Normal Distribution

Considering the aspect of the continuous probability distribution, the normal distribution is an important factor and is applied in the fields of probability theories. Also referred to as the Gaussian distribution, this is a key factor in approximating a particular probability distribution given. Even accurate levels of measurement are possible for a large set of variables using this mathematical concept. 

The Definition and Characteristics of Normal Distribution

The major point of defining a normal distribution lies in the fact that this mathematical property falls under the category of the Probability density function. Consider a probability random variable function “f(x)”. Here, X is denoted to be the random variable for the probability function f. For a given interval of range between (x to x + dx), the probability function for the given random variable is represented as the following:

  • f(x) ≥ 0 ∀ x ϵ (− , + )

  • - ∫ + f(x) = 1

Therefore, in a normal distribution, a group of numbers and random variables can be precisely denoted using this method. Moreover, this is also a reason why the normal distribution is also called the Bell Curve. This is particularly more relevant in the fields of economics and other physical sciences.  

Since the Gaussian Distribution curve is also helpful to draw approximations with other distribution properties, the term ‘normal’ is hence preferred, supporting its feature. 

Normal Distribution and the General Formula

As we have now got a clear idea about what is normal distribution, let us move onto the important aspect of learning the formula. 

To define the normal distribution for a variable X, having the mean μ and variance as σ2, the statistic distribution as per its probability density function can be represented as:

(Image to be added soon)

What Does a Normal Distribution Curve Mean?

In a given set of a normal distribution, the random variables that follow the pattern are possibly used to study and evaluate the unknown values as per a given range sequence. 

One of the best examples of this condition is that of human height. When a doctor wants to assess and estimate the total height of all the IPs (in-patients) of a specific ward, then the clinician is already forced with the query of having the patient’s height between 0 and 6 feet. 

As a general rule, there are normal distributions that are not limited to a specific range. Such a distribution is called the continuous variables since the range can extend up to – ∞ to + ∞. This will again result in a smooth bell curve only. Also, note that you can still prefer to use the normal distribution calculator to evaluate the probability density function by making note of the standard deviation value and its mean number. 

Table Representing the Normal Distribution

(Image to be added soon)


For a set of real-value random digit sequences, the Gaussian distribution or the normal distribution is said to be the continuous probability distribution. The formula is derived using the probability function f(x) using the mean value and standard deviation score. The Gaussian or normal distribution is also called a Bell Curve, referring to its shape geometrically. From statistics and mathematics to social sciences and physical sciences, many areas of study and research focus on using the normal distribution to approximate large collections of variables, to the near or exact value as they are presented. 

FAQ (Frequently Asked Questions)

1. State The Law Of Central Limit Theorem.

The law of central limit theorem represents that the average value for several collected observations or samples for a random variable (with a finite mean and variance), is already an existing random variable, under specific given conditions. Here, as the size of a sample extends, the normal distribution is said to be more converged.

2. What Is Meant By The Standard Normal Distribution?

Regarded as the simplest form to the regular Gaussian curve, the standard normal distribution is a special case under the normal distribution scenarios. Both the mu value and the sigma value here is expected to be zero (0). There were controversies in deciding which distribution to be referred to as ‘standard’ by various authors in the field.  

3. Mention Any 3 Properties Of The Normal Distribution Curve.

Apart from the 1st two (variance and mean), the normal distribution is the only type of distribution that has cumulants beyond the mentioned 0. For the given set of specifications in the variance and mean value, this continuous distribution set presents the maximum value of entropy. The normal distribution also falls under the classification called  the ‘elliptical distributions.’

4. What Are The Other Forms Of Distribution That Can Be Approximated Using The Normal Distribution?

Possession distribution, the Binomial distribution, Student’s t-distribution, and even the Chi-squared distribution can be approximated to a great extent using the normal or Gaussian distribution.

5. Give A Few Real-life Examples Where Normal Distribution Is Observed Or Used.

Measuring the heights of a person, assigning the marks or grades for a student’s test, errors that occur during scaling, detecting the size of products and machines and even checking your blood pressure involves normal distribution in real-life scenarios.

6. How To Convert A Value To Its Z-score?

To convert a value to its z-score, firstly, take the mean and subtract it. Then, by dividing the value obtained using the standard deviation, you can get the z-score number. This process is termed ‘standardizing’.