The word **standardization **may sound a little weird at first but understanding it in the context of statistics is not brain surgery. It is something that has to do with **distributions**. In fact, every **distribution** can be standardized. Say the **mean** and the **variance** of a variable are ** mu** and

**squared respectively.**

*sigma***Standardization**is the process of transforming a variable to one with a

**mean**of 0 and a

**standard deviation**of 1.

You can see how everything is denoted below along with the formula that allows us to standardize a **distribution**.

## Standard Normal Distribution in Statistics: Definition and Formulas

Logically, a **normal distribution** can also be standardized. The result is called a **standard normal distribution.**

You may be wondering how the **standardization** goes down here. Well, all we need to do is simply shift the **mean** by ** mu**, and the

**standard deviation**by

**.**

*sigma*We use the letter ** Z** to denote it. As we already mentioned, its

**mean**is 0 and its

**standard deviation**: 1.

The standardized variable is called a **z-score**. It is equal to the original variable, minus its **mean**, divided by its **standard deviation**.

**A Case in Point**

Let’s take an approximately normally distributed set of numbers: 1, 2, 2, 3, 3, 3, 4, 4, and 5.

Its **mean** is 3 and its **standard deviation**: 1.22. (You can quickly compute them by inputing the data into our mean and standard deviation calculators.) Now, let’s subtract the **mean** from all data points.

As shown below, we get a new data set of: -2, -1, -1, 0, 0, 0, 1, 1, and 2.

The new **mean** is 0, exactly as we anticipated.

Showing that on a graph, we have shifted the curve to the left, while preserving its shape.

**The Next Step of the Standardization**

So far, we have a new **distribution**. It is still normal, but with a mean of 0 and a **standard deviation** of 1.22. The next step of the **standardization** is to divide all data points by the **standard deviation**. This will drive the **standard deviation** of the new data set to 1.

Let’s go back to our example.

The original dataset has a **standard deviation** of 1.22. The same goes for the dataset which we obtained after subtracting the **mean** from each data point.

**Important:** Adding and subtracting values to all data points does __not__ change the **standard deviation**.

Now, let’s divide each data point by 1.22. As you can see in the picture below, we get: -1.6, -0.82, -0.82, 0, 0, 0, 0.82, 0.82, and 1.63.

If we calculate the **standard deviation** of this new data set, we will get 1.

And the **mean** is still 0!

In terms of the curve, we kept it at the same position, but reshaped it a bit, as shown below.

## Standardization of Normal Distribution: Next Steps

This is how we can obtain a **standard normal distribution** from any normally distributed data set.

Using it makes predictions and inferences much easier. This is exactly what will help us a great deal in the next tutorials. So, if you want to use the knowledge you gained here, feel free to jump into the linked tutorial.

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**Next Tutorial: **Getting Familiar with the Central Limit Theorem and the Standard Error