# Shaun White won the gold medal with a score of 46.80, what is the corresponding z-score?

## In honor of the Winter Olympics beginning this week, let’s consider scores of athletes in the Men’s Half Pipe Snowboarding Final in Vancouver in 2010. Scores had a mean of 33.87 and a standard deviation of 9.19. For this exercise let’s consider the men in the finals competition to be the population of interest. Gregory Bretz scored 18.30 in the competition, what is the corresponding z-score? The 9th place finisher scored a z-score of -0.3667, what is the corresponding raw score? The bronze medalist had a z-score of 0.9717, what is the corresponding raw score?

Sep 15, 2017

Shaun White z-score$\approx 1.407$
Gregory Betz z-score $\approx - 1.6942$
9th place raw score $\approx 30.5$
Bronze raw score $\approx 42.8$

#### Explanation:

The z-score is the amount of standard deviations an element, which means the absolute value of $| z |$ represents the distance from the raw score and the population mean, in units of the standard deviation. To calculate the z-score, you use this nifty formula:
$z = \frac{X - \mu}{\sigma}$,
where $z$ is the z-score, $\sigma$ is the standard deviation, $X$ is the value of the element and $\mu$ is the population mean. Using this formula, we can look at the first question and see that all values except for $z$ are accounted for.

$X = 46.8$, $\mu = 33.87$ and $\sigma = 9.19$
Therefor, we plug these values into the equation to get
$z = \frac{46.8 - 33.87}{9.19}$,
$z = \frac{12.93}{9.19}$,
$z \approx 1.407$, or the element $\left(X\right)$ is 1.407 standard deviations greater then the mean. We repeat this formula for Gregory Betz, to find that $z \approx - 1.6942$, or the element $\left(x\right)$ is 1.6942 standard deviations less then the mean.

For the next two questions, we no longer know the value of the raw score $\left(X\right)$, but we now know the z-score, $- 0.3667$ and $0.9717$ respectively. Once again, we plug these numbers into the formula, which for the 9th place person is
$- 0.3667 = \frac{X - 33.87}{9.19}$ From here we multiply both sides by 9.19,

$- 3.369973 = X - 33.87$, finally add 33.87 to both sides to get
$30.500027 = X$, or $30.5 \approx X$, the raw score. We do the same thing for the bronze medallist to find that
$42.8 \approx X$

http://stattrek.com/statistics/dictionary.aspx?definition=z%20score

Thanks for reading, I hope that helped!