Psyc202 Week3B

Z-Scores (Standardized Scores)

  • Definition: Z-scores are used to describe the location of individual raw scores (X) on a distribution in terms of standard deviation units.
    • Necessary components to calculate a z-score include:
    • Raw Score (X)
    • Mean (μ)
    • Standard Deviation (σ)
    • The raw score (X) alone does not indicate its position on the distribution.

Understanding Z-Scores in Context of Distributions

  • Key Characteristics:
    • When all raw scores are converted to z-scores:
    • The mean of z-scores becomes 0.
    • The standard deviation of z-scores becomes 1.
    • The overall shape of the distribution remains the same, whether it is normal, skewed, etc.
    • Advantages:
    • Enables comparison of scores from different distributions (e.g., scores in PSYC101 vs. PSYC202).

Figures Illustrating Z-Scores

Figure 5-1: Two Distributions of Exam Scores

  • For both distributions:
    • Mean (µ) = 70
    • One distribution has standard deviation (σ) = 12
    • Position of raw score (X) = 76 varies between the two distributions.

Figure 5-2: Relationship Between Z-Score Values and Positions

  • Graphically represents the correlation between z-score values and positions in a population distribution:
    • Mean (μ), Standard Deviations (σ) are shown with scales of -2, -1, 0, +1, +2.

Figure 5-3: Population of Scores Transformed into Z-Scores

  • Illustrates the transformation of an entire population of scores into z-scores:
    • This transformation does not alter the shape of the population distribution.
    • The mean is changed to 0 and the standard deviation to 1.

Distribution Characteristics

Distribution for "Marks" Variable

  • Depicted through frequencies associated with marks:
    • Example shows frequency distributions for raw scores with the highest mark noted at 96.

Marks Variable Converted to a Z-Score

  • Demonstrates the transformation of mark frequencies to z-scores:
    • Z-score values range from -3.00000 to 3.00000.
    • Frequency distributions of z-scores illustrate how scores relate to their mean and standard deviation.

No Change in Distribution Shape

Figure 5-5: Shape Consistency During Transformation

  • Reiterates that transforming a distribution of raw scores into z-scores maintains the shape of the distribution:
    • Raw scores are depicted in the upper section and z-scores in the lower section of the figure.