Standard Normal Distribution in Statistics

In the previous segment, you learnt the concept of a normal distribution. One key takeaway from the previous segment was that the probability values depend on how far away the observed value is from the mean of the distribution. Or, more precisely, the probability values depend on how many standard deviations away is the observed value from the mean of the distribution.

Irrespective of the means and the standard deviations of two normal distributions, if the observed values for the two distributions lie, say, one standard deviation to the right of their respective means, their probability values (for example, cumulative probability) are going to be exactly the same (84%, in this case).

This observation has massive implications for normal distributions and their use, as you will learn in the upcoming video.

In this video, you learnt that a standard normal distribution is a normal distribution with a mean (µ) of 0 and a standard deviation (σ) of 1. You also learnt that the points in any normal distribution can be represented as equivalent points in a standard normal distribution. The formula to convert any point in a normal distribution into its equivalent point (referred to as the z-score) on the standard normal distribution is as follows:

• $Z-score=\frac{x-\mu }{\sigma }$

An important point to note is that the properties of the original point on the normal distribution are retained when it is translated on to the standard normal distribution. For instance, if the cumulative probability of 'x' is 0.813 in a normal distribution, the cumulative probability of its equivalent 'z-score' will also be 0.813 in the standard normal distribution.

One way of looking at the z-score is that it basically tells you how many standard deviations away from the mean is your observed value. This is extremely helpful in making probability calculations and comparing points on two different normal distributions.

You also need to know about the empirical rule. According to the empirical rule, 95% of the area lies between two standard deviations from the mean. Critical z-scores are defined as the boundary between the values that lie inside the 95% zone and those that lie outside the 95% zone. In this case, the critical z-scores are ±2 (or, more precisely, ±1.960). Critical z-scores will be explored in greater detail in the module on 'hypothesis testing'.