Normal Distribution | Definition, Formula, Properties (2024)

Normal Distribution is the most common or normal form of distribution of Random Variables, hence the name “normal distribution.” It is also called Gaussian Distribution in Statistics or Probability. We use this distribution to represent a large number of random variables. It serves as a foundation for statistics and probability theory.

It also describes many natural phenomena, forms the basis of the Central Limit Theorem, and also support numerous statistical methods.

In this article we will learn about Normal Distribution in detail, including its formula, characteristics, uses , and examples.

What is Normal Distribution?

Normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean.

We define Normal Distribution as the probability density function of any continuous random variable for any given system. Now for defining Normal Distribution suppose we take f(x) as the probability density function for any random variable X.

Also, the function is integrated between the interval, (x, {x + dx}) then,

f(x) ≥ 0 ∀ x ϵ (−∞,+∞),

_-∞∫^+∞ f(x) = 1

We observe that the curve traced by the upper values of the Normal Distribution is in the shape of a Bell, hence Normal Distribution is also called the “Bell Curve”.

Check: Python – Normal Distribution in Statistics

Key Features of Normal Distribution

• Symmetry: The normal distribution is symmetric around its mean. This means the left side of the distribution mirrors the right side.
• Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.
• Bell-shaped Curve: The curve is bell-shaped, indicating that most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.
• Standard Deviation: The spread of the distribution is determined by the standard deviation. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Normal Distribution Examples

We can draw Normal Distribution for various types of data that include,

• Distribution of Height of People
• Distribution of Errors in any Measurement
• Distribution of Blood Pressure of any Patient, etc.

Normal Distribution Formula

The formula for the probability density function of Normal Distribution (Gaussian Distribution) is added below,

where,

• x is Random Variable
• μ is Mean
• σ is Standard Deviation

Normal Distribution Curve

In any Normal Distribution, random variables are those variables that take unknown values related to the distribution and are generally bound by a range.

An example of the random variable is, suppose take a distribution of the height of students in a class then the random variable can take any value in this case but is bound by a boundary of 2 ft to 6 ft, as it is generally forced physically.

• Range of any normal distribution can be infinite in this case we say that normal distribution is not bothered by its range. In this case, range is extended from –∞ to + ∞.
• Bell Curve still exist, in that case, all the variables in that range are called Continuous variable and their distribution is called Normal Distribution as all the values are generally closed aligned to the mean value.
• The graph or the curve for the same is called the Normal Distribution Curve Or Normal Distribution Graph.

Normal Distribution Standard Deviation

We know that mean of any data spread out as a graph helps us to find the line of the symmetry of the graph whereas, Standard Deviation tells us how far the data is spread out from the mean value on either side. For smaller values of the standard deviation, the values in the graph come closer and the graph becomes narrower. While for higher values of the standard deviation the values in the graph are dispersed more and the graph becomes wider.

Empirical Rule of Standard Deviation

Generally, the normal distribution has a positive standard deviation and the standard deviation divides the area of the normal curve into smaller parts and each part defines the percentage of data that falls into a specific region This is called the Empirical Rule of Standard Deviation in Normal Distribution.

Empirical Rule states that,

• 68% of the data approximately fall within one standard deviation of the mean, i.e. it falls between {Mean – One Standard Deviation, and Mean + One Standard Deviation}
• 95% of the data approximately fall within two standard deviations of the mean, i.e. it falls between {Mean – Two Standard Deviation, and Mean + Two Standard Deviation}
• 99.7% of the data approximately fall within a third standard deviation of the mean, i.e. it falls between {Mean – Third Standard Deviation, and Mean + Third Standard Deviation}

Normal Distribution Graph

Studying the graph it is clear that using Empirical Rule we distribute data broadly in three parts. And thus, empirical rule is also called “68 – 95 – 99.7” rule.

