Posted on : 28-01-2019 Posted by : Admin

If our actually observed data do not match the data expected on the basis of assumptions, we would have serious doubts about our assumptions. Such data of assumption often lead to theoretical frequency distributions also known as probability distribution. This distribution is not based on actual experimental data but on certain theoretical considerations. This may be simple two valued distribution like 3:1 as in Mendelian cross or it may be more complicated. Some of the most important probability distributions are,

- Gaussian/Normal distribution
- Binomial distribution
- Poisson distribution

Binomial and Poisson distribution apply to the discontinuous random variables and are together known as discontinuous distributions. Normal distribution applies to continuous random variables and is called as continuous distribution.

Normal distribution is also known as normal probability distribution which is very useful for continuous random variables. Many statistical data concerned with business and economic problems are displayed in the form of normal distribution. Normal distribution is the cornerstone of the modern biostatistics. It is important for the reason that it plays a vital role in the theoretical and applied statistics.

In many natural processes, random variation matches to a particular probability distribution which is known as the normal distribution. In 1733, English mathematicians deMoivre and Laplace first discovered normal distribution. Later in 1812, German mathematician Gauss rediscovered it to analyze astronomical data, and it consequently became to be known as the Gaussian distribution.

A continuous random variable x is said to have random distribution with parameters of mean (μ) and standard deviation (σ2) if its density function is,

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{}\mathbf{\sigma}\mathbf{}\sqrt{\mathbf{2}\mathbf{\pi}}}\mathbf{.}{\mathit{e}}^{\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}{\left[\frac{\mathbf{x}\mathbf{-}\mathbf{\mu}}{\mathbf{\sigma}}\right]}^{\mathbf{2}}}$

Here,

e and π are mathematical constants

μ is mean

σ is standard deviation

If μ=0 and σ=1 then the variate is called as standard normal variate

- Normal distribution is a limiting event of the binomial distribution under the following conditions.

- Number of trials “n” is indefinitely large or we can say that number of trials is infinite (n→∞)
- Neither p nor q is very small.

- Normal distribution can also be acquired as a limiting case of Poisson distribution with parameter m→∞
- Constants of normal distribution are mean = m, variation =σ2, Standard deviation =σ.

Normal probability curve is the curve representing the normal distribution. Normal probability curve is balanced or symmetrical at the mean (m), bell-shaped and the two tails on the right and left sides of the mean extends to the infinity. The shape of the curve is shown in the following figure.

A normal distribution is important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc.

- The curve of normal distribution is bell-shaped, unimodal, symmetric about the mean and extends to infinity in both directions. It is divided into two equal parts by the coordinate μ. The curve on one side of the coordinate is the mirror image of the coordinate on the other side.
- Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
- The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
- Since the distribution is symmetrical and has maximum height at mean,

**Mean=median=mode**

- The total area under normal curve is 1. The area under the normal curve is distributed as

**Area under μ + σ is 68.27% **

**Area under μ + 2σd is 95.45% **

**Area under μ + 3σ is 99.73%**

- Since the height of the curve is 1, hence for a given mean μ, as standard deviation σ increases the height of the curve decreases and approaches nearer and nearer to base but never touches it.
- The points of inflexion are μ + σ
- The first and third quartiles are equidistant from median
- Median deviation = standard deviation
- The spread of a normal distribution is controlled by the standard deviation, σ.
- The smaller the standard deviation, the more concentrated the data.

- It’s application goes beyond describing distributions
- It is used by researchers and modellers.
- The major use of normal distribution is the role it plays in statistical inference.
- The z-score along with the t –score, chi-square and F-statistics is important in hypothesis testing.
- It helps managers/management make decisions.

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