Standard Scores, Z-Scores, and Effect Size in Context

Standard Scores

  • Definition of Standard Scores: A standard score is a way to assess an individual's score relative to a broader population, providing context to raw scores.

  • Contextual Example:

    • Questionnaire measures "grumpiness" with scores ranging from 0-50.
    • Sample Size: 1,000,000 people surveyed.
    • Mean Grumpiness Score: 17 out of 50.
    • Standard Deviation: 5.

  • Grumpiness Score Interpretation:

    • Classmate scores 35 out of 50 (70% grumpy).
    • Raw score alone is uninformative; context needed.
    • If 20% of respondents have a score of 35, that score is above average and in the top 20%.

  • Standardization:

    • Helps compare scores against the population rather than using raw percentages.
    • Problem: Limited extreme scores can affect percentile calculation.

Z-Scores

  • Definition: A z-score indicates how many standard deviations away from the mean a score lies.

  • Z-Score Calculation:
    zi=X−XˉSz_i = \frac{X - \bar{X}}{S}
    Where:

    • $X$: raw score
    • $\bar{X}$: mean
    • $S$: standard deviation

  • Example Calculation:

    • For grumpiness:
      z=35−175=3.6z = \frac{35 - 17}{5} = 3.6
    • Meaning: Classmate is 3.6 standard deviations above the mean.
    • Significance: Corresponds to the highest 1% of grumpiness scores.

  • Cross-Variable Comparison:

    • Z-scores allow comparison between different questionnaires.
    • Example with Extraversion:
    • Mean = 13, Standard Deviation = 4, Score = 2
    • Calculate z-score:
      z=2−134=−2.75z = \frac{2 - 13}{4} = -2.75
    • Result: Classmate is 2.75 standard deviations below average in extraversion; indicates extreme scores in personality assessment.

Effect Size (Cohen's d)

  • Importance: Used to quantify the strength of a phenomenon (e.g., treatment effects).

  • Study Example:

    • Medication Group: Mean score = 6
    • Placebo Group: Mean score = 7
    • Standard Deviation of cravings = 2.

  • Effect Size Formula:
    d=(mean<em>1−mean</em>2)stddevd = \frac{(mean<em>1 - mean</em>2)}{std dev}

  • Example Calculation:

    • d=(6−7)2=−0.5d = \frac{(6 - 7)}{2} = -0.5
    • Interpretation: Difference of 0.5 standard deviations in craving scores between groups.

  • Cohen's d Interpretation:

    • Rough guidelines:
    • About 0.2 = small effect
    • About 0.5 = moderate effect
    • About 0.8 = large effect

  • Practical Significance:

    • Small effects can be important depending on context.
    • Cohen's d allows understanding how significant differences are in standardized terms.

  • Considerations:

    • Clarification on which standard deviation to use: pooled, control, or treatment group?
    • Tools (like jamovi) can calculate effect size conveniently for analyses.