2.2: Data Transformations and Z-Scores

Measures of center include the mean, median, quartiles, minimum, and maximum of a set of data

Measures of spread include the IQR, range, and standard deviation

When a constant is added to every number in a list,

The measures of center increase by that amount and the measures of spread remain the same

When every number in a list is multiplied by a constant,

The measures of center are multiplied by that amount and the measures of spread also get multiplied by that amount

Suppose we have two values that we want to compare, but they don’t come from the same distribution

Eg. getting a 610 on SAT math vs. a 24 on ACT math

Eg. being a 80” guy vs. a 76.5” girl

We can compare these figures by getting a standardized score called a z-score

z = (x-μ)/σ

z: z-score

x: value

μ: population mean

σ: population standard deviation

A z-score shows exactly how many standard deviations above or below the mean a value is

Even if two data points come from different distributions, they can be directly compared

*if*converted to z-scoresWhen comparing two z-scores, the higher z-score is the better relative score

A chart which can be used to convert proportions to z-scores and vise-versa

Available on the “cheat sheet” during the AP exam

Find percent of values less than 54

μ = 65

σ = 9

X~N (μ,σ) → Z~N (65,9)

Finding P(x<54) → x = 54, put into equation z = (x-μ)/σ

Locate z = -1.2222 on z-chart

P(x < 54) = P(z < -1.22) = 0.1112 (11.12%)

Measures of center include the mean, median, quartiles, minimum, and maximum of a set of data

Measures of spread include the IQR, range, and standard deviation

When a constant is added to every number in a list,

The measures of center increase by that amount and the measures of spread remain the same

When every number in a list is multiplied by a constant,

The measures of center are multiplied by that amount and the measures of spread also get multiplied by that amount

Suppose we have two values that we want to compare, but they don’t come from the same distribution

Eg. getting a 610 on SAT math vs. a 24 on ACT math

Eg. being a 80” guy vs. a 76.5” girl

We can compare these figures by getting a standardized score called a z-score

z = (x-μ)/σ

z: z-score

x: value

μ: population mean

σ: population standard deviation

A z-score shows exactly how many standard deviations above or below the mean a value is

Even if two data points come from different distributions, they can be directly compared

*if*converted to z-scoresWhen comparing two z-scores, the higher z-score is the better relative score

A chart which can be used to convert proportions to z-scores and vise-versa

Available on the “cheat sheet” during the AP exam

Find percent of values less than 54

μ = 65

σ = 9

X~N (μ,σ) → Z~N (65,9)

Finding P(x<54) → x = 54, put into equation z = (x-μ)/σ

Locate z = -1.2222 on z-chart

P(x < 54) = P(z < -1.22) = 0.1112 (11.12%)