Normal Distribution Formula - What Is Normal Distribution Formula? (2024)

The normal distribution or bell curve or the gaussian distribution is the most significant continuous probability distribution in probabilityand statistics. Inphysical science and economics, avast number of random variables of interest are either nearly or exactly described by the normal distribution. The normal distribution formula can be used to approximate other probability distributions as well.

The random variables which follow the normal distribution are ones whose values can assume any known value in a given range.

What Is Normal Distribution Formula?

The normal distribution is defined by the probability density function f(x) for the continuous random variable X considered in the system. It is a function whose integral across an interval (say x to x + dx) gives the probability of the random variable X, by considering the values between x and x + dx.Since there will be infinite values between x and x + dx, thus, a range of x is considered, and a continuous probability density function is defined as

f\left( x \right) \ge 0 \ \forall \ x \in \left( { - \infty , + \infty } \right) \\
\int_{ - \infty }^{ + \infty } {f\left( x \right) = 1} \\

For a normal distribution of a random variable X with the mean = μ and the variance = σ2, the probability density f(x) is given by; :

\(f\left( x \right) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{\frac{{ - \left( {x - \mu } \right)^2 }}{{2\sigma ^2 }}}\)

An equivalent representation:

\( X\sim N( {\pi ,\sigma ^2 }) \)

Normal Distribution Formula - What Is Normal Distribution Formula? (1)

For a specific μ = 3 and a σ ranging from 1 to 3, the probability density function (p.d.f.) is as:

The probability density function of normal or gaussian distribution is given by;
\(f(x, \mu, \sigma)=\frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^{2}}{2 \sigma^{2}}}\)
\(\mathrm{x}\)is the variable
\(\mu\)is the mean
\(\sigma\)is the standard deviation

Normal Distribution Formula - What Is Normal Distribution Formula? (2)

Normal Distribution Formula - What Is Normal Distribution Formula? (3)

Have questions on basic mathematical concepts?

Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts

Book a Free Trial Class

Examples using Normal Distribution Formula

Example 1:If \( X\sim N( 4 ,9) \), find \( P(X>6) \) using normal distribution formula.

When a variable X follows a normal distribution, with mean μ and variance σ2, this is denoted by:
\( X\sim N( {\pi ,\sigma ^2 }) \)
To use normal distribution formula, let \(Z = \frac{{X - \mu }}{\sigma }=\frac{{X - 4 }}{3}\)
\(P\left( {X > 6} \right) = 1 - P\left( {X < 6} \right) \\ = 1 - \varphi \left( {\frac{{6 - 4}}{3}} \right) \\ = 1 - \varphi \left( {0.67} \right) \\ = 1 - 0.74857 \\ = 0.25143\)
Answer: \( P(X>6)=0.25143 \)

Example 2:The working lives of a particular brand of electric light bulb are distributed with a mean of 1200 hours and a standard deviation of 200 hours. What is the probability of a bulb lasting more than 1150 hours? Use normal distribution formula.

Let X, the working life, is distributed usingnormal distribution formulaas,
\(X\sim N( {1200,200^2 }) \)

\(P\left( {X > 1150} \right) = 1 - P\left( {X < 1150} \right) \\ = 1 - \varphi \left( {\frac{1150 - 1200}{200}} \right) \\ = 1 - \varphi \left( {-0.25} \right) \\ = 0.59871\)

Answer: The probability of a bulb lasting more than 1150 hours is 0.59871.

Example 3:What will be the probability density function of normal distribution for the data;\(x=3, \mu=\)4 and \(\sigma=2\)?
Given, variable, x=3
Mean =4and
Standard deviation =2
By the formula of the probability density of normal distribution, we can write;
\(f(3,4,2)=\frac{1}{2 \sqrt{2 \pi}} e^{\frac{-(3-2)^{2}}{2 \times 2^{2}}}\)
Answer: Hence, \(f(3,4,2)=1.106\).

FAQs on Normal Distribution Formula

What Is a Normal DistributionFormula in Statistics?

In order to be considered a normal distribution, a data set (when graphed) must follow a bell-shaped symmetrical curve centered around the mean. The normal distribution formula in statistics is given by,\(f(x, \mu, \sigma)=\frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^{2}}{2 \sigma^{2}}}\)
\(\mathrm{x}\)is the variable
\(\mu\)is the mean
\(\sigma\)is the standard deviation

What Are the Characteristics of a Normal distribution?

The important characteristics of a normal distribution are:

  • It is symmetric andunimodal
  • The mean, median, and mode are all equal.
  • A normal distribution is very symmetrical about its center. The left side of the center of the peak is a mirror image of the right side.

What Is Standard Normal Distribution?

