Standardization: The Standard Normal Variate (Z)

In practice, business and economic data can have any mean and any standard deviation — wages might average ₹8,000 with an SD of ₹1,200, while exam marks average 55 with an SD of 10. It would be impossible to print a separate area-table for every possible combination of $\mu$ and $\sigma$. The solution is standardization: convert any normal variable $X \sim N(\mu,\sigma^2)$ into a single universal variable $Z$, called the Standard Normal Variate, using the formula

The variable $Z$ always follows the Standard Normal Distribution, $Z \sim N(0, 1)$ — that is, a normal distribution with mean 0 and standard deviation (and variance) 1, regardless of what $\mu$ and $\sigma$ the original $X$ had. This is precisely why only one standard normal table is needed for every normal-distribution problem in this syllabus, whatever the original units (rupees, marks, hours, centimetres) may be.

How to read $Z$: the value of $z$ tells you how many standard deviations a particular value of $x$ lies away from the mean $\mu$ — a positive $z$ means $x$ is above the mean, a negative $z$ means $x$ is below the mean, and $z = 0$ corresponds exactly to $x = \mu$.

Worked illustration of the standardization step itself: if wages of factory workers follow $X \sim N(8000, 1200^2)$ (mean ₹8,000, SD ₹1,200), a worker earning ₹9,200 has

$z = \dfrac{9200 - 8000}{1200} = \dfrac{1200}{1200} = 1.00$ …