Normal Distribution Formula

In the theoretical discussion of probability, the normal or Gaussian distribution is a very common type of distribution. It is a continuous probability distribution. It is a very important statistical data distribution pattern. Also, it occurs in many natural phenomena some examples are like height, blood pressure, lengths of objects produced by some machines. In this topic, we will see the concept of it. Let us now discuss the Normal distribution formula here with examples. Let us learn it.

Normal Distribution Formula

What is Normal Distribution?

The normal distribution also popularly known as the Gaussian distribution. It is a probability distribution that is symmetric about the mean value. It will show that data near the mean is more frequent in occurrence than data far away from the mean. In the graphical form, normal distribution will appear as a bell curve.

The normal distribution is the most common type of distribution considered in technical stock market analysis. Also, we use it in other types of statistical analyses. The standard normal distribution has two essential parameters which are the mean and the standard deviation.

For a normal distribution, 68% portion of the observations is within + or – of one standard deviation of the mean. Similarly, 95% are within + or – of two standard deviations. And  99.7% are within the + or – of three standard deviations.

The normal distribution model is guided by the Central Limit Theorem. This theory says that averages calculated from independent but identically distributed random variables have approximately normal distributions. It does not take regards to the type of distribution from which the variables are sampled. On the other hand,  we sometimes confuse Normal distribution with the symmetrical distribution. As we know that symmetrical distribution is one where a dividing line produces two mirror images.

Source:  en.wikipedia.org

How Normal Distribution is useful in Finance?

Traders can use the standard deviations to suggest potential trade values by using the formula. This type of trading generally performs on very short time frames. This is because larger timescales make it much harder to pick entry and exit points.

The spread of a normal distribution is controlled by the standard deviation represented as \sigma. Lower the standard deviation more will be the concentrated data. The popular formula for a normal probability distribution is:

\(\large P(x)=\frac{1}{\sqrt{2\pi \sigma^{2}}}\:e^{\frac{(x-\mu)^{2}}{2\sigma ^{2}}}\)

P(x)	Probability Distribution
x	Normal random variable
\(\mu\)	Mean of the data
\(\sigma\)	Standard Distribution of the data.

When mean \((\mu) = 0\) and standard deviation \((\sigma) = 1\), then that distribution is normal distribution.

Solved Examples Normal Distribution Formula

Q.1: An average light bulb lasts 300 days with a standard deviation value of 50 days. Here assume that bulb life is normally distributed. Then what is the probability that the light bulb will last at most 365 days?

Solution: As given in the problem:

A mean value is of 300 days and a standard deviation is of 50 days.

We have to find the cumulative probability which bulb life is less than or equal to 365 days. Therefore, we know the following:

• The value of the normal random variable will be 365 days.
• The mean value is equal to 300 days.
• The standard deviation is 50 days.

Formula for probability distribution is as follows:

\(\large P(x)=\frac{1}{\sqrt{2\pi \sigma^{2}}}\:e^{\frac{(x-\mu)^{2}}{2\sigma ^{2}}}\)

Substituting the known values , we will get,

= 0.90. This is obtain from the normal distribution table.

Therefore, there will be a 90% chance that a light bulb will burn out within a period of 365 days.