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Normal Distribution and Z-Scores (Ch 6)

Oct 2, 2025

Overview

This lecture explains how to find proportions from raw scores using the standard normal distribution, z-scores, and the unit normal table. It includes step-by-step methods, practical examples, and related probability concepts, as well as applications for comparing groups and interpreting results.

Steps to Find Proportion from a Raw Score

  • Calculate the z-score:
    Use the formula z = (X - mean) / standard deviation, where X is the raw score.
  • Plot the z-score on the normal curve:
    Visualize where the z-score falls on the standard normal distribution.
  • Determine the region of interest:
    Decide if you are interested in the body (main central region), tail (extreme end), or middle (area between two values) of the distribution, based on the question.
  • Look up the z-score in the unit normal table:
    Use the appropriate column (body, tail, or middle) in the table to find the corresponding proportion.
  • Multiply by the number of cases (optional):
    If you need the actual count, multiply the proportion by the total number of cases in your population or sample.

Example Calculations

  • Example 1: Income Distribution

    • Population: mean = $50,000, SD = $20,000, N = 1,000,000
    • Find: Number of people earning more than $100,000
      • z = (100,000 - 50,000) / 20,000 = 2.5
      • Look up z = 2.5 in the tail column: proportion = 0.0062
      • Number of people: 0.0062 × 1,000,000 = 6,200

  • Example 2: Exam Scores

    • Test: mean = 75, SD = 10, N = 100
    • Find: Number of students scoring above 90
      • z = (90 - 75) / 10 = 1.5
      • Tail proportion for z = 1.5: 0.0668
      • Number of students: 0.0668 × 100 = 6.68

Finding Proportion Between Two Scores

  • Convert both scores to z-scores:

    • Use z = (X - mean) / SD for each score.

  • Find the area between the two z-scores:

    • Use the unit normal table to get the middle area for each z-score, then add them together.
    • Alternatively, subtract the tail area of the lower z-score from the body area of the higher z-score.

  • Example:

    • Mean = 75, SD = 5, scores between 65 and 83
      • z for 83: (83 - 75) / 5 = 1.6
      • z for 65: (65 - 75) / 5 = -2
      • Middle values: 0.4452 (for 1.6) + 0.4772 (for -2) = 0.9224
      • Or: body for 1.6 (0.9452) - tail for 2 (0.0228) = 0.9224

  • Another Example:

    • Mean = 58, SD = 10, scores between 55 and 65
      • z for 55: (55 - 58) / 10 = -0.3
      • z for 65: (65 - 58) / 10 = 0.7
      • Middle values: 0.1179 (for 0.3) + 0.2580 (for 0.7) = 0.3759
      • Or: body for 0.7 (0.7580) - tail for 0.3 (0.3821) = 0.3759

Finding Bounds for a Given Proportion

  • To find raw score boundaries for a central proportion (e.g., middle 90%):
    • Divide the remaining percentage equally between both tails (e.g., 5% in each tail for 90% in the middle).
    • Use the unit normal table to find the z-score corresponding to the tail proportion (e.g., z ≈ ±1.65 for 0.05).
    • Convert z-scores back to raw scores: X = mean + (z × SD).
  • Example:
    • Mean = 24.3, SD = 10, middle 90% boundaries
      • Lower bound: 24.3 + (-1.65 × 10) = 7.8
      • Upper bound: 24.3 + (1.65 × 10) = 40.8

Application and Interpretation

  • Comparing groups:
    These methods help compare a treated group to a control group to see if there is a meaningful difference.
  • Significance:
    Scores in the extreme 5% (tails) are unlikely to come from the original population, suggesting a treatment effect.
  • Interpretation:
    If a treated group’s scores fall in the extreme ends, it indicates the treatment likely had an effect.

Practice and Probability Questions

  • Mensa Qualification Example:
    • IQ test: mean = 100, SD = 15, qualifying score = 130
      • z = (130 - 100) / 15 = 2
      • Tail proportion for z = 2: 0.0228 (2.28% of the population qualifies)
  • True/False:
    • You can find the raw score (x) for a given percentile rank in a normal distribution: True
    • Z-scores give probabilities only for normal distributions, not for other shapes: True
  • Standardizing a distribution:
    Transforming raw scores to z-scores does not change the shape of the distribution.
  • Probability examples:
    • Drawing a black pen from 10 pens (5 black, 3 blue, 1 red, 1 green): 5/10 = 0.5
    • Drawing a blue or red pen: (3 blue + 1 red) / 10 = 4/10 = 0.4
    • Drawing a red pen, then a green pen (with replacement): 1/10 × 1/10 = 1/100 = 0.01
  • Assumptions of random sampling:
    • Selections are independent.
    • Probability of selection remains constant (requires replacement after each draw).

Key Terms & Definitions

  • Z-score:
    A standardized value showing how many standard deviations a raw score is from the mean.
  • Unit Normal Table:
    A chart that provides proportions under the normal curve for given z-scores.
  • Body:
    The main central region of the distribution.
  • Tail:
    The extreme ends of the distribution, beyond a certain z-score.
  • Middle:
    The area between two z-scores, representing the proportion between two raw scores.

Action Items / Next Steps

  • Practice more problems using these steps to reinforce understanding.
  • Review how to use the unit normal table for different types of questions.
  • Complete assigned homework and revisit concepts of percentiles and probability.
  • Apply these methods to real-life scenarios, such as comparing groups or interpreting test results.

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