Before discussing the core statistical tests, I will briefly discuss some concepts that will help us to make sense out of statistical analyses. These concepts are important, but are largely ignored in class room discussions. Even if you are an advanced user of statistics, you will find these discussions valuable.
MATHEMATICAL AND EMPIRICAL PROBABILITIES: In statistical analyses, we are interested in assessing the probabilities for occurrence of certain events. For example, suppose that we are interested in finding out the probability of getting a 4 in cast of a die. This question can be answered in two ways: first, through MATHEMATICAL or also called classical probability, and second, through EMPIRICAL probability. In the realm of MATHEMATICAL probability, our approach to answer will be through the use of the formula: number of favourable event/ total number of possible events. In this case, this is 1/6 (‘4’ is one favourable event, and the total possible events are 1 through 6 i.e. 6). MATHEMATICAL probability is an a priori probability. It exists irrespective of the number of throws of die actually made.
Apart from the MATHEMATICAL probability, there is also the concept of EMPIRICAL PROBABILITY. This refers to the probability based on a particular sample. The EMPIRICAL probability is a posteriori. It changes from sample to sample. In the example of the die, we may cast the die, lets say 10 times, and we may get a face of 4 lets say, three times. Then, our EMPIRICAL probability of getting a 4 is 3/10. Again, this probability will change with the change in the sample i.e. number of times the die is cast.
In statistical analysis, we are concerned with EMPIRICAL probability. So we draw a sample, and use the sample values as an estimate of the probability. In fact, we chose EMPIRICAL probability because of the law of large numbers, which says that EMPIRICAL probability becomes equal to MATHEMATICAL probability, provided a sufficiently large sample is drawn. I will explain this with an example. Lets take the case of flip of coin. MATHEMATICAL probability suggests that the probability of getting a ‘head’ is 1/2. This type of probability is a priori and fixed. On the other hand, EMPIRICAL probability will depend on the number of flips of coin actually made. The EMPIRICAL probability will change from sample to sample. For example, if we flip the coin 5 time and find that the ‘head’ appears only once, then, our EMPIRICAL probability for getting a head is 1/5 (we only had 1 head, out of the total 5 flips). If the ‘head’ appears 3 times, our EMPIRICAL probability for getting a ‘head’ would be 3/5. EMPIRICAL probability is thus the relative frequency of ‘heads’ obtained. On the other hand, MATHEMATICAL probability, being fixed, was 1/2. According to the Law of Large Numbers, if our sample is large enough, the EMPIRICAL probability becomes equal to MATHEMATICAL Probability. In other words, the EMPIRICAL probability of 1/5, eventually converges to a value of 1/2, provided the coins are flipped a large number of times. In statistical analysis, we assume that, since our sample size was large enough, the EMPIRICAL probability obtained from the sample, represents the a priori MATHEMATICAL probability.