Finding a raw score from a given probability

Sometimes only the probability of an outcome occurring is known. To determine the specific value of the variable (X) that corresponds to a given probability in a normal distribution, the steps are as follows:

• Express the problem using a formula. For example, P(X < a) = p or P(X > a) = p.
• Find the appropriate percentile for z in the z-value tables.
• Convert the z value to X using the formula X = µ + zσ, which is derived from the z-score formula z = (X - µ) / σ.

We will illustrate the method of finding a score that corresponds to a certain probability using statistical tables, as follows:

• using a statistical table containing areas under the normal curve from µ to z and
• using a statistical table containing areas under the normal curve from -∞ to z.