Calculating and Interpreting Z-Scores: Transformations and the Empirical Rule

Definition and Understanding of Raw Scores

• Raw Score ($x$): A raw score is a measurement exactly in the form it was collected. It represents the original data point before any transformations (like converting to a $z$-score) have been applied.
• Nature of Raw Scores: Once a score is transformed into something else, it is no longer considered a raw score.

Problem 5: Solving for Standard Deviation ($s$)

• Given Information:
  • Mean ($\mu$ or $\bar{x}$) = $20$
  • Raw score ($x$) = $19.5$
  • $z$-score ($z$) = $-0.25$
• Interpreting the $z$-score ($z = -0.25$):
  • The Sign: The negative sign indicates the score is below the average (mean).
  • The Magnitude: The score is $0.25$ standard deviations below the mean.
  • Comparison: A $z$-score of $-3$ is very far below the mean, $-2$ is pretty far, and $-1$ is somewhat far. A $z$-score of $-0.25$ is not very far below the mean; it is relatively close to the center.
• The Mathematical Formula:
  • $z = \frac{x - \text{mean}}{s}$
• Calculation Steps:
  1. Substitute the known values: $-0.25 = \frac{19.5 - 20}{s}$
  2. Isolate the variable $s$ by multiplying both sides by $s$: $-0.25 \times s = 19.5 - 20$
  3. Simplify the numerator: $19.5 - 20 = -0.5$
  4. The equation becomes: $-0.25 \times s = -0.5$
  5. Divide by $-0.25$ to solve for $s$: $s = \frac{-0.5}{-0.25}$
  6. Final result: $s = 2$

Problem 6: Conceptualizing Deviations and Calculating $z$

• Given Information:
  • Standard deviation ($s$) = $4$
  • Akira scored $7$ points above the mean.
• Constraint: The problem does not provide the specific raw score ($x$) or the specific mean (e.g., the mean could be $100$ and the score $107$, or the mean $50$ and the score $57$). It only provides the relationship between the two.
• Defining the ‘Deviation’:
  • The numerator of the $z$-score formula ($x - \text{mean}$) is formally called the deviation.
  • The deviation tells you if a score was above or below the mean and by how many real-world units (points, dollars, inches, etc.).
  • In this problem, the deviation is given as positive $7$ (since it is 7 points above the mean).
• Calculating the $z$-score:
  • Conceptual Formula: $z = \frac{\text{deviation}}{\text{standard deviation}}$
  • Plugging in values: $z = \frac{7}{4}$
  • Final result: $z = 1.75$
• Interpretation: Akira is $1.75$ standard deviations above the mean. Because this value is close to $2$, she performed very well. While $7$ points might seem small or large depending on the specific test, a $z$-score of $1.75$ consistently indicates a high relative standing.

Interpreting $z$-scores: Magnitude and Direction

• Generic Interpretation: A $z$-score represents how many standard deviations a score is above or below the average.
• Examples:
  • $z = 1.2$: $1.2$ standard deviations above average.
  • $z = -1.2$: $1.2$ standard deviations below average.
  • $z = 0.1$: $0.1$ standard deviations above average.
  • $z = -2.3$: $2.3$ standard deviations below average.
• Who Scored Above Average? Those with positive $z$-scores ($1.2$ and $0.1$).
• Furthest Above Average: The highest positive value ($z = 1.2$).
• Furthest Below Average: The negative value with the largest magnitude ($z = -2.3$).
• Most Average: The score closest to zero ($z = 0.1$). The sign does not matter for being "average"; only the proximity to zero (absolute value) matters.
• Least Average: The score farthest from zero ($z = -2.3$). This represents the furthest distance from the mean in any direction.
• Exam Preferences:
  • Between positive $z$-scores: It is better to have a higher score (e.g., $1.2$ is better than $0.1$).
  • Between negative $z$-scores: It is better to have a smaller magnitude (e.g., $-1.2$ is better than $-2.3$ because being slightly below average is preferable to being significantly below average).

The Utility of $z$-scores

1. Relative Standing: $z$-scores specify where a score sits within its distribution. A raw score (e.g., a height of $67\text{ inches}$) does not reveal if you are tall or short relative to a group, but a $z$-score does.
2. Cross-Distribution Comparison: $z$-scores allow for the direct comparison of scores from entirely different distributions (different tests, teachers, or subjects).

Case Study: Heather vs. Jasmine

• Scenario: Two students take exams from different professors.
• Heather's Class: Mean ($\mu$) = $70$, Standard Deviation ($\sigma$) = $10$. Heather scored $95$.
  • Deviation: $95 - 70 = 25$
  • $z$-score: $z = \frac{25}{10} = 2.5$
• Jasmine's Class: Mean ($\mu$) = $80$, Standard Deviation ($\sigma$) = $5$. Jasmine scored $95$.
  • Deviation: $95 - 80 = 15$
  • $z$-score: $z = \frac{15}{5} = 3.0$
• Conclusion: Even though both earned a raw score of $95$, Jasmine performed better relative to her class ($z = 3.0$) than Heather did relative to hers ($z = 2.5$). Jasmine is three full standard deviations above her class mean, likely represent the very top of her class.

$z$-score Transformations

• Definition: Transformation is the process of converting every single raw score in a sample or population into its corresponding $z$-score.
• Three Constant Characteristics of a $z$-score Distribution:
  1. The Mean is always $0$: Since a raw score equal to the mean results in a $z$-score of $0$, the average of all transformed $z$-scores will always be $0$ ($\mu_z = 0$).
  2. The Standard Deviation is always $1$: The process of standardizing the scores scales the distribution so that the standard deviation is exactly $1$ ($\sigma_z = 1$).
  3. The Shape remains the same: Transformational scaling does not change the shape of the distribution. If the raw scores follow a normal distribution, the $z$-scores will be normal. If the raw scores are skewed, the $z$-score distribution remains skewed.
• Common Misconception: Changing scores to $z$-scores does NOT "fix" or "normalize" a distribution. It only standardizes it.

The Empirical Rule (68-95-99.7 Rule)

• Condition: This rule applies only if you are working with a Normal Distribution (bell-shaped curve).
• The Rule Breakdown:
  • 68% within 1 SD: Approximately $68\%$ of scores fall between $z = -1$ and $z = 1$.
  • 95% within 2 SD: Approximately $95\%$ of scores fall between $z = -2$ and $z = 2$.
  • 99.7% within 3 SD: Approximately $99.7\%$ of scores fall between $z = -3$ and $z = 3$.
• Area Under the Curve: Conceptually similar to calculus, we look at the area between points on a number line as a percentage of the total area.
• Outliers: Only $0.3\%$ of people fall beyond three standard deviations from the mean in either direction ($100\% - 99.7\% = 0.3\%$). This means scoring higher than a $z = 3$ or lower than a $z = -3$ is extremely rare (less than half of one percent of the population).
• Summary of Magnitude:
  • $0$: Exactly at the mean.
  • $\pm 1$: Solid middle majority ($68\% \text{ to } 70\%$
  • $\pm 2$: Pretty far out (in the outer $5\%$
  • $\pm 3$: Extreme outlier.