What is the Normal distribution? Definition and example

Graphic of the normal distribution

Suppose you have wondered about the most important continuous probability distribution in probability and economics. In that case, Normal Distribution may satisfy your thoughts. Some people refer to it as the Gaussian Distribution and the bell curve.

With Normal Distribution, many random variables are represented in economics and trading. Furthermore, traders can also utilize it to obtain other probability distributions. Hence, it is of great importance when it comes it analyzing data. Let us learn more about Normal Distribution, its definition, and example. 

Normal distribution definition

To understand the Normal Distribution, let us assume a density function where f(x) is its probability and X is a random variable. Therefore, it defines a function that is integrated between the range (x to x + dx). It further gives X’s probability by considering values between x and x+dx.

f(x) ≥ 0 ∀ x ϵ (−∞,+∞)

And -∞+∞ f(x) = 1

Finally, we can say that Normal Distribution is the probability density function for a continuous random variable in a system. 

Normal distribution formula

You can find the probability density function of gaussian distribution using the following formula:

Normal Distribution Formula


  • x is the variable
  • μ represents the mean
  • σ refers to standard deviation

Normal distribution curve

The normal distribution curve shown on a laptop device

The Normal Distribution Curve is usually bell-shaped. That is why the bell curve is another name for this distribution curve. In Normal Distribution, random variables can take up any unknown value from a given range. You can also refer to these random variables as continuous variables. 

For example, the students in a school have heights in the range, say, 0 to 6 ft. However, this range is influenced by the physical limitations of a human being.

In reality, the range of variables can even extend from –∞ to + ∞. The Normal Distribution here will provide the probability of the value lying in the particular range of the experiment. Suppose you can’t spare much time for making all calculations. In that case, you can use the Normal Distribution Calculator to find probability density by providing standard deviation and mean value. 

Normal distribution standard deviation

Usually, the standard deviation is positive in the Normal Distribution. In the Normal Distribution Curve, the Mean determines the line of symmetry. In contrast, the standard deviation defines how far the data is spread. 

A smaller standard deviation would result in a narrower graph and vice versa. 

If we use standard deviation, the verifiable rule states:

  • One standard deviation of the Mean contains approximately 68% of the data.
  • Approximate data of 95% fall within two standard deviations of the Mean. 
  • Three mean standard deviations have about 99.7% of the total data.
Normal distribution curve

Thus, we also refer to the empirical rule as the 68 – 95 – 99.7 rule.

Example of normal distribution

Suppose the value of the random variable is 2. If the Mean is 5 and the standard deviation is 4, you can find Normal Distribution by the formula of probability density:

normal distribution problem and solution

f(2,2,4) = 1/(4√2π) e0

f(2,2,4) = 0.0997

Mean, and standard deviation are two vital parameters of Normal Distribution. Without them, you cannot find Normal Distribution and use it with other trading indicators to make accurate decisions. 

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