As we discussed in lecture, in order to use Z-scores to calculate proportions and probabilities, we must know that the data is disributed normally (or at least is close to normal).

A) Why is normality relevant to the calculation of proportions using Z-scores?

B) What would happen if our data was very "not normal"?

C) What use are Z-scores with "not normal" data?

Step-by-step explanation:

Introduction : Z scores are used to compute probabilities. for example if X is a norally ditributed variable and we want to know that robability of x being less than or greater tha a certain real no, say a, then we compute the z score for. and probability of ' x being less than a' is equal to probability of 'a standard normal variable being less than the z score of a '.

a) The above statement that : "probability of ' x being less than a' is equal to probability of 'a standard normal variable being less than the z score of a '. " is true because the z transformation on X gives us a standard normal variable. had X not been normal the z transformation would not follow standard normal exactly.

b) then, the probabilities will lead to wrong conclusion. because the probability based on z score is calculated from the standard normal PDF. but id the real PDF is not standard normal then calculating probabilities from z score using standard normal PDF will give erroneous answers.

c) z score for non normal data : z scores for non normal data can standardize a variable. it's like : if there is a data where the observations are all big numbers, the we calculate the z score for them. then calculate median, mode, GM etc. then we do the reverse transformation and get the actual median, mode etc.