Obtaining Standard Normal Distribution Step-By-Step

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Iliya Valchanov 12 Apr 2023 5 min read

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 sigma squared respectively. 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.

Formula that allows us to standardize a distribution, standardization

Standard Normal Distribution in Statistics: Definition and Formulas

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

Standard normal distribution, standardization

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.

Standard deviation by sigma, standardization

We use the letter Z to denote it. As we already mentioned, its mean is 0 and its standard deviation: 1.

Its mean is 0 and its standard deviation: 1, standardization  

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

Z-score, standardization

A Case in Point

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

1, 2, 2, 3, 3, 3, 4, 4, and 5, standardization

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.

A new data set of: -2, -1, -1, 0, 0, 0, 1, 1, and 2, standardization

The new mean is 0, exactly as we anticipated.

The new mean is 0, standardization

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.

The original dataset has a standard deviation of 1.22, standardization

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.

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!

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.

The same position but reshaped

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.


Interested in learning more? You can take your skills from good to great with our statistics course!

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

Iliya Valchanov

Co-founder of 365 Data Science

Iliya is a finance graduate with a strong quantitative background who chose the exciting path of a startup entrepreneur. He demonstrated a formidable affinity for numbers during his childhood, winning more than 90 national and international awards and competitions through the years. Iliya started teaching at university, helping other students learn statistics and econometrics. Inspired by his first happy students, he co-founded 365 Data Science to continue spreading knowledge. He authored several of the program’s online courses in mathematics, statistics, machine learning, and deep learning.