ch 3

Z SCORES

Learning Objectives

Computing and Interpreting Z for a Raw Score

• The mean balances the positive and negative deviation scores of a distribution.

• The standard deviation describes the variability within a set of numbers.

• Together, the mean and standard deviation provide insights into the distribution of scores; the mean indicates the center, and the standard deviation shows how much scores deviate from the center.

Example of Z Score Calculation: ACT Score

• Consider an ACT score of 25.

• Mean ACT score (µ) = 21, with a standard deviation (σ) = 4.70.

• Interpretation:

  • Your score of 25 is 4 points above the mean.

  • Since the standard deviation is 4.70, this score deviate is less than 1 standard deviation above average.

• This indicates that while the performance is above average, it is not substantially so.

Purpose of Z Scores

• The z score serves two main purposes:

  1. Locates how a score ranks in a distribution, helping determine if it is good, bad, or average.

  2. Compares scores from different distributions (e.g., ACT vs. SAT).

  • Understanding how a score like 22 on the ACT compares to a score of 1,100 on the SAT, using z scores allows for comparison despite differing scales.

Computing a Z for an Individual Score

Formula

• The z score calculation involves knowing the raw score (X), the mean (µ), and the standard deviation (σ or SD).

• The formula for a z score is:
  $z = rac{X - ext{µ}}{ ext{σ}}$

Example Calculation

• For an ACT score of 22:

  • Mean (µ) = 21,

  • Standard deviation (σ) = 4.70.

• Calculation:
  $z = rac{22 - 21}{4.70} = rac{1}{4.70} imes 1 = 0.21$

Interpreting the Z Score for a Single Score

• A positive z score (e.g., +0.21) means the score is above average, while a negative z score indicates it's below average.

• The absolute value of the z score reflects how significantly the score deviates from the mean.

  • For example, an absolute z score of 1 translates to 1 standard deviation from the mean.

Finding Raw Score Cut Lines

• To find a raw score above or below average (cut line):

  • For example, if you want a score that is 2 standard deviations above the ACT mean (µ = 21).

  • Use the z formula with z = 2:
    $X = ext{µ} + z imes ext{σ}$

  • Formula applied:
    $X = 21 + 2 imes 4.70 = 21 + 9.4 = 30.40$

Finding Probability of Z Scores Using the Standard Normal Curve

• Z scores allow both ranking and comparison among scores.

• If the raw scores are normally distributed, z scores also enable precise probability statements regarding any score.

Characteristics of a Normal Distribution

• The normal distribution is symmetric with the following properties:

  • It follows the 68-95-99 rule:

  • 68.26% of scores fall within 1 standard deviation of the mean.

  • 95.44% fall within 2 standard deviations.

  • 99.72% fall within 3 standard deviations.

Commonality of Normal Distributions

• Many populations generate a normal distribution, and due to the Central Limit Theorem, this property is prevalent in research settings.

Converting Raw Scores to Z Scores

• Converting raw scores into z scores standardizes them:

  • This transformation results in a new distribution with a mean of 0 and a standard deviation of 1.

Examples of Using Z Scores

Positive Z Score Example

• Harriet's ACT Score:

  • Score = 22, therefore:
    $z = rac{22 - 21}{4.70} = 0.21$

  • This indicates she scored somewhat above average.

  • Percentile rank can be computed using the unit normal table for z = 0.21, which suggests 58.32% of scores are less than her score.

Negative Z Score Example

• Antonio's SAT Score:

  • Score = 1,005 with µ = 1,008,  = 114.

  • Calculate z:
    $z = rac{1005 - 1008}{114} = -0.026$

  • Find the percentile rank by referencing the unit normal table and determining whether to use the tail or body column. - For the negative z score, refer only to the absolute value when looking up in the z table.

  • Result: Approximately 48.80% of students scored below his score of 1,005.

Comparing Performance Using Z Scores

• Harriet's performance in the ACT distribution is significantly better than Antonio's position in the SAT distribution, with a difference of 10 percentile points.

Proportion Between Two Z Scores Example

• To find the proportion of ACT scores between +1 and -1 standard deviations:

  • Sketch a normal curve.

  • Recognize the areas corresponding to z scores.

  • Calculate using the body and tail columns accordingly.

  • Result: Approximately 68.26% of students scored between one standard deviation above and below the mean.

Key Takeaways

• Z scores are a valuable tool for evaluating and comparing scores across different distributions.

• They provide a standardized way to interpret and communicate performance metrics, applicable in educational assessments and various fields requiring statistical analysis.