Research Methods and Statistics: z Scores, Probability, and Normal Distribution

z-Scores
  • Definition:
    • z-scores are also called standardized scores.
    • They specify the precise location of any individual score X within a normal distribution.
  • Purpose:
    1. Make Individual Raw Scores Meaningful:
    • Example: A score of X = 78 on a quiz is meaningful relative to the distribution.
    • If the mean (M) is 70 and standard deviation (SD) is known, it helps in evaluating the score.
    • If SD = 4, a score of X = 78 indicates higher performance compared to SD = 12, where the same score may not be as impressive.
    1. Standardize Entire Distributions:
    • Different IQ tests can be standardized to ensure μ = 100 and σ = 15, allowing comparisons across varied tests.
Calculating a z-Score
  • Formula:
    z=X−μσz = \frac{X - \mu}{\sigma}
  • Components of z-Scores:
    1. Sign:
    • Positive (+) if above the mean, negative (-) if below the mean.
    1. Magnitude:
    • Indicates the distance in standard deviations from the mean.
  • Examples:
    • If a score is below the mean by ¾ standard deviation, then z = -0.75.
    • If above the mean by 1 standard deviation, then z = +1.00.
Standardizing Distributions
  • Standardization Process:
    • Convert each score into a z-score, maintaining the original distribution's characteristics (e.g., skew, symmetry).
    • The mean of the standardized distribution is μ = 0 and standard deviation is σ = 1.
  • Purpose of Standardization:
    • Facilitates direct comparisons across different distributions.
      • Example: Check which is better:
      • Xtest1 = 56, μ1 = 48, σ1 = 4
      • Xtest2 = 60, μ2 = 50, σ2 = 10.
Transforming z-Scores
  • You can also create a new distribution from original scores.
  • Transformation:
    • Using z=X−μσz = \frac{X - \mu}{\sigma} to convert scores, then use
      X=z⋅σ+μX = z \cdot \sigma + \mu with new values (e.g., μ = 100, σ = 15).
Probability Basics
  • Definition:
    • Probability of an event indicates the likelihood of that event occurring versus all possible events.
  • Importance:
    • Understanding probability is essential for defining relationships between samples and populations.
  • Formula for Probability:
    p(A)=number of outcomes classified as Atotal number of possible outcomesp(A) = \frac{\text{number of outcomes classified as A}}{\text{total number of possible outcomes}}
  • Examples:
    • For M&Ms in a jar:
      • If there are 84 M&Ms and 23 are blue,
      • Prob of blue: p(blue)=2384≈0.2738 or 27.38%p(blue) = \frac{23}{84}\approx 0.2738 \text{ or } 27.38\%.
Probability Distribution Examples
  • Cumulative Probability Examples:
    • If p(X > 4) where 7 of 20 scores are above 4:
      • p(X > 4) = \frac{7}{20} = 0.35 \text{ or } 35\% .
    • For p(X ≤ 2) from frequency distribution, with 4 of 20:
      • p(X≤2)=420=0.20 or 20%p(X \leq 2) = \frac{4}{20} = 0.20 \text{ or } 20\%.
Normal Distribution Properties
  • Normal Distribution Characteristics:
    • Symmetrical distribution with specific area percentages related to standard deviations from the mean.
      • Proportions for standard deviations:
        • 34.13% (1 SD from the mean),
        • 13.59% (2 SDs),
        • 2.28% (3 SDs).
  • Probability Calculation:
    • Standardizing Xi values into z-scores makes them easier to analyze with the unit normal table.
Applied Examples of Probability
  • Example 1: IQ less than X = 85 from distribution μ = 100, σ = 15:
    • To find p(X < 85) .
  • Example 2: IQ between 115 and 130:
    • To find p(115 < X < 130) .
  • Example 3: Minimum score for top 5% in SAT (μ = 500, σ = 100):
  • Use z-scores to determine corresponding raw scores and probabilities.