Overview
This lecture covers measures of relative position, focusing on z-scores (standard scores) and how they allow comparisons between different data sets or scales.
Measures of Relative Position
- Measures of relative position assess where a value stands compared to others within a data set.
- Z-scores (standard scores) are commonly used to compare values from different distributions or scales.
- A z-score tells how many standard deviations a value is from the mean.
- Z-scores enable fair comparisons, such as between ACT and SAT scores.
Calculating Z-Scores
- Population z-score formula: ( z = \frac{X - \mu}{\sigma} ), where ( X ) is the data value, ( \mu ) the mean, and ( \sigma ) the standard deviation.
- Sample z-score formula: ( z = \frac{X - \bar{x}}{s} ), where ( \bar{x} ) is the sample mean and ( s ) the sample standard deviation.
- Always round z-scores to two decimal places.
Interpreting Z-Scores
- A positive z-score means the value is above the mean; negative means below the mean.
- For test scores, higher (more positive) z-scores usually indicate better performance.
- When comparing two positive z-scores, the higher one is better (farther above the mean).
- When comparing two negative z-scores, the one closer to zero (less negative) is better (closer to the mean).
- Context matters: for some measures (e.g., cholesterol), a lower z-score might be preferable.
Worked Examples
- Example: SAT score of 630, mean 500, standard deviation 150 → z-score = 0.87 (above mean).
- Comparing two calculus test scores, convert each to z-scores to determine who performed better relative to their class.
- When comparing scores from different tests with different scales, use z-scores for valid comparisons.
Key Terms & Definitions
- Relative Position — How a value compares to others in the data set.
- Z-score (Standard Score) — Number of standard deviations a value is from the mean.
- Mean (( \mu ) for population, ( \bar{x} ) for sample) — The average value in a data set.
- Standard Deviation (( \sigma ) for population, ( s ) for sample) — Measure of spread or variability in a data set.
Action Items / Next Steps
- Download and review guided notes for section 3.3 from Canvas.
- Practice calculating and interpreting z-scores with provided examples.
- Prepare to study percentiles, the next measure of relative position.