Normal Distribution

In probability theory and statistics, the Normal Distribution, also called the Gaussian Distribution, is the most significant continuous probability distribution. Sometimes it is also called a bell curve. 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.

Table of Contents:

Normal Distribution Definition

The Normal Distribution is defined by the probability density function for a continuous random variable in a system. Let us say, f(x) is the probability density function and X is the random variable. Hence, it defines a function which is integrated between the range or interval (x to x + dx), giving the probability of random variable X, by considering the values between x and x+dx.

f(x) ≥ 0 ∀ x ϵ (−∞,+∞)

And -∞+∞ f(x) = 1

Normal Distribution Formula

The probability density function of normal or gaussian distribution is given by;

Normal Distribution Formula

Where,

  • x is the variable
  • μ is the mean
  • σ is the standard deviation

Also, read:

Normal Distribution Curve

The random variables following the normal distribution are those whose values can find any unknown value in a given range. For example, finding the height of the students in the school. Here, the distribution can consider any value, but it will be bounded in the range say, 0 to 6ft. This limitation is forced physically in our query.

Whereas, the normal distribution doesn’t even bother about the range. The range can also extend to –∞ to + ∞ and still we can find a smooth curve. These random variables are called Continuous Variables, and the Normal Distribution then provides here probability of the value lying in a particular range for a given experiment. Also, use the normal distribution calculator to find the probability density function by just providing the mean and standard deviation value.

Normal Distribution Table

The table here shows the area from 0 to Z-value.

Z-Value 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

Normal Distribution Problems and Solutions

Question 1: Calculate the probability density function of normal distribution using the following data. x = 3, μ = 4 and σ = 2.

Solution: Given, variable, x = 3

Mean = 4 and

Standard deviation = 2

By the formula of the probability density of normal distribution, we can write;

normal distribution example

Hence, f(3,4,2) = 1.106.

Question 2: If the value of random variable is 2, mean is 5 and the standard deviation is 4, then find the probability density function of the gaussian distribution.

Solution: Given,

Variable, x = 2

Mean = 5 and

Standard deviation = 4

By the formula of the probability density of normal distribution, we can write;

normal distribution problem and solution

f(2,2,4) = 1/(4√2π) e0

f(2,2,4) = 0.0997

There are two main parameters of normal distribution in statistics namely mean and standard deviation. The location and scale parameters of the given normal distribution can be estimated using these two parameters. 

Normal Distribution Properties

Some of the important properties of the normal distribution are listed below:

  • The normal distribution is a subclass of the elliptical distributions.
  • The normal distribution is the only distribution whose cumulants surpassing the two parameters, i.e., other than the mean and variance are zero. 
  • It is also the continuous distribution with the highest entropy for a particularized mean and variance. 
  • It is symmetric about its mean and is non-zero across the complete real line.
  • The normal distribution value is substantially zero when the value x lies more than a few standard deviations away from the mean. For example, a spread of four standard deviations comprises all but 0.37% of the total distribution.


Frequently Asked Questions on Normal Distribution – FAQs

What is a normal distribution in statistics?

A probability function that specifies how the values of a variable are distributed is called the normal distribution. It is symmetric since most of the observations assemble around the central peak of the curve. The probabilities for values of the distribution are distant from the mean narrow off evenly in both directions.

What does normal distribution mean?

In statistics (and in probability theory), the Normal Distribution, also called the Gaussian Distribution, is the most important continuous probability distribution. Sometimes it is also called a bell curve.

What is a normal distribution used for?

A normal distribution is significant in statistics and is often used in the natural sciences and social arts to describe real-valued random variables whose distributions are unknown.

What are the characteristics of a normal distribution?

The essential characteristics of a normal distribution are:
It is symmetric, unimodal (i.e., one mode), and asymptotic.
The values of mean, median, and mode are all equal.
A normal distribution is quite symmetrical about its center. That means the left side of the center of the peak is a mirror image of the right side. There is also only one peak (i.e., one mode) in a normal distribution.

How do you know if data is normally distributed?

A histogram presents a useful graphical representation of the given data. When a histogram of distribution is superimposed with its normal curve, then the distribution is known as the normal distribution.

How do you use a normal distribution table?

As we know, the label for rows contains the integer part and the first decimal place of z. In contrast, the title for columns comprises the second decimal place of z. The values within the table are the probabilities corresponding to the table type. Hence, to get the value of 0.56 from the z-table, identify the probability value corresponding to the 0.5 row and 0.06 column (=0.2123).

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