Notes on Transformations, Z-Scores, and Percentiles

Transforming Data: Adding/Subtracting and Multiplying/Dividing

• Adding/Subtracting a constant 'a':
  • Measures of center/location (mean, median, percentiles) change by $+a$ or $-a$.
  • Measures of variability (range, IQR, standard deviation) do not change.
  • Shape of the distribution does not change. Shifts distribution left/right.
• Multiplying/Dividing by a positive constant 'b':
  • Measures of center/location and variability are multiplied/divided by $b$.
  • Shape of the distribution does not change. Scales the distribution.
• Combined Transformations (linear transformation $Y = bX + a$):
  • Mean: $\mu_Y = b\mu_X + a$
  • Standard Deviation: $\sigma_Y = |b|\sigma_X$
  • Shape remains unchanged.

Standardizing Distributions: Z-scores

• Definition: A z-score measures how many standard deviations a data value $X$ is from the mean $\mu$.
• Formula: $Z = \frac{X - \mu}{\sigma}$
• Effects on Distribution:
  • Shape: Unchanged.
  • Center: Mean of z-scores is $0$.
  • Variability: Standard deviation of z-scores is $1$.
• Purpose: Allows comparison of values from different distributions on a common scale.

Percentiles and Cumulative Relative Frequency

• p-th percentile: The value below which $p$ percent of the observations fall.
  • Formula: $P(X \le x_p) = \frac{p}{100}$
• Cumulative Relative Frequency Graphs: Used to estimate percentiles for individual values and vice versa.

Check Your Understanding (Knoebels Amusement Park Example)

• Initial: Mean $\mu = 1.705$ dollars, Std Dev $\sigma = 0.447$ dollars.

1. Convert to Cents (Multiply by 100):
   • Shape: Unchanged.
   • Mean: $1.705 \times 100 = 170.5$ cents.
   • Std Dev: $0.447 \times 100 = 44.7$ cents.
2. Increase by 25 Cents (Add 25):
   • Shape: Unchanged.
   • Mean: $170.5 + 25 = 195.5$ cents.
   • Std Dev: $44.7$ cents (unchanged by addition).
3. Convert to Z-scores:
   • Shape: Unchanged.
   • Mean: $0$.
   • Std Dev: $1$.
   • The new mean (195.5 cents) and SD (44.7 cents) would be used for the z-score calculation.$Z' = \frac{X' - 195.5}{44.7}$

Summary: Key Concepts and Formulas

• Transformations:
  • Add/Subtract 'a': Location measures change by 'a'. Variability (range, IQR, SD) and shape unchanged.
  • Multiply/Divide by 'b': Location measures and variability scale by 'b'. Shape unchanged.
  • Linear Transformation $Y = bX + a$:
  • Mean: $\mu_Y = b\mu_X + a$
  • Std Dev: $\sigma_Y = |b|\sigma_X$
• Z-scores: $Z = \frac{X - \mu}{\sigma}$
  • Standardized distribution has mean $0$, standard deviation $1$, and original shape.
• Percentiles: The $p$-th percentile is the value $x_p$ such that $P(X \le x_p) = \frac{p}{100}$.
  • Cumulative relative frequency graphs help visualize