Normal Distribution

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Presentation on Normal Distribution in Statistical Quality Control

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NORMAL DISTRIBUTION :

NORMAL DISTRIBUTION Priyanka Deka - 1RV09ME077 Trisha Gopalakrishna-1RV09ME106

Normal Distribution:

Normal Distribution The normal distribution is pattern for the distribution of a set of data which follows a bell shaped curve. This distribution is sometimes called the Gaussian distribution in honour of Carl Friedrich Gauss, a famous mathematician.

PowerPoint Presentation:

It is an important continuous distribution curve. Review of continuous random variable, for examples: ( i ) heights and weights of adults (ii) length and width of leaves of the same species (iii) distance jumped by a C grade boy in several times (iv) distance jumped by a group of C grade boys (v) actual weights of rice in 5 kg bags sold in supermarkets

Mathematically, :

Mathematically, Let μ & σ be two arbitrary real constants such that -∞< μ<∞ and σ> 0. Then the continuous probability distribution for which N( μ , σ ,x)=1/ σ (2 π ) 1/2 exp{-1/2(x- μ / σ )} is the Probability Distribution Function which is called Normal Distribution ; the corresponding continuous variable x is called the Normal Variate . The constants μ & σ are called the parameters of the distribution

PowerPoint Presentation:

For a population that is normally distributed : approx. 68% of the data will lie within 1 standard deviation of the mean; approx. 95% of the data will lie within 2 standard deviations of the mean, and approx. 99.7% of the data will lie within 3 standard deviations of the mean.

Standard Normal distribution:

Standard Normal distribution Many situations will involve data that is normally distributed. We will often want to find probabilities of events occurring or percentages of nonconformities, etc.. A standardized normal random variable is:

PowerPoint Presentation:

Z is normally distributed with mean 0 and standard deviation, 1. Use the standard normal distribution to find probabilities when the original population or sample of interest is normally distributed. Tables, calculators are useful.

Properties of the Bell Shaped Curve:

Properties of the Bell Shaped Curve The curve concentrated in the centre and decreases on either side. This means that the data has less of a tendency to produce unusually extreme values, compared to some other distributions. The bell shaped curve is symmetric. This tells you that he probability of deviations from the mean are comparable in either direction. When you want to describe probability for a continuous variable, you do so by describing a certain area. A large area implies a large probability and a small area implies a small probability. Some people don't like this, because it forces them to remember a bit of geometry (or in more complex situations, calculus). But the relationship between probability and area is also useful, because it provides a visual interpretation for probability. Here's an example of a bell shaped curve. This represents a normal distribution with a mean of 50 and a standard deviation of 10.

Applications to Engineering:

Applications to Engineering There are two principle applications of normal distribution to engineering & reliability . One application deals with analysis of items which exhibit failure due to wear, such as mechanical devices. Frequently, the wear-out failure distribution is sufficiently close to normal that the use of this distribution for predicting or assessing reliability is valid.

Example 1

PowerPoint Presentation:

Another application is in the analysis of manufactured items and their ability to meet specifications. No two parts made to the same specifications are exactly alike. The variability of the parts leads to variability in the systems composed of these parts. The design must take this part variability into account, otherwise the system may not meet the requirement due to the combined effect of part variability. Another aspect of this application is quality control procedures. The basis for the use of normal distribution in this application is central limit theorem which states that the sum of a large number of identically distributed random variables, each with finite mean & variance, is normally distributed. Thus, the variations in value of mechanical component parts, for example, due to manufacturing are considered normally distributed.

Example 2

Applications in behavioral studies :

Applications in behavioral studies The normal distribution is a convenient model of quantitative phenomena in the natural and behavioural sciences. A variety of psychological test scores and physical phenomena like photon counts have been found to approximately follow a normal distribution

Applications to Business & Management:

Applications to Business & Management In the field of operations management, results of many processes fall along the Normal Distribution Curve. The Normal Probability Distribution governs many aspects of human performance. Human resource professionals often use the Normal Distribution to describe employee performance. A diversified portfolio will typically have returns that fall in a Normal Distribution.

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