In Probability and Statistics, the normal distribution or Gaussian Distribution is one of the most important and most studied concepts. Many theories and studies in Statistics are based upon Normal Distribution. It is a continuous probability distribution that gives a good description of data which is centered in and around the mean. Normal distribution is well represented by the graph. It is bell-shaped, with the mean at its peak.
The Gaussian Distribution is one of many things that is associated with Carl Friedrich Gauss, who used it to analyze astronomical data.
The normal distribution is often used to describe, at least approximately, any variable that tends to be closer to the mean. For ex: the height of adults in the United States is roughly normally distributed, with a mean height of 70-inches (1.8m). Most men height is close to the mean, though a small number of people whose height is above or below the mean. This can be said as generalization in normal terms.
Central limit theorem states that regardless of the distribution of the population, the distribution of the means of random samples approaches a normal distribution for a large sample size, under certain conditions. The sum of a number of random variables with finite means and variances approaches a normal distribution as the variable increases. For this reason, normal distribution is commonly used throughout in statistics, natural science as a sample model for complex phenomena.
Bell Curve characteristics
The bell curve has the following characteristics
• Extends to +/- infinity
• Area under the curve = 1
Completely described by two parameters
The normal distribution can be completely specified by two parameters:
• Standard deviation
Basic rules governing the normal distribution.
The rule is a quick estimate of the spread of the data given the mean and standard deviation of data set that follows the normal distribution.
The rule states that for a normal distribution
• 68% of the data falls within 1 standard deviation of the mean
• 95% of the data falls within 2 standard deviation of the mean
• Almost all (99.7%) of the data falls within 3 standard deviation of the mean
It should be kept in mind that the above values are approximations
The concepts of normal distribution have practical application in business administration:
• Modern portfolio theory commonly assumes that the returns of a diversified asset portfolio follow the pattern of normal distribution.
• In operations management, process variations are generally in normal distribution.
• In Human Resource Management, employee performance is sometimes is considered as normally distributed.
The normal distribution often is used to describe random variables, especially those having symmetrical, unimodal distributions. In many cases however, the normal distribution is only a rough approximation of the actual distribution. For example, the physical length of a component cannot be negative, but normal distribution extends both positive and negative directions. Nonetheless, the resulting errors may be negligible or within acceptable limits, allowing one to solve problems with sufficient accuracy by assuming a normal distribution.
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