# Question 3c68b

May 6, 2017

A z-score is a multiple of the Std. Dev. for a standard normal distribution.

#### Explanation:

The standard deviation of any distribution measures how spread out ("dispersed") the scores are.

For any normal distribution, expect approximately 68% of the data to lie within 1 standard deviation of the mean. Expect about 95% to lie within 2 standard deviations of the mean. Expect 99.7% to lie within 3 standard deviations of the mean.

For a standard normal curve, a z-score indicates the number of standard deviations that a given score is above or below the mean for that distribution. The z-score does not identify the standard deviation of the original distribution, but for the standard normal distribution the standard deviation is $\sigma = 1$.

May 7, 2017

Standard distribution is calculated for a given distribution,
z-score is calculated for an $x$ value of a given distribution.

$\sigma = 6.67$

#### Explanation:

Standard distribution is calculated for a given distribution,
z-score is calculated for an $x$ value of a given distribution.

For each $x$ value in the given distribution, there is one corresponding $z$score.

There are as many $z s c \mathmr{and} e s a s t h e r e a r e$x values. But there is only one standard deviation value.

SD along with the mean of a series is used to calculate $z$ score of a given $x$ value.

Given -
Mean $\overline{x} = 45$
An element of this data set $x = 50$

$z = 0.75$
$\sigma =$?

$z = \frac{x - \overline{x}}{\sigma}$
$0.75 = \frac{50 - 45}{\sigma}$
$0.75 \sigma = 50 - 45$
$\sigma = \frac{5}{0.75} = 6.67$