Standardised Scores & Z-Score Applications

The Normal (Gaussian) Distribution

• Mathematically defined, symmetrical, bell-shaped curve.
• Follows the 68 – 95 – 99.7 rule:
  • $68\%$ of data lie within $\pm1\sigma$ of the mean.
  • $95\%$ within $\pm2\sigma$.
  • $99.7\%$ within $\pm3\sigma$.
• Consequences of differing spreads:
  • Two distributions can share the same mean but have different $\sigma$; an identical raw distance from the mean is a smaller relative deviation in the wider curve.
  • Outliers are defined relative to their own distribution’s $\sigma$, not absolute raw units.

Why We Standardise

• Raw scores from different tests (heights, weights, IQ, exam marks, etc.) are on incompatible scales.
• Standardisation rescales any normal distribution to a common frame with:
  • Mean $\mu =0$.
  • Standard deviation $\sigma =1$.
• Gives a direct way to:
  • Express distance from a mean in standard deviation units.
  • Compare ostensibly different measurements (e.g.  cm vs. 6 kg).
  • Look up any fractional $\sigma$ in a table.

Families of Standard Scores

• Standard scores provide a unified way to interpret individual performance across different tests or contexts.
• Multiple conventions exist (different educational & clinical contexts):
  • Z-scores (mean $0$, $\sigma=1$): The foundational standard score, direct comparison to the standard normal distribution.
  • T-scores (mean $50$, $\sigma=10$): Commonly used in psychological testing, designed to avoid negative scores and decimals found in Z-scores.
  • IQ (mean $100$, $\sigma=15$ or $16$ depending on test): Specific to intelligence measures, scaled for broader interpretability.
  • Stanines (standard nines), percentiles, C-scores, etc.: Other scales used for specific assessment or reporting purposes.
• Clinicians must recognise and translate among scales when interpreting cognitive-ability tests.

Z-Scores: The Core Transformation

• Formula: $Z = \frac{x-\mu}{\sigma}$, where:
  • $x$ = observed/raw score.
  • $\mu$ = population (or sample) mean.
  • $\sigma$ = population (or sample) standard deviation.
• Properties:
  • $\mu_Z = 0$.
  • $\sigma_Z = 1$.
  • Shape is unchanged (normal stays normal; skew stays skew).
• Interpretative cues:
  • Sign indicates direction (positive = above mean, negative = below).
  • Magnitude indicates extremity (|Z|>2 roughly top/bottom 2.5 %).

Example 1: Single Test Score

• Raw distribution: $\mu=34$, $\sigma=8.5$.
• Frank: $x=45$.
  • $Z = \frac{45-34}{8.5}=1.29$ (1.29 sd above mean).
• Paul: $x=18$.
  • $Z = \frac{18-34}{8.5}=-1.88$ (1.88 sd below).

Example 2: Comparing Across Tests

• Stephanie:
  • Statistics: $x=76,\;\mu=70,\;\sigma=4$ $\Rightarrow Z=1.50$.
  • Philosophy: $x=82,\;\mu=75,\;\sigma=8$ $\Rightarrow Z=0.875$.
• Even though 82>76, her relative performance is stronger in Statistics.

Example 3: Sporting Legends

• Calculate Z of performance metric within each sport:
  • Don Bradman $Z\approx4.4$ (cricket batting average).
  • Pelé $Z\approx3.7$, etc.
• Allows “greatest ever” debate on a common yardstick.

Using Z-Tables to Find Proportions

• A Z-table provides P(Z<z) (area left of z).
• Key features:
  • P(Z<0)=0.5 (curve is symmetric).
  • Right-tail proportion =1-P(Z<z).
  • Table usually ranges roughly $-3.49\le z\le3.49$ (tails never truly reach 0 because the curve extends to $\pm\infty$).

Three Typical Tasks

1. Proportion below a value.
2. Proportion above a value.
3. Proportion between two values.

Worked Problem A (Below a Cut-off)

• Cut-off: $x=16$, $\mu=34$, $\sigma=8.5$.
• $Z=\frac{16-34}{8.5}=-2.12$.
• P(Z<-2.12)=0.017\Rightarrow1.7\% need remedial help.

Worked Problem B (Above a Cut-off)

• Exceptional: $x\ge54$.
• $Z=\frac{54-34}{8.5}=2.35$.
• $P(Z<2.35)=0.9906\Rightarrow P(Z>2.35)=0.0094$.