Question

Think of a situation in which it would be useful to transform a set of measurements of something into a set of z-scores. Try to think of a real-world example & explain how it would be useful to work with these z-scores instead of the original raw scores.

Answer 1

There is an infinite number of normal distributions. There is a single standard normal distribution. The goal of calculating a z - the score is to relate a particular normal distribution to the standard normal distribution. The standard normal distribution has been well-studied, and there are tables that provide areas underneath the curve, which we can then use for applications.

Due to this universal use of the standard normal distribution, it becomes a worthwhile endeavour to standardize a normal variable. All that this z-score means is the number of standard deviations that we are away from the mean of our distribution.

Suppose that we know about a population of a particular breed of cats having weights that are normally distributed. Furthermore, suppose we know that the mean of the distribution is 10 pounds and the standard deviation is 2 pounds. Consider the following questions:

1. What is the z-score for 13 pounds?
2. What is the z-score for 6 pounds?
3. How many pounds corresponds to a z-score of 1.25?

The z-score in the middle of the curve is zero. The z-scores to the right of the mean are positive and the z-scores to the left of the mean are negative. If you look up the score in the z-table, you can tell what percentage of the population is above or below your score

it gives us normality to calculate any percentage