3.2 Normal distribution: z-scores

Suppose $X$ ~ $\mathrm{N(5,\; 6)}$ . This says that $X$ is a normally distributed random variable with mean $\mathrm{\mu \; =\; 5}$ and standard deviation $\mathrm{\sigma \; =\; 6}$ . Suppose $\mathrm{x\; =\; 17}$ . Then:

This means that $\mathrm{x\; =\; 17}$ is 2 standard deviations $\mathrm{(2\sigma )}$ above or to the right of the mean $\mathrm{\mu \; =\; 5}$ . The standard deviation is $\mathrm{\sigma \; =\; 6}$ .

Notice that:

Now suppose $\mathrm{x=1}$ . Then:

This means that $\mathrm{x\; =\; 1}$ is 0.67 standard deviations $\mathrm{(-\; 0.67\sigma )}$ below or to the left of the mean $\mathrm{\mu \; =\; 5}$ . Notice that:

$5+\left(-0.67\right)\left(6\right)$ is approximately equal to 1 $$ (This has the pattern $\mu +\left(-0.67\right)\sigma =1$ )

Summarizing, when $z$ is positive, $x$ is above or to the right of $\mu$ and when $z$ is negative, $x$ is to the left of or below $\mu$ .

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 anormal distribution. Let $X$ = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. $X$ ~ $\mathrm{N(5,\; 2)}$ . Fill in the blanks.

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

The z-score for $\mathrm{y\; =\; 4}$ is $\mathrm{z\; =\; 2}$ . This means that 4 is $\mathrm{z\; =\; 2}$ standard deviations to the right of the mean. Therefore, $\mathrm{x\; =\; 17}$ and $\mathrm{y\; =\; 4}$ 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 $X$ ~ $\mathrm{N(5,\; 6)}$ represents weight gains for one group of people who are trying to gain weight in a 6 week period and $Y$ ~ $\mathrm{N(2,\; 1)}$ measures the same weight gain for a second group of people. A negative weight gain would be a weight loss.Since $\mathrm{x\; =\; 17}$ and $\mathrm{y\; =\; 4}$ are each 2 standard deviations to the right of their means, they represent the same weight gain relative to their means .

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

• About 68.27% of the $x$ values lie between -1 $\sigma$ = (-1)(6) = -6 and 1 $\sigma$ = (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 -2 $\sigma$ = (-2)(6) = -12 and 2 $\sigma$ = (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 -3 $\sigma$ = (-3)(6) = -18 and 3 $\sigma$ = (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.