Chapter 5: z-Scores: Location of Scores and Standardized Distributions.

Chapter 5: z-Scores: Location of Scores and Standardized Distributions

Learning Outcomes

  • Understand z-score as location in distribution
  • Transform X value into z-score
  • Transform z-score into X value
  • Describe effects of standardizing a distribution
  • Transform scores to standardized distribution

Tools You Will Need

  • The mean (discussed in Chapter 3)
  • The standard deviation (discussed in Chapter 4)
  • Basic algebra (Math Review, Appendix A)

5-1 Introduction

  • z-scores: Also referred to as standard scores, they identify and describe the location of every score in the distribution.
  • Standardization: Involves standardizing an entire distribution such that different distributions become equivalent and comparable.

5-2 z-Scores and Locations in a Distribution

  • Definition of z-Score: The exact location of a score within a distribution described as a z-score.
    • The sign (+ or -) indicates if the score is above (positive) or below (negative) the mean.
    • The numerical value indicates the distance between the score and the mean in terms of standard deviations.
z-Score Formula for a Population
  • The formula can be expressed as: z=(X−μ)σz = \frac{(X - \mu)}{\sigma}
    • Numerator: Deviation score (difference between the individual score and the mean).
    • Denominator: Expresses deviation in standard deviation units, where ( \mu ) is the population mean and ( \sigma ) is the standard deviation.

5-3 Other Relationships between z, X, the Mean, and the Standard Deviation

  • Establishing Relationships: A z-score establishes a relationship among the raw score, the mean, and the standard deviation.
    • Given: mean (( \mu )), raw score (X), and z-score, one can compute the standard deviation (( \sigma )) using:
    • σ=(X−μ)z\sigma = \frac{(X - \mu)}{z}
    • Conversely, given standard deviation, raw score, and z-score, one may compute the mean:
    • μ=X−(z×σ)\mu = X - (z \times \sigma)

5-4 Using z-Scores to Standardize a Distribution

  • Characteristics of z-score transformation:
    • The shape of the distribution remains unchanged after transformation.
    • The mean of the z-score distribution is always 0.
    • The standard deviation is always 1.00.
  • The resulting distribution after this transformation is termed a standardized distribution.

5-5 Other Standardized Distributions Based on z-Scores

  • The process of standardization is widely employed in assessments. For example:
    • SAT scores are distributed with a mean (( \mu )) of 500 and a standard deviation (( \sigma )) of 100.
    • IQ scores have a mean (( \mu )) of 100 and a standard deviation (( \sigma )) of 15.
  • Standardization Steps:
    1. Original scores transformed into z-scores.
    2. z-scores transformed into new X values to obtain specific mean (( \mu )) and standard deviation (( \sigma )).

5-6 Looking Ahead to Inferential Statistics

  • Inferential statistics are methodologies employed to utilize sample information to draw conclusions about populations.
  • Interpretation of research results is predicated on determining whether the sample is "noticeably different" from the population.
  • One approach to assess this is via z-scores.

Learning Checks and Answers

Learning Check 1
  • A z-score of z = +1.00 indicates a position in a distribution:
    • Correct Answer: Above the mean by a distance equal to 1 standard deviation.
  • True/False Statements:
    • A negative z-score always indicates a location below the mean: True.
    • A score close to the mean has a z-score close to 1.00: False.
Learning Check 2
  • For a population with ( \mu = 50 ) and ( \sigma = 10 ), what is the X corresponding to z = 0.4?
    • Correct Answer: 54 (derived using the z-score formula)
  • True/False Statements:
    • If ( \mu = 40 ) and 50 corresponds to z = +2.00, then ( \sigma = 10 ) points: False (( \sigma = 5 )).
    • If ( \sigma = 20 ), a score above the mean by 10 points will have z = 1.00: False (z = 0.5).
Learning Check 3
  • A score of X = 59 from a distribution with ( \mu = 63 ) and ( \sigma = 8 ) is standardized to a new distribution with ( \mu = 50 ) and ( \sigma = 10
    • Correct Answer: 55 (using transformation calculations).
Learning Check 4
  • Andy’s performance comparison in chemistry and Spanish based on their respective means and standard deviations indicates:
    • Correct Answer: Need for calculation of z-scores to determine better performance.
  • True/False Statements:
    • Transforming an entire distribution of scores into z-scores will not change the shape of the distribution: True.
    • If a sample of n = 10 scores is transformed into z-scores, it will have five positive z-scores and five negative z-scores: False (number may vary based on actual scores).