12 December 2017

Important properties of the Normal distribution


The normal distribution was first described by Abraham Demoivre (1667-1754) as the limiting form of binomial model in 1733. Normal distribution was rediscovered by Gauss in 1809 and by Laplace in 1812. Both Gauss and Laplace were led to the distribution by their work on the theory of errors of observations arising in physical measuring processes particularly in astronomy. Here I will show you some important properties of Normal distribution

1. The normal curve is “bell shaped” and symmetrical in nature. The distribution of the frequencies on either side of the maximum ordinate of the curve is similar with each other.

2. The maximum ordinate of the normal curve is atclip_image002. Hence the mean, median and mode of the normal distribution coincide.

3. It ranges between clip_image004 to clip_image006

4. The value of the maximum ordinate is clip_image008

5. The points where the curve change from convex to concave or vice versa is at clip_image010

6. The first and third quartiles are equidistant from median.

7. The area under the normal curve distribution are:

a. clip_image012 covers 68.27% area

b. clip_image014 covers 95.45% area

c. clip_image016 covers 99.73% area


8. When μ = 0 and σ = 1, then the normal distribution will be a standard normal curve. The probability function of standard normal curve is


The following table gives the area under the normal probability curve for some important value of Z.

Distance from the mean ordinate in

Terms of ± σ

Area under the curve

Z = ± 0.6745


Z = ± 1.0


Z = ± 1.96


Z = ± 2.00


Z = ± 2.58


Z = ± 3.0


9. All odd moments are equal to zero.

10. Skewness = 0 and Kurtosis = 3 in normal distribution.

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