Standardizing with z-Scores and Percentiles

Flashcards reviewing key vocabulary, formulas, interpretations, and calculations for percentiles and z-scores based on Pixel Run and Soccer career statistics.

Percentile

The percent of values in a distribution that are less than or equal to a given value.

z-Score (Standardized Score)

A measure that describes a value's position relative to the mean, calculated using the formula $z = \frac{\text{value} - \text{mean}}{\text{SD}}$.

Standardized Score Interpretation Template

A standard phrasing for interpreting a z-score: "[The context] is $[z]$ standard deviations [above/below] the mean."

Patrick's Pixel Run z-Score

For a score of $500$ with mean $700$ and SD $100$, $z = \frac{500 - 700}{100} = -2$, meaning the score is $2$ standard deviations below the mean.

SpongeBob's Pixel Run z-Score

For a score of $800$ with mean $700$ and SD $100$, $z = \frac{800 - 700}{100} = 1$, meaning the score is $1$ standard deviation above the mean.

SpongeBob's Relative Game Performance

SpongeBob performed better relative to his peers in Star Catch ($z = 1.5$) than in Pixel Run ($z = 1$) because $1.5 > 1$.

Lionel Messi's Career Goals Percentile

$96$\% (calculated as $\frac{24}{25} = 96\text{\%}$), indicating that $96$\% of the top $25$ soccer players scored less than or equal to Messi's total of $819$ goals.

Lionel Messi's Career Goals z-Score

$z = \frac{819 - 722}{60} = 1.617$, meaning Messi's career goals total is $1.617$ standard deviations above the mean of $722$.

Romário's Career Goals

$780$ goals, corresponding to the $80\text{th}$ percentile ($0.8 \times 25 = 20\text{th}$ value in ordered data).

Lionel Messi's Career Assists z-Score

$z = \frac{392 - 158}{73} = 3.205$, showing his assists performance ($3.205$ SD above mean) was more impressive than his goal performance ($1.617$ SD above mean).