One type of problem that is typical in an introductory statistics course is to find the z-score for some value of a normally distributed variable. After providing the rationale for this, we will see several examples of performing this type of calculation.

### Reason for Z-scores

There are an infinite number of normal distributions. There is a single standard normal distribution. The goal of calculating a *z* - score is to relate a particular normal distribution to the standard normal distribution. The standard normal distribution has been well-studied, and there are tables that provide areas underneath the curve, which we can then use for applications.

Due to this universal use of the standard normal distribution, it becomes a worthwhile endeavor to standardize a normal variable. All that this z-score means is the number of standard deviations that we are away from the mean of our distribution.

### Formula

The formula that we will use is as follows: *z* = (*x* - μ)/ σ

The description of each part of the formula is:

*x*is the value of our variable- μ is the value of our population mean.
- σ is the value of the population standard deviation.
*z*is the*z*-score.

### Examples

Now we will consider several examples that illustrate the use of the *z*-score formula. Suppose that we know about a population of a particular breed of cats having weights that are normally distributed. Furthermore, suppose we know that the mean of the distribution is 10 pounds and the standard deviation is 2 pounds. Consider the following questions:

- What is the
*z*-score for 13 pounds? - What is the
*z*-score for 6 pounds? - How many pounds corresponds to a
*z*-score of 1.25?

For the first question, we simply plug *x* = 13 into our *z*-score formula. The result is:

(13 – 10)/2 = 1.5

This means that 13 is one and a half standard deviations above the mean.

The second question is similar. Simply plug *x* = 6 into our formula. The result for this is:

(6 – 10)/2 = -2

The interpretation of this is that 6 is two standard deviations below the mean.

For the last question, we now know our *z* -score. For this problem we plug *z* = 1.25 into the formula and use algebra to solve for *x*:

1.25 = (*x* – 10)/2

Multiply both sides by 2:

2.5 = (*x* – 10)

Add 10 to both sides:

12.5 = *x*

And so we see that 12.5 pounds corresponds to a *z*-score of 1.25.