Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 74

1 If X is a non-standard normal with mean µ and standard deviation σ, then Z=(X−μ)/σ.

2 If Z is a standard normal, then X=σ⋅Z+μ is a non-standard normal with mean µ and standard deviation σ.

Note that the value of a non-standard normal X can be converted into a Z-value and vice versa. The first equation shows that centering and scaling the values of X converts them to Z-values, and the second equation shows how to convert a Z-value into an X-value. The relationship between a standard normal and a non-standard normal is shown in Figure 2.29.

Figure 2.29 The correspondence between the values of a standard normal and a non-standard normal.

To determine the cumulative probability for the value of a non-standard normal, say x, convert the value of x to its corresponding z value using z=z−μ/σ; then determine the cumulative probability for this z value. That is,

To compute probabilities other than a cumulative probability for a non-standard normal, note that the probability of being in any region can also be computed from the cumulative probabilities associated with the standard normal. The rules for computing the probabilities associated with a non-standard normal are given below.