Check: Mathematics | Probability Distribution s Set 3 (Normal Distribution)

Normal Distribution Table

Normal Distribution Table which is also called, Normal Distribution Z Table is the table of z-value for normal distribution. This Normal Distribution Z Table is given as follows:

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Normal Distribution in Statistics

• Normal distribution, also known as Gaussian distribution, is a bell-shaped curve that describes a large number of real-world phenomena. It’s one of the most important concepts in statistics because it pops up in many areas of study.
• Bell-Shaped Curve: Imagine a symmetrical bell where the middle is the highest point and the tails taper off on either side. That’s the basic shape of a normal distribution. Most data points cluster around the center, and as you move further away from the center, the data points become less frequent.
• Central Tendency: The center of the bell curve represents the central tendency of the data, which means it shows where most of the values are concentrated. This could be the mean, median, or mode, depending on the specific data set.
• Spread of Data: The width of the bell curve indicates how spread out the data is a wider curve means the data points are more dispersed, while a narrower curve signifies the data points are closer together.
• Random Variables: Normal distribution is typically used with continuous random variables, which can take on any value within a specific range. Examples include heights, weights, IQ scores, or exam grades.

Check: Normal Distribution in Business Statistics

Normal Distribution Problems with Solutions

Let’s solve some problems on Normal Distribution

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

Solution:

Given,

• Variable (x) = 2
• Mean = 3
• Standard Deviation = 4

Using formula of probability density of normal distribution

[Tex] f(x,\mu , \sigma ) =\frac{1}{\sigma \sqrt{2\pi }}e^\frac{-(x-\mu)^2}{2\sigma^{2}}[/Tex]

Simplifying,

f(2, 3, 4) = 0.09666703

Example 2: If the value of the random variable is 4, the mean is 4 and the standard deviation is 3, then find the probability density function of the Gaussian distribution.

Solution:

Given,

• Variable (x) = 4
• Mean = 4
• Standard Deviation = 3

Using formula of probability density of normal distribution

[Tex] f(x,\mu , \sigma ) =\frac{1}{\sigma \sqrt{2\pi }}e^\frac{-(x-\mu)^2}{2\sigma^{2}}[/Tex]

Simplifying,

f(4, 4, 3) = 1/(3√2π)e⁰

f(4, 4, 3) = 0.13301

Conclusion – Normal Distribution

Normal Distribution, also known as the Gaussian distribution, is a fundamental concept in statistics and probability theory. It is characterized by its bell-shaped curve, which is symmetrical and centered around the mean.

The properties of the normal distribution, such as its mean and standard deviation, play crucial roles in many statistical analyses and applications. Normal distributions are widely used in fields such as finance, engineering, natural sciences, and social sciences to model and analyze a wide range of phenomena.

Normal Distribution – FAQs

What is Normal Distribution?

In statistics, the normal 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.

Why is Normal Distribution Called “Normal?”

Normal Distribution also called the Gaussian Distribution is called “Normal” because it is shown that various natural processes normally follow the Gaussian distribution and hence the name “Normal Distribution”.

What is Normal Distribution Graph?

A normal distribution graph, also known as a Gaussian distribution or bell curve, is a specific type of probability distribution. It is characterized by its symmetric, bell-shaped curve when plotted on a graph.

What is Normal Distribution Z Table?

Z table, also known as a standard normal distribution table or a Z-score table, is a reference table used in statistics to find the probabilities associated with specific values in a standard normal distribution.

What are characteristics of Normal Distribution?

Properties of Normal Distribution are,

• Normal Distribution Curve is symmetric about mean.
• Normal Distribution is unimodal in nature, i.e., it has single peak value.
• Normal Distribution Curve is always bell-shaped.
• Mean, Mode, and Median for Normal Distribution is always same.
• Normal Distribution follows Empirical Rule.

What is Mean of Normal Distribution?

Mean (denoted as μ) represents the central or average value of data. It is also the point around which the data is symmetrically distributed.

What is Standard Deviation of Normal Distribution?

Standard deviation (denoted as σ) measures the spread or dispersion of data points in distribution. A smaller σ indicates that data points are closely packed around mean, while a larger σ indicates more spread.

What is Empirical Rule (68-95-99.7 Rule)?

Empirical rule for normal distribution states,

• Approximately 68% of data falls within one standard deviation of mean.
• Approximately 95% falls within two standard deviations of mean.
• About 99.7% falls within three standard deviations of mean.

What are Uses of Normal Distribution?

Various uses of Normal Distribution are,

• For studying vrious Natural Phenomenon
• For studying of Financial Data.
• In Social Sciense for studying and predicting various parameters, etc.

What are Limitations of Normal Distribution?

Normal Distribution is an extremely important Statical concept, but even it has some limitations such as,

• Various distribution of data does not follow Normal Distribution and thus it is unable to comment on these data.
• To much relliance of Normal Distriution or Bell curve is not a good way to prdict data as it is not 100% accurate, etc.