Thestandard normal distributionis anormal distributionwith ameanof zero and a standard deviationof 1. For every standard normal distribution, 68% of the observations lie within 1standard deviationof themean; 95% lie within twostandard deviationsof themean, and 99.9% lie within 3standard deviationsof themean.

What Is the Use of Normal Distribution?

The normal distribution is the most commonly known and used of all distributions. Because the normal distribution relates to many natural phenomena so well, it has become a standard of reference for many probability problems.

Normal Distribution Formula - What Is Normal Distribution Formula? (2024)


What is the formula for the normal distribution? ›

What is the normal distribution formula? For a random variable x, with mean “μ” and standard deviation “σ”, the normal distribution formula is given by: f(x) = (1/√(2πσ2)) (e[-(x-μ)^2]/^2).

How to define a normal distribution? ›

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The normal distribution appears as a "bell curve" when graphed.

How do you calculate if it is normal distribution? ›

Let X be a continuous random variable. Then X takes on a standard normal distribution if its probability density function is f(x)=1√2πexp(−12x2). f ( x ) = 1 2 π e x p ( − 1 2 x 2 ) . In other words, the standard normal distribution is the normal distribution with mean μ=0 and standard deviation σ=1 .

What is the meaning of standard normal distribution formula? ›

The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1. Any normal distribution can be standardized by converting its values into z scores. Z scores tell you how many standard deviations from the mean each value lies.

What is the formula for the approximately normal distribution? ›

μ = n 1 ( n 1 + n 2 + 1 ) ∕ 2 and σ 2 = n 1 n 2 ( n 1 + n 2 + 1 ) ∕ 12 . The sample size n 1 should be taken to correspond to whichever value, T or T ′ , has been selected as the test statistic. These parameter values are used to compute a test statistic having a standard normal distribution.

Why do we calculate normal distribution? ›

A large number of random variables are either nearly or exactly represented by the normal distribution, in every physical science and economics. Furthermore, it can be used to approximate other probability distributions, therefore supporting the usage of the word 'normal 'as in about the one, mostly used.

What is normal distribution in your own words? ›

A normal distribution is commonly referred to as the bell shaped curve and it describes the frequency of something that you are measuring, such the SAT scores, or the size of sand. The center of the curve is the average (mean) and the curve width the variation (the standard deviation).

What is an example of a normal distribution? ›

Height, birth weight, reading ability, job satisfaction, or SAT scores are just a few examples of such variables. Because normally distributed variables are so common, many statistical tests are designed for normally distributed populations.

What is the normal distribution described by? ›

Answer:mean and standard deviation A normal distribution is completely described by it's mean and standard deviation(spread). A normal distribution is symmetric.

What does it mean when data is normally distributed? ›

What is normal distribution? A normal distribution is a type of continuous probability distribution in which most data points cluster toward the middle of the range, while the rest taper off symmetrically toward either extreme. The middle of the range is also known as the mean of the distribution.

What is the formula for the normal distribution of samples? ›

For samples of any size drawn from a normally distributed population, the sample mean is normally distributed, with mean μˉX=μ and standard deviation σˉX=σ/√n, where n is the sample size. The effect of increasing the sample size is shown in Figure 6.4 "Distribution of Sample Means for a Normal Population".

How do you know if a distribution will be normal? ›

A normal quantile plot shows a normal distribution as a straight line instead of as a bell curve. If your data are normal, then the data values will fall close to the straight line. If your data are non-normal, then the data values will fall away from the straight line.

What is the formula for the normal distribution table? ›

To facilitate a uniform standard method for easy calculations and applicability to real-world problems, the standard conversion to Z-values was introduced, which form the part of the Normal Distribution Table. Z = (X – mean)/stddev, where X is the random variable.

What is the normal distribution function? ›

See the figure. The normal distribution is produced by the normal density function, p(x) = e(x μ)2/2σ2/σ √2π. In this exponential function e is the constant 2.71828…, is the mean, and σ is the standard deviation.

Top Articles
Latest Posts
Article information

Author: Ouida Strosin DO

Last Updated:

Views: 5988

Rating: 4.6 / 5 (76 voted)

Reviews: 83% of readers found this page helpful

Author information

Name: Ouida Strosin DO

Birthday: 1995-04-27

Address: Suite 927 930 Kilback Radial, Candidaville, TN 87795

Phone: +8561498978366

Job: Legacy Manufacturing Specialist

Hobby: Singing, Mountain biking, Water sports, Water sports, Taxidermy, Polo, Pet

Introduction: My name is Ouida Strosin DO, I am a precious, combative, spotless, modern, spotless, beautiful, precious person who loves writing and wants to share my knowledge and understanding with you.