Question

a data set has the following characteristics:
mean: 4.9
median: 6
mode: 6
variance: 4
using the formula, calculate the z - score for the listed data points.
z - score: $z_{x}=\frac{x - mu}{sigma}$
$z_{1}=square$
$z_{5}=square$
$z_{8.1}=square$

Explanation:

First, we need to find the standard deviation ($\sigma$) from the variance. The standard deviation is the square root of the variance.

Step 1: Find the standard deviation ($\sigma$)

Given variance = 4, so $\sigma = \sqrt{4} = 2$.

Step 2: Calculate the z - score for $x = 1$

Using the z - score formula $z_{x}=\frac{x-\mu}{\sigma}$, where $\mu = 4.9$ (mean), $\sigma = 2$, and $x = 1$.
$z_{1}=\frac{1 - 4.9}{2}=\frac{- 3.9}{2}=-1.95$

Step 3: Calculate the z - score for $x = 5$

Using the z - score formula with $x = 5$, $\mu = 4.9$, and $\sigma = 2$.
$z_{5}=\frac{5 - 4.9}{2}=\frac{0.1}{2}=0.05$

Step 4: Calculate the z - score for $x = 8.1$

Using the z - score formula with $x = 8.1$, $\mu = 4.9$, and $\sigma = 2$.
$z_{8.1}=\frac{8.1 - 4.9}{2}=\frac{3.2}{2}=1.6$

Answer:

$z_{1}=-1.95$, $z_{5}=0.05$, $z_{8.1}=1.6$