As you can imagine, it’s not enough to be able to estimate between 1, 2, or 3 standard deviations. What if we wanted to go 2.5, or -1.75 standard deviation from the mean? To do this, we use a z-score. A z-score is a measurement of how many standard deviations above or below the mean a given value is. The formula below is used to calculate the z-score. In the formula, ‘x’ is an observation, μ ("mu") is the mean of the distribution, and σ (lower case "sigma") is the standard deviation of the distribution.

$z = \frac{x - μ}{σ}$ Let's use a situation and some examples to illustrate how useful z-scores can be.

The Situation:
Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat intake and by exercising consistently. Suppose the number of pounds lost by a person has a normal distribution. Let X = the amount of weight lost(in pounds) by a person in a month. Use a standard deviation of two pounds

Read the examples and include them in your journal for reference.

Example 1: Suppose a person lost ten pounds in a month. The z-score is _____. This z-score tells you that the value "10" is ________ (how many?) standard deviations to the ________ (right or left?) of the mean, _____ (what is the mean?).

(1) Label the known values.

x = 10, μ = 5, σ = 2

(2) Plug these values into the z-score formula and solve for z.

$z = \frac{10 - 5}{2} = \frac{5}{2} = 2.5$

You should have written:

"The z-score is 2.5. This z-score tells you that the value "10" is 2.5 standard deviations to the right of the mean, 5."

Example 2: Suppose a person gained three pounds (a negative weight loss). The z-score = _____ This z-score tells you that x = –1 is ________ standard deviations to the __________ (right or left) of the mean _____.

(1) Label the known values.

x = –1, μ = 5, σ = 2

(2) Plug these values into the z-score formula and solve for z.

$z=\frac{-1-5}{2}=-3$

normal curve

You should have written:

"The z-score is –3. This z-score tells you that the value "–1" is 3 standard deviations to the left of the mean 5."

This image adapted from OpenStax College. Download for free at http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de@18.13