Areas under the Normal Curve

The curve of any continuous probability distribution or density function is constructed so that the area under the curve bounded by the two ordinatos x — x\ and x — x2 equals the probability that the random variable X assumes a value between x = x\ and x = x2. Thus, for the normal curve in Figure 6.6,

In Figures 6.3, 6.4, and 6.5 we saw how the normal curve is dependent on the mean and the standard deviation of the distribution under investigation. The area under the curve between any two ordinates must then also depend on the values p and a. This is evident in Figure 6.7, where we have shaded regions corresponding

to P(xi < X < x-2) for two curves with different means and variances. The P{x\ < X < x2), where X is the random variable describing distribution A, is indicated by the darker shaded area. If X is the random variable describing distribution B, then P(x.\ < X < x2) is given by the entire shaded region. Obviously, the two

shaded regions are different in size; therefore, the probability associated with each distribution will be different for the two given values ol X. The difficulty encountered in solving integrals of normal density functions necessitates the tabulation of normal curve areas for quick reference. However, it 

Definition

The distribution of a normal random variable with mean 0 and variance 1 is called

a standard normal distribution.

The original and transformed distributions are illustrated in Figure 6.8. Since

all the values of X falling between x\ and x2 have corresponding z values between

z\ and z2, the area under the X-curve between the ordinates x = x\ and x = x2 in

Figure 6.8 equals the area under the Z-curve between the transformed ordinates

z = z\ and z — z2.

We have now reduced the required number of tables of normal-curve areas to

one, that of the standard normal distribution. Table A.3 indicates the area under

the standard normal curve corresponding to P(Z < z) for values of z ranging from

—3.49 to 3.49. To illustrate the use of this table, let us find the probability that Z

is less than 1.74. First, we locate a value of z equal to 1.7 in the left column, then

move across the row to the column under 0.04, where we read 0.9591. Therefore,

P(Z < 1.74) = 0.9591. To find a z value corresponding to a given probability, the

process is reversed. For example, the z value leaving an area of 0.2148 under the

curve to the left of z is seen to be —0.79