If **X** is a normally distributed random variable and X∼N(µ, σ), then the z-score is:

**The z-score tells you how many standard deviations that the value x is above (to the right of) or below (to the left of) the mean**, µ. Values of x that are larger
than the mean have positive z-scores and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of 0.

Example 3.1

Suppose **X** **∼** **N (5, 6)**. This says that **X** is a normally distributed random
variable with mean µ = 5 and standard deviation σ = 6. Suppose x = 17. Then:

This means that x = 17 is **2** **standard deviations** (2σ) above or to the right of the mean µ = 5. The standard deviation is σ = 6.

Notice that:

5+2 · 6 = 17 (The pattern is µ + zσ = x.)

Now suppose x = 1. Then:

(rounded to two decimal places)

**This means that x = 1 is 0.67 standard deviations (****−** **0.67****σ****) below or to the left of the mean** **µ** **= 5. Notice that:**

5+(−0.67) (6) is approximately equal to 1 (This has the pattern µ +(−0.67) σ = 1 ) Summarizing, when z is positive, x is above or to the right of µ and when z is negative, x is to the left of
or below µ.

Example 3.2

Some doctors believe that a person can lose 5 pounds, on the average, in a month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has a normal distribution.
Let **X** = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. **X****∼****N (5, 2)**. Fill in the blanks.

Problem 1

Suppose a person lost 10 pounds in a month. The z-score when x = 10 pounds is z = 2.5 (verify). This z-score tells you that x = 10 is ________ standard deviations to the ________ (right or left) of the mean ________ ( What is the mean?).

Problem 2

Suppose a person **gained** 3 pounds (a negative weight loss). Then z ________. This z-score tells you that x = −3 is ________ standard deviations to the ________
(right or left) of the mean.

Suppose the random variables X and Y have the following normal distributions: X ∼ N (5, 6) and Y ∼ N (2, 1). If x = 17, then z = 2. (This was previously shown.) If y = 4, what is z?

where µ = 2 and σ = 1.

The z-score for y = 4 is z = 2. This means that 4 is z = 2 standard deviations to the right of the mean. Therefore, x = 17 and y = 4 are both 2 (of **their**)
standard deviations to the right of **their** respective means.

**The z-score allows us to compare data that are** **scaled differently**. To understand the concept, suppose **X**
**∼****N (5, 6)** represents weight gains for one group of people who are trying to gain weight in a 6 week period and **Y** **∼****N (2, 1)** measures the same weight gain for a second group of people. A negative weight gain would be a weight
loss. Since **x = 17** and **y = 4** are each **2** standard deviations to the right of their means, they
represent the same weight gain **relative to their means**.

**The Empirical Rule**

If **X** is a random variable and has a normal distribution with mean µ and standard deviation σ then the **Empirical Rule** says (See the
Figure 3.1 below)

- About 68.27% of the x values lie between -1σ and +1σ of the mean µ (within 1 standard deviation of the mean).
- About 95.45% of the x values lie between -2σ and +2σ of the mean µ (within 2 standard deviations of the mean).
- About 99.73% of the x values lie between -3σ and +3σ of the mean µ (within 3 standard deviations of the mean). Notice that almost all the x values lie within 3 standard deviations of the mean.
- The z-scores for +1σ and 1σ are +1 and -1, respectively.
- The z-scores for +2σ and 2σ are +2 and -2, respectively.
- The z-scores for +3σ and 3σ are +3 and -3 respectively.

The Empirical Rule is also known as the 68-95-99.7 Rule.

**Example 3.3**

Suppose X has a normal distribution with mean 50 and standard deviation 6.

- About 68.27% of the x values lie between -1σ = (-1)(6) = -6 and 1σ = (1)(6) = 6 of the mean 50. The values 50 - 6 = 44 and 50 + 6 = 56 are within 1 standard deviation of the mean 50. The z-scores are -1 and +1 for 44 and 56, respectively.
- About 95.45% of the x values lie between -2σ = (-2)(6) = -12 and 2σ = (2)(6) = 12 of the mean 50. The values 50 -12 = 38 and 50 + 12 = 62 are within 2 standard deviations of the mean 50. The z-scores are -2 and 2 for 38 and 62, respectively.
- About 99.73% of the x values lie between -3σ = (-3)(6) = -18 and 3σ = (3)(6) = 18 of the mean 50. The values 50 - 18 = 32 and 50 + 18 = 68 are within 3 standard deviations of the mean 50. The z-scores are -3 and +3 for 32 and 68, respectively.

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