) are truly significant (in which
case the two entire populations are truly different), or if the
differences between the two samples simply occurred by chance (in which
case you cannot draw a conclusion as to the similarity/dissimilarity
of the two entire populations).
Limitations of the Comparison of Two Means:
Method:
Limitations of the Comparison of Two Means:
 If both samples have less than 30 values, use the TTest
 If the sample is NOT normally distributed, or for ordinal (=ranked) data, use the Mann Whitney UTest
Method:
 The Null Hypothesis is that there is NO significant difference between the two samples
 Calculate (D) = the difference between the means of the two samples (use only the positive value of the result)
 Calculate (SE) = Standard Error of Difference, with the formula above, where (s) is the standard deviation of each sample, and (n) the number of values in each sample
 If D / SE > 2, then the Null Hypothesis can be rejected with 95% certainty, which means there is only a 95% probability that the entire populations from which the samples are taken are significantly different.
Example: comparison between the IB grades obtained at two schools
We have sampled only 10% of the student body in each school. School A has 400 students (sample = 40) and school B has 500 students (sample = 50). The results are as follows:
We have sampled only 10% of the student body in each school. School A has 400 students (sample = 40) and school B has 500 students (sample = 50). The results are as follows:

Clearly students at School B seem to perform better than at School A: is the a coincidence resulting from the students that were sampled (ie: another sample might show otherwise), or is this difference representative of a significant difference between the results for all students in both schools?
