Properties of Normal Distribution

The normal distribution thus indentified above has a good deal of mathematical properties for which it is considered as the most important of all the theoretical distributions developed so far. Some such properties are enumerated here as under:

1. It gives a bell shaped curve when the data are plotted on a graph paper.
2. The values of Mean, Median and Mode of such a distribution are identical. This means,
3. X = M = Z
4. The curve obtained from such a distribution is symmetrical about the Mean, which implies that the number of cases above the Mean and below the Mean are equal.
5. The first, and the third quartiles are equidistant from the Median of such a distribution. This Means (Q₃ – M) = (M – Q₁).
6. The height of the curve obtained from such a distribution is maximum at its Mean value which implies that the Mean ordinate divides the curve into two equal parts.
7. There is only one Mode in such a distribution as there is only one point of maximum frequency i.e. 6. Thus, in the example cited above, Z = 8 only, and nothing else. In other words, such distributions are unimodal.
8. The Mean Deviation (δ) of such a distribution is 3/4th, or more precisely 0.7979 of its Standard Deviation.
9. Semi-inter quartile range of such a distribution is nearly equal to the probable error which is 0.6745 σ
10. The curve drawn from such a distribution is asymptotic to the base which means that the curve continues to approach the base line but never touches it. The figure exhibited under the point 5 above bears a testimony to this effect.
11. The points of inflexion (i.e. the points at which a curve changes its direction) are each at a distance of one standard devation from the Mean of such a distribution.
12. The various ordinates at different distances of Standard devation from the mean ordinate, stand in a fixed proportion of the total area of the curve. Thus, the area enclosed between the two ordinates at 1 distance on either side of the mean ordinate is 60.653% of the height of the mean ordinate.
13. The area of the normal distribution curve enclosed between the mean ordinate, and an ordinate at a certain standard deviation distance from the Mean remains always in a fixed proportion of the total area of the curve. Thus, the area enclosed between the two ordinates at 1σ distance from the Mean on either side would always be 68.268% of the total area of the curve (i.e. 31.134% this side and 31.134% that side). Similarly, the area between the always be 95.45% of the total area of the curve (i.e. 47.725% this side and 47.715% that side). Likewise, the area between the two ordinates at 3σ distance from the mean ordinate on either side would be 99.73% of the total area of the curve. Such area relationship in a normal curve is very important from the view pint of testing the various hypotheses.

The chief points of importance of a normal distribution may be enumerated as under:

1. It possesses a lot of mathematical properties for which it is extensively used in a wide variety of fields of physical, natural and social sciences for making various types of analysis.

2. It is highly useful in statistical quality control in industries for setting up of the control limits.

3. It is very much used in finding out the probabilities of various sample results as the sample Mean has a theoretical property.

4. It is highly used in large sampling theory to find out the estimates of the parameters from statics and confidence limits.

5. It is confirmed by almost all the exact sampling distribution viz : Chi-squire distribution, t-distribution, F-distribution, Z –distribution etc for large degree of freedom.

6. According to the central limit theorem, the normal distribution is used to draw inferences abont a universe through sample studies. With this, we can also, estimate the upper and lower limits within which a value in the universe would lie.

7. This distribution can be regarded as a limiting case of the Poisson distribution, if the value of Mean(m) is infinitely large.

8. This distribution can be regarded as a limiting case of the binomial distribution, if the value of ‘n’ (number of trials) is very large, and neither p nor q is very small.

9. When n is very large, computation of probability for most of the discrete distributions e.g. binomial, Poisson etc. becomes quite arduous, and time taking. In such cases normal distribution can be advantageously used with great case and convenience.