Normal Distribution

Figure 1

The normal distribution reflects the various values taken by many real life variables like the heights and weights of people or the marks of students in a large class. In all these cases a large number of observations are found to be clustered around the mean value m and their frequency drops sharply as we move away from the mean in either direction. For example, if the mean height of an adult in a city is 6 feet then a large number of adults will have heights around 6 feet. Relatively a few adults will have heights of 5 feet or 7 feet.

Further, if we draw samples of size n (where n is a fixed number over 30) from any population, then the sample mean    will be (approximately) normally distributed with a mean equal to m i.e. the mean of the population.

The characteristics of normal probability distribution with reference to the above figure are

1. The curve has a single peak; thus it is unimodal.

2. The mean of a normally distributed population lies at the center of its normal curve.

3. Because of the symmetry of the normal probability distribution, the median and the mode of the distribution are also at the center.

4. The two tails of the normal probability distribution extend indefinitely and never touch the horizontal axis.

If s is the standard deviation of the normal distribution, 80% of the observation will be in the interval m -1.28s to m + 1.28s.

Figure 2

95% of the observations will be in the interval m - 1.96s to m + 1.96s.

Figure 3

98% of the observations will lie in the interval m - 2.33 s to m + 2.33 s.

Figure 4

Standard Normal Distribution

The Standard Normal Distribution is a normal distribution with a mean m = 0 and a standard deviation s = 1. The observation values in a standard normal distribution are denoted by the letter Z.