I understand how to standardize a normal distribution, but I’m wondering why we would use it? I guess it’s to ease the analysis but I would like to have more insight of the reason why.
You’re right in saying that standardized normal distribution makes the analysis easier. Here’s why.
With normally distributed data, It is easier to find the probabilities within a certain bounds and hence easier to make inferences.
The probabilities (or area) of normally distributed data are obtained by getting the integral within limits of the Gaussian function below.
When this function has mean equal to 0 and variance 1, the Gaussian function is simplified as
The probabilities of different z values were already computed and organized as Z-table. The probabilities found in Z-table are also the probabilities of the corresponding sample mean (or difference of means, or proportions). Hence, with the z-scores and Z-table, we can find the probabilities for sample statistics (sample mean, mean difference, or proportions) without doing the integration.
Hope this helps and gave you more insight why we use the standardized normal distribution.
The 365 Team