Importance of Normal Distribution

The normal distribution has long occupied a central place in the theory of statistics. Its importance will be clear from the following points:

1.    The normal distribution has the remarkable property stated in the so-called central limit theorem. According to this theorem as the sample size n increase, the distribution or mean, X of a random sample taken from practically any population approaches a normal distribution (with mean µ and standard deviation σ / √n). Thus, if samples of large size, n, are drawn from a population that is not normally distributed, nevertheless the successive sample means will form themselves distribution that is approximately normal. As the size of the sample is increased the sample means will tend to be normally distributed. The central limit theorem applies to the distribution of most other statistics as well, such as the median and standard deviation (but nor range). The central limit theorem gives the normal distribution its central place in the theory of sampling since many important problems can be solved by this single patter of sampling variability. As a result the work on statistical inferences is made simpler.

2.    As n becomes large the normal distribution serves as a good approximation of many discrete (such as the Binomial of the Poisson model) whenever the exact discrete probability is laborious to obtain or impossible to calculate accurately.

3.    In theoretical statistics many problems can be solved only under the assumptions of the normal population. In applied work as well, we often find that methods developed under the normal probability law yield satisfactory results, even when the assumption of a normal population is not fully met, despite the fact that the problem can have a formal solution only if such a premise is hypothesized.

4.    The normal distribution has numerous mathematical properties which make it popular and comparatively easy to manipulate.

5.    The normal distribution is used extensively in statistical quality control in industry in setting up of controls limits.

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