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normal distribution

The document discusses the concept of normal distribution, also known as Gaussian distribution, and its significance in applied mathematics within the field of chemical engineering. It illustrates properties, graphical representation, and examples of standardization, highlighting its application in statistical analysis, including quality control and probability calculations. Additionally, it provides references for further reading on the topic.

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APPLIED MATHEMATICS IN CHEMICAL ENGG SEMINAR ON : SUBMITTED BY MAHASWARI JOGIA 5/2/2017DSCE,CHEM ENGG DEPT
5/2/2017DSCE,CHEM ENGG DEPT 2 CONTENTS  NORMAL DISTRIBUTION IMPORTANCE OF NORMAL DISTRIBUTION PROPERTIES NORMAL DISTRIBUTION GRAPHICALLY AREA UNDER THE CURVE EXAMPLE STANDARDIZATION EXAMPLE REFERENCE
NORMAL DISTRIBUTION •Normal Distribution, also called Gaussian Distribution, is one of the widely used continuous distributions existing which is used to model a number of scenarios such as marks of students, heights of people, salaries of working people etc. •Each binomial distribution is defined by n, the number of trials and p, the probability of success in any one trial. Each Poisson distribution is defined by its mean. •In the same way, each Normal distribution is identified by two defining characteristics or parameters: its mean and standard deviation. 5/2/2017 3DSCE,CHEM ENGG DEPT
Mean = Median = Mode. Skewness = zero Total area under the curve = 1 •The Normal distribution has three distinguishing features: - It is unimodal, in other words there is a single peak. - It is symmetrical, one side is the mirror image of the other. -It is asymptotic, that is, it tails off very gradually on each side but the line representing the distribution never quite meets the horizontal axis. - It is bell-shaped curve 5/2/2017 4DSCE,CHEM ENGG DEPT
Fig: NORMAL DISTRIBUTION CURVE When number of trials increase , probability distribution tends to normal distribution hence , majority of problems and studies can be analyzed through normal distribution. Used in statistical quality control for setting quality standards. IMPORTANCE OF NORMAL DISTRIBUTION: 5/2/2017 5 DSCE,CHEM ENGG DEPT
PROPERTIES: •It is symmetric around the point x = μ, which is at the same time the mode, the median and the mean of the distribution. •It is unimodal: its first derivative is positive for x < μ, negative for x > μ, and zero only at x = μ. •It has two inflection points (where the second derivative of f is zero and changes sign), located one standard deviation away from the mean, namely at x = μ − σ and x = μ + σ. •It is log-concave •It is infinitely differentiable. 5/2/2017 6DSCE,CHEM ENGG DEPT
THE NORMAL DISTRIBUTION: GRAPHICALLY Normal Curve is Symmetrical Two halves identical Mean, Median and Mode are equal. Theoretically, curve extends to - ∞ Theoretically, curve extends to + ∞ 5/2/2017 7DSCE,CHEM ENGG DEPT
AREA UNDER THE CURVVE P(X<x1) P(X>x2) P(x1<X<x2) If X is Normally distributed with mean µ and standard deviation 5/2/2017 8DSCE,CHEM ENGG DEPT
The probability density of the Normal distribution is given by: Calculating probabilities from the Normal distribution: For a continuous probability distribution we calculate the probability of being less than some value x, i.e. P(X < x), by calculating the area under the curve to the left of x. 5/2/2017 9DSCE,CHEM ENGG DEPT
EXAMPLE: 1) Suppose Z~N (0,1) then what is P(Z<0) ? Symmetry P(Z < 0) = 0:5 2) What about P(Z < 1:0)? Calculating this area is not easy and so we use probability tables. Probability tables are tables of probabilities that have been calculated on a computer. All we have to do is identify the right probability in the table and copy it down! 5/2/2017 10 DSCE,CHEM ENGG DEPT
The N(0, 1) distribution is called the standard Normal distribution Only one special Normal distribution, N(0, 1), has been tabulated. The tables allow us to read off probabilities of the form P(Z < z). 5/2/2017 11DSCE,CHEM ENGG DEPT
From this table we can identify that P(Z < 1:0) = 0:8413 5/2/2017 12DSCE,CHEM ENGG DEPT
•If Z~ N(0, 1) what is P(Z > 0:92)? We know that P(Z > 0:92) = 1 - P(Z < 0:92) and we can calculate P(Z < 0:92) from the tables. Thus, P(Z > 0:92) = 1 - 0:8212 = 0:1788 5/2/2017 13DSCE,CHEM ENGG DEPT
STANDARDIZATION All of the probabilities above were calculated for the standard Normal distribution N(0, 1). If we want to calculate probabilities from different Normal distributions we convert the probability to one involving the standard Normal distribution. We convert this probability to one involving the N(0, 1) distribution by: (i) Subtracting the mean (ii) (ii) Dividing by the standard deviation 5/2/2017 14DSCE,CHEM ENGG DEPT
EXAMPLE: Suppose we know that the birth weight of babies is Normally distributed with mean 3500g and standard deviation 500g. What is the probability that a baby is born that weighs less than 3100g? That is X ~ N(500, 5002) and we want to calculate P(X < 3100)? We can calculate the probability through the process of standardization. Draw a rough normal distribution curve to make it easier to understand the distribution 5/2/2017 15DSCE,CHEM ENGG DEPT
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REFERENCES: •Engineering mathematics IV by Dr. K.S.C •Normal distribution by Subhrat Sharma- CUHP13MBA85 •Normal distribution and hypothesis testing by Mr. Roderico Y. Dumaug jr •Math statistics by A.J Hilde brad •The normal distribution by Jonathan Marchini •Advanced engineering mathematics by Erwin Kreyszig 5/2/2017 17DSCE,CHEM ENGG DEPT
5/2/2017DSCE,CHEM ENGG DEPT 18 THANK YOU!