Essentials of Statistics for the Behavioral Sciences - z-Scores

Chapter 5 Learning Outcomes

  • Understand z-score as location in distribution.
  • Transform X value into z-score.
  • Transform z-score into X value.
  • Describe effects of standardizing a distribution.
  • Transform scores to standardized distribution.

Concepts to Review

  • The mean (Chapter 3).
  • The standard deviation (Chapter 4).
  • Basic algebra (math review, Appendix A).

5.1 Purpose of z-Scores

  • Identifies location of every score in the distribution.
  • Standardizes an entire distribution.
  • Makes different distributions equivalent and comparable.

5.2 z-Scores and Their Locations

  • Exact location described by z-score:
  • Sign: Indicates position relative to the mean (above or below).
  • Number: Distance from the mean in standard deviation units.
Learning Check

  • Q: A z-score of z = +1.00 indicates:

  • Answer: Above the mean by a distance equal to 1 standard deviation.

  • Q: True or False:

  • Negative z-score indicates below the mean: True.

  • Scores close to the mean have z-scores close to 1.00: False (scores close to the mean have z-scores close to 0.00).

5.3 Equation for z-Scores

  • Formula: z = (X - μ) / σ
  • Numerator: Deviation score (X - μ).
  • Denominator: Standard deviation (σ).
Determining Raw Score from z-Score
  • Use the z-score formula to find the original X value based on the z-score.

5.4 Standardizing a Distribution

  • Every X value can be transformed into a z-score.
  • Characteristics of z-score transformation:
  • Same shape as the original distribution.
  • Mean of z-score distribution is always 0.
  • Standard deviation is always 1.00.
  • A z-score distribution is termed a standardized distribution.

Visualization of Transformations

  • Figures illustrate transformations of populations of scores into z-scores and re-labeling axes for comparison.
  • After transformation, the distribution's shape remains unchanged but is centered at 0 with a standard deviation of 1.

5.5 z-Scores for Comparisons

  • All z-scores are comparable.
  • Scores from different distributions can be converted to z-scores for meaningful comparisons.

5.6 Using z-Scores in Inferential Statistics

  • Z-scores are fundamental in determining if a sample is significantly different from the population based on statistical analysis.
Learning Check Example
  • Question: Given A score of X=59, μ=63, σ=8, what is the new standardized value?

5.7 Computing z-Scores for Samples

  • z-scores can be computed for samples to indicate their relative positions within that sample context:
  • Indicates how far a score is from the sample mean.
  • The sample distribution can also be transformed into z-scores.
Learning Check Example (Grades)
  • Question: Compare Andi's scores and decide for which class (Chemistry vs. Spanish) he should expect a better grade based on z-scores.

5.8 True or False Statements

  • Transforming scores into z-scores does not alter the distribution shape: True.
  • If a sample of n = 10 is transformed into z-scores, expect five positive and five negative z-scores: False (could vary based on data distribution).

Conclusion

  • Understanding z-scores and their implications in statistics is crucial for analyzing data distributions and making comparisons between different datasets.