What Is Normal Distribution Theory in Statistics?

The normal distribution is one of the most common and practically useful types of statistical distribution, as many phenomena naturally follow it. This makes it an excellent starting point if you’re looking to develop a fuller understanding of statistical methods. Let’s explore the concept thoroughly now:

What Is Normal Distribution?

In statistics, the normal distribution is a bell-shaped curve that is symmetric about the mean. This means that the left and right sides of the curve are mirror images of each other, centered around the mean value. The data points are distributed in such a way that most of them cluster around the mean, with fewer points appearing as you move away from the center in either direction.

Why Is It Referred to as Normal?

Rather than referring to something ordinary, the term “normal” in statistics has a more specific meaning. In essence, it’s used to illustrate the fact that the normal distribution occurs frequently just about everywhere-from the natural world to human society and industry. Many useful data sets tend to cluster around a central value in this manner, including but not limited to:

Heights of people
IQ scores
Quality control test results
Income distributions

Thus, the normal distribution is referred to as such because it has historically been seen as a standard way for many random variables to behave. The fact that it’s so common is also what makes it so important in statistics, as many statistical methods and theories are based on the assumption of normality. If data follows a normal distribution, analysts will likely have an easier time interpreting and drawing conclusions about it.

Key Terms for Understanding Normal Distribution

To interpret and utilize the normal distribution in a wide variety of business contexts, you’ll need to be familiar with the following terms:

Mean ( $(\mu)$ ): The mean, or average, is the central value of the data set.
Median (M): The median is the value found in the middle of the data set when you order it from least to greatest.
Mode (Z): The mode is the value in the data set that occurs most frequently.
Standard Deviation ( $(\sigma)$ ): This is a measure of ow much the values in a distribution spread out or disperse from the mean.

In a normal distribution, you’ll find the mean at the center of the curve, as it’s also the point of highest probability. The mean, median, and mode should also all be equal. In terms of standard deviation, a normal distribution will generally behave according to the following principle, which is known as the Empirical Rule:

68% of the data will fall within one standard deviation of the mean $(\mu \pm 1\sigma)$
95% will fall within two $(\mu \pm 2\sigma)$
7% within three $(\mu \pm 3\sigma)$

You can think in particular of the mean $(\mu)$ and standard deviation $(\sigma)$ as the two most essential parameters for understanding the normal distribution. This is because, taken together, they completely define the shape and position of the bell-shaped curve. When you know these two parameters, you can readily understand where your data is centered and how it is spread out.

Illustrating Normal Distribution and the Empirical Rule

Now let’s see how normal distribution works in practice by examining a practical example. Suppose the test scores of a large class are normally distributed with a mean score of 75 and a standard deviation of 5. You’ll find the data behaves as follows:

68% of students: Scores will range between 70 (75 – 5) and 80 (75 + 5).
95% of students: Scores will range between 65 (75 – 2(5)) and 85 (75 + 2(5)).
7% of students: Scores will range between 60 (75 – 3(5)) and 90 (75 + 3(5)).

From this data, you can infer that most of the class’s lowest scorers will have scored around 60, while most of its highest scorers will have scored around 90, though there may be a few outliers in both cases. The majority of scores, meanwhile, will likely fall between 70 and 80.

Understanding Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution that always has a mean of 0 and a standard deviation of 1. It is represented by the variable 𝑍.

In statistics, the standard normal distribution is used to simplify calculations by converting any normal distribution to this standard form. This makes it easier to compare and interpret data across different normal distributions using the same reference. You can convert your distribution into a standard normal distribution by applying the following formula, which is known as the Z-score formula:

where:

X is the value from the original normal distribution,
$(\mu)$ is the value of the original distribution’s mean, and
$(\sigma)$ is the original distribution’s standard deviation.

Applying Normal Distribution in Business Contexts

Normal distribution theory is well-used in finance, marketing, and a host of other various business functions. Here are just a few practical applications for you to consider:

Sales Analysis – Analyzing historical sales data can reveal patterns that help in forecasting future sales. If you understand these patterns, you can predict future sales more accurately and improve inventory management at your business.
Customer Insights – Customer satisfaction scores, gathered through surveys, typically follow a normal distribution. These scores, when analyzed, can yield insights into customers’ impressions of the business and identify areas for improvement.
Employee Performance – Performance evaluations often result in scores that follow a normal distribution. This data can help your HR managers identify top performers, average performers, and those who may need additional support or training. It can also guide decisions on promotions, bonuses, and professional development programs.

In sum, normal distribution theory underpins many statistical systems and tests. It thus gives statisticians a solid foundation for making predictions and understanding data variability. As a business owner, your familiarity with normal distribution can help you make more informed decisions, unlock new operational efficiencies, and plan more strategically.

About Glen Dimaandal

Glen Dimaandal is a data scientist from the Philippines. He has a post-graduate degree in Data Science and Business Analytics from the prestigious McCombs School of Business in the University of Texas, Austin. He has nearly 20 years of experience in the field as he worked with major brands from the US, UK, Australia and the Asia-Pacific. Glen is also the CEO of SearchWorks.PH, the Philippines' most respected SEO agency.