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 non-ranked 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:

• If both samples have less than 30 values, use the T-Test
  
• If the sample is NOT normally distributed, or for ordinal (=ranked) data, use the Mann Whitney U-Test

The Comparison of Two Means is used when at least one of the samples has more than 30 values AND both samples are normally distributed.

Method:

1. The Null Hypothesis is that there is NO significant difference between the two samples
   
2. Calculate (D) = the difference between the means of the two samples (use only the positive value of the result)
3. 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
4. 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:

	School A	School B
Size of Sample (n)	40 Students	50 Students
Mean IB Grade (x̄)	4.1	5.3
Standard Deviation (σ)	1.1	1.3

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?

• D = 5.3 - 4.1 = 1.2 and SE = 0.25
  
• D / SE = 4.74 (> 2)
  
• We can therefore conclude, with 95% certainty, that the fact that there is a difference between the sample is NOT a coincidence, and that the ENTIRE population of students in school B does better, on average, than the ENTIRE population of students in school B