Explain Standardised Normal Distribution and Z- Score in Statistics

A unique combination of mean (μ) and standard deviation (σ) represents or defines a unique normal distribution. So, to analyse or compare different normal distributions, you make use of a standardised normal distribution. A standardised normal distribution is a special type of normal distribution where μ = 0 and σ = 1.

A normal distribution is converted into a standardised normal distribution with the help of the Z score.

The formula for Z score is as follows:

$ZScore=\frac{x-\mu }{\sigma }$

As it’s evident from the formula, for every value of x (or the values on the X-axis), we will calculate the corresponding Z scores using the formula above and plot these Z scores against their respective probabilities on the Y-axis.

For example, for a normal distribution with μ= 35 and σ = 5, the normal distribution curve and the standard normal distribution curve will look like this:

So, Z scores can be used to:  

• Calculate the probability of the occurrence of a particular random variable, and

• Compare normal distributions.

Let’s try to understand both the scenarios above:

1. Calculating the probability of a random variable’s occurrence is done with the help of a Z table (watch this YouTube video). It can also be done in Excel using the ‘NORMDIST’ formula.

2. Suppose that the marks obtained by the students of a class are normally distributed. In the mid-term exam, the mean score out of 100 was 50 and the standard deviation was 10; and in the end-term, the mean score was 60 and the standard deviation was 20. A student, Ram, scored 70 in the mid-term exam and 72 in the end-term exam. In which exam was his relative performance better?
       To answer this question, you can make use of Z scores.
       Mid-term’s Z score = 2
       End-term’s Z score = 0.6
   Looking at the Z scores, we can conclude that his relative performance was better in the mid-term exam even though the marks he obtained were less compared to his end-term marks.