Statistical Methods (1+1)

Unit 2 - Probability distributions

2.7.2.Area under the normal curve
The area bounded by the normal curve on the x - axis is 1. Quite frequently the area under this curve that falls between two points on the x - axis, say, x = a and x = b is required. This area can be worked out using integral calculus. However, it is not necessary to work out the area by this method as tables giving the areas under the normal curve are available for ready use. These tables give the area under the normal curve which has mean zero and standard deviation one (called standard normal curve). Hence to make use of this table, the normal variable X has to be transformed to normal variable Z by the following relation,

As the standard normal curve is symmetric (Fig. below) about Z = 0, the area between Z = 0 and any negative Z value, say, Z = -a, is equivalent to the area between Z = 0 and Z = +a.

Fig : Standard normal curve

Example 7
Weight of a particular species of fish was found to be distributed normally with the mean 400 grams and standard deviation 50 grams. Find the standard normal variate of fishes with weights (i) 300, (ii) 450 and (iii) 430.