Significance tests

Significance is often given as a p value where the p value is the probability of getting a result which is as extreme by chance if the null hypothesis was true (that there is no difference/correlation between 2 groups). For example if I wanted to see whether tomatoes grown in the open vs in the green house had different sizes, the null hypothesis would be that there is no difference in size and the alternative hypothesis would be that the tomatoes are either bigger or smaller. If I then I picked 5 tomatoes out of each group and measured their diameter, I cannot be sure whether any difference in average size was caused by me accidentally picking bigger tomatoes in one sample compared to the other. Also the number of tomatoes I picked to measure would affect how accurate the measurement represents the true average of all the tomatoes I have grown. The P value calculated would represent the probability of me getting the same result by chance if there was no difference in the size of the tomatoes. The difference in size is said to be significant if it is less than a certain value usually set at either 0.1 or 0.05. For example if the calculated P value of the tomato sizes above was 0.03 then the size difference would be said to be significant and the null hypothesis would be rejected. 

Different significant tests are used depending on the type of data which is collected. These can be broadly divided into parametric tests (Where the parameter is measurable and normally distrubuted- for example measuring the heights of a group of people) and non parametric tests.
Parametric tests: 

• Student T test 
•  paired (compares the same subject at 2 different times) - e.g. comparing the reaction speed of adults before and after drinking alchohol.
• unpaired (compared 2 different people/groups) e.g. comparing the heights of chinese v.s. english adults. 

• Pearson's product-moment coefficient - Looks for correlations between 2 measures - e.g. comparing weight with waist size.  

Non parametric tests:

• Mann-Whitney U test - unpaired data - e.g. Comparing the blood sugar control of diabetics randomised to carb count v.s. traditional methods (assuming that blood sugar control is not normally distributed) 
• Wilcoxon signed-rank test - Paired data -e.g. comparing BM control of the same group of diabetics before and after learning how to carb count. 
• chi-squared test - For proportions and ratios - e.g. the number of students enrolling into each subject after an event to promote the arts. 
• Spearman, Kendall rank - For correlation with nonparametric data

References:
www.passmedicine.com

http://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/BS704_Nonparametric4.html

http://onlinestatbook.com/2/describing_bivariate_data/pearson.html

http://www.stat.yale.edu/Courses/1997-98/101/sigtest.htm