Properties of Normal Distribution

The following are the important properties of the normal curve and the normal distribution:
1.    The normal curve is symmetrical about the mean. If the curve were folded along its vertical axis, the two halves would coincide. The number of cases below the mean in a normal distribution is equal to the number of cases above the mean, which makes the mean and median coincide. The height of the curve for a positive deviation of 3 units is the same as the height of the curve of negative deviation of 3units.

2.    The height of the normal curve is at its maximum at the mean. Hence the mean and mode of the normal distribution coincide. Thus for a normal distribution mean, median and mode are all equal.

3.    There is one maximum point of the normal curve which occurs at the mean. The height of the curve declines as we go in either direction from the mean. The curve approaches nearer and nearer to the base but it never touches it, i.e., the curve is asymptotic to the base on either side. Hence its range is unlimited or infinite in both directions.

4.    Since there is only one maximum point, normal curve is unimodal, i.e., it has only one mode.

5.    The points of inflexion, i.e., the points where the change in curvature occurs are X + σ.

6.    As distinguished form Binomial and Poisson distributions where the variable is discrete, the variable distributed according to the normal curve is a continuous one.

7.    The first and third quartiles are equidistant from the median.

8.    The mean deviation is 4/5th or more precisely 0.7979 of the standard deviation.

9.    The area under the normal curve is distributed  as follows:

(a)    Mean + 1 σ covers 68.27% area; 31.135% area will lie on either side of mean.
(b)    Mean + 2 σ covers 95.45% area.
(c)    Mean + 3 σ covers 99.73% area.

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