Chapter 5: Z-Scores and Standardized Distributions

LEARNING OUTCOMES

1. Understand z-score as location in distribution.

2. Transform X value into z-score.

3. Transform z-score into X value.

4. Describe effects of standardizing a distribution.

5. Transform scores to standardized distribution.

THE PURPOSE OF Z-SCORES

• Definition and Importance of Z-Scores:

  • Z-scores identify and describe the location of every score in a distribution.

  • The sign of the z-score indicates whether the score is located above or below the mean.

  • The absolute value indicates the distance between the score and the mean in standard deviation units.

  • Z-scores facilitate the standardization of an entire distribution which allows for equating and comparing different distributions.

  • Standardization is crucial because it offers a method to make different distributions comparable.

• Normal Distribution and Z-Scores:

  • In a normal distribution, approximately 99.7% of the data will lie within ±3σ (three standard deviations) of the mean.

INTERPRETATION OF Z-SCORES

• Z-Score Examples:

  • A z-score of z = +1.00 indicates a position in a distribution:

  • Correct Explanation:

    • Above the mean by 1 point.

    • Above the mean by a distance equal to 1 standard deviation.

  • Additional options not correct:

  • Below the mean by 1 point.

  • Below the mean by a distance equal to 1 standard deviation.

• **True/False Statements to Consider: **

  1. A negative z-score always indicates a location below the mean (True/False).

  2. A score close to the mean has a z-score close to 1.00 (True/False).

EQUATION FOR Z-SCORE

• Standardized Score Formula:

  • The formula for z-scores is given as:
    $z = \frac{(X - \mu)}{\sigma}$

  • Where:

    • X = raw score

    • $\mu$= mean of the population (deviation score)

    • $\sigma$ = standard deviation of the population

    • z = Standardized score

DETERMINING A RAW SCORE FROM A Z-SCORE

• Manipulation of Z-Score Equation:

  • To find a raw score ($X$) from a z-score, the equation can be rearranged:
    $X = z \cdot \sigma + \mu$

  • Example:

  • Given ${\mu}=40$ and $\sigma = 2$, find $X$ for:

    • z = +1.50:
      $X = 1.50 \cdot 2 + 40 = 43$

    • z = -2.00:
      $X = (-2.00) \cdot 2 + 40 = 36$

STANDARDIZING A DISTRIBUTION

• Characteristics of Z-Score Transformation:

  • Every X value can be converted to z-scores.

  • The z-score distribution retains the same shape as the original distribution.

  • The mean of the z-score distribution will always be 0.

  • The standard deviation of the z-score distribution will always be 1.

  • The new z-score distribution is termed a standardized distribution.

• Example with IQ Scores:

  • Raw scores have $\mu = 50$ and $\sigma = 10$. The characteristics of the standardized distribution follow the aforementioned principles.

COMPARISON OF Z-SCORES

• Comparability of Z-Scores:

  • All z-scores from different distributions can be directly compared to each other when standardized.

  • This comparability arises because the standardization process aligns diverse scores onto the same scale.

CREATING A STANDARDIZED DISTRIBUTION

• Standardization Process:

  • The standardization involves two steps.

  • Example Scenario:

  • Joe's original test score: 43.

  • New standardized distribution desired to have $\mu = 50$ and $\sigma = 10$.

• Goal:

  • To determine how Joe’s original score fits within the new distribution.

MEASURES OF VARIABILITY: VARIANCE

• Raw Score Calculation:

  • For a score of $X=59$ from a distribution with $\mu=63$ and $\sigma=8$, standardize it:

  • New distribution with $\mu=50$ and $\sigma=10$ requires determination of the new value.

  • Answer options:

  • A. 59

  • B. 45

  • C. 46

  • D. 55

COMPUTING Z-SCORES FROM A SAMPLE

• Context of Z-Scores in Samples:

  • While the calculation of z-scores is predominantly done in populations, it can also apply to samples.

  • They indicate:

  • The relative position of a score within the sample.

  • The distance of the score from the sample mean.

  • Sample distributions can be transformed into z-scores:

  • They keep the same shape as the original distribution.

  • Retain an equivalent mean ($M$) and standard deviation ($s$).

LOOKING AHEAD TO INFERENTIAL STATISTICS

• Research Interpretation:

  • The interpretation of research results hinges on identifying whether a “treated” sample is “noticeably different” from the population using z-scores.

  • Questions to explore:

  • Are the sample statistics equal to the population parameters?

STANDARD DEVIATION AND VARIANCE FOR A SAMPLE

• Example Scenarios:

  • Analyzing performance in two subjects:

  • Chemistry: ${}\mu=30$ , ${}\sigma=5$ , Joe's score: $X = 45$.

  • Spanish: ${}\mu=60$ , ${}\sigma=6$ , Joe's score: $X = 65$.

  • Decision Point: In which class should Joe expect the better grade?

  • A. Chemistry

  • B. Spanish

  • C. Insufficient information

TRUE OR FALSE EXERCISES

• Z-Score Distribution Statements:

  • Transforming a distribution into z-scores does not change its shape (True/False).

  • For a sample with $n = 10$, when transformed into z-scores, expect to find five positive z-scores and five negative z-scores (True/False).