Question

How many score between 0.5 to 1.5 of normal curve?

Answer 1

As we saw, the Standard Deviation Rule is very limited in helping us answer probability questions, and basically limited to questions involving values that fall exactly 1, 2, and 3 standard deviations away from the mean. How do we answer probability questions in general? The key is the position of the value relative to the mean, measured in standard deviations.

We can approach the answering of probability questions two possible ways: a table and technology. In the next sections, you will learn how to use the “standard normal table,” and then how the same calculations can be done with technology.

Standardizing Values

The first step to assessing a probability associated with a normal value is to determine the relative value with respect to all the other values taken by that normal variable. This is accomplished by determining how many standard deviations below or above the mean that value is.

EXAMPLE: Foot Length

How many standard deviations below or above the mean male foot length is 13 inches? Since the mean is 11 inches, 13 inches is 2 inches above the mean.

Since a standard deviation is 1.5 inches, this would be 2 / 1.5 = 1.33 standard deviations above the mean. Combining these two steps, we could write:

(13 in. – 11 in.) / (1.5 inches per standard deviation) = (13 – 11) / 1.5 standard deviations = +1.33 standard deviations.

In the language of statistics, we have just found the z-score for a male foot length of 13 inches to be z = +1.33. Or, to put it another way, we have standardized the value of 13.

In general, the standardized value z tells how many standard deviations below or above the mean the original value is, and is calculated as follows:

z-score = (value – mean)/standard deviation

The convention is to denote a value of our normal random variable X with the letter “x.”

Notice that since the standard deviation (sigma, σ) is always positive, for values of x above the mean (mu, μ), z will be positive; for values of x below the mean (mu, μ), z will be negative.