8.2 Score Types and Statistical Foundations
Key Takeaways
- z-scores have mean 0 and SD 1; T-scores have mean 50 and SD 10; deviation IQ/standard scores have mean 100 and SD 15.
- The normal curve: ~68% of cases fall within ±1 SD, ~95% within ±2 SD, ~99.7% within ±3 SD.
- Percentile rank is the percentage of the norm group at or below a score; it is not a percent-correct score.
- Mean, median, and mode coincide in a symmetric normal distribution; skew pulls the mean toward the tail.
8.2 Score Types and Statistical Foundations
Appraisal items lean heavily on basic statistics because score interpretation is impossible without them. Master the normal distribution, central tendency, variability, and the family of standard scores. Many questions give one score and ask you to translate it into another, or to state what percentage of people fall above or below it.
The normal curve and the empirical rule
The normal (Gaussian) curve is symmetric and bell-shaped. The 68-95-99.7 rule is the backbone of most score questions:
| Range from mean | Percent of cases captured |
|---|---|
| ±1 SD | ~68% |
| ±2 SD | ~95% |
| ±3 SD | ~99.7% |
Because it is symmetric, a score exactly +1 SD above the mean sits at roughly the 84th percentile (50% below the mean plus 34% between the mean and +1 SD). A score at -1 SD sits near the 16th percentile. These two anchors solve a large share of items.
Central tendency and skew
- Mean — arithmetic average; sensitive to outliers.
- Median — middle value; resistant to extreme scores, best for skewed data.
- Mode — most frequent value.
In a perfectly normal distribution mean = median = mode. In a positively (right) skewed distribution the long tail and the mean sit to the right of the median; in a negatively (left) skewed distribution the mean is pulled left. Mnemonic: the mean chases the tail.
Measures of variability
Variability describes how spread out scores are, and the exam expects you to know the hierarchy:
- Range — highest minus lowest score; crude and outlier-sensitive.
- Variance — the average squared deviation from the mean.
- Standard deviation (SD) — the square root of the variance, expressed in the original score units; the workhorse for standard scores.
A larger SD means a flatter, wider distribution; a smaller SD means scores cluster tightly around the mean. Because all standard scores are built from the SD, misreading the SD value in a stem (for example, treating a Wechsler SD of 15 as if it were 10) is a frequent self-inflicted error.
Kurtosis
Kurtosis describes the peakedness of a distribution. A leptokurtic curve is tall and narrow with heavy tails; a platykurtic curve is flat and wide. The normal curve is mesokurtic. While less commonly tested than skew, recognizing the vocabulary prevents an easy miss.
The standard-score family
All standard scores are linear transformations of z; learn the three anchors and you can convert between any of them.
| Score type | Mean | SD | Typical use |
|---|---|---|---|
| z-score | 0 | 1 | Universal reference scale |
| T-score | 50 | 10 | MMPI-2, personality and clinical scales |
| Standard / deviation IQ | 100 | 15 | Wechsler (WAIS/WISC), Stanford-Binet |
| Stanine | 5 | ~2 | 9-point band scores, achievement testing |
| SAT-type scaled | 500 | 100 | College admissions subtests |
Worked conversion: A WAIS Full Scale IQ of 115 is exactly +1 SD (since SD = 15). That equals a z of +1.0, a T-score of 60, and roughly the 84th percentile. An IQ of 70 is -2 SD, a z of -2.0, around the 2nd percentile, and the conventional threshold for intellectual disability.
Percentiles versus percent-correct
A percentile rank is the percentage of the norm group scoring at or below a given score. The 75th percentile means the person outscored 75% of the norm group; it says nothing about how many items were answered correctly. Confusing percentile rank with percent-correct is a classic distractor.
Correlation and its squared meaning
A correlation coefficient (r) ranges from -1.0 to +1.0. The coefficient of determination (r-squared) is the proportion of variance shared. An r of .60 means .60-squared = 36% of the variance in one variable is explained by the other. Correlation never proves causation, a point exam writers test directly.
Reading correlation direction and strength
The sign tells direction: a positive r means variables move together (more study time, higher scores); a negative r means they move oppositely (more absences, lower scores). The magnitude tells strength regardless of sign, so r = -.80 is a stronger relationship than r = +.30. A negative correlation is not a weak or bad correlation. Exam stems exploit this by pairing a large negative coefficient with distractors that call it weak.
Statistical significance versus practical meaning
A correlation can be statistically significant (unlikely due to chance, often reported as p less than .05) yet practically trivial if the effect is tiny. With a large enough sample, even an r of .10 can reach significance while explaining only 1% of variance. The exam rewards candidates who separate "unlikely to be chance" from "large enough to matter," and who remember that significance never establishes causation.
Worked percentile problem
A client scores at the 16th percentile on a reading test. Because 16% of the norm group scored at or below this point, and the 16th percentile corresponds to roughly -1 SD, the counselor knows the client is about one standard deviation below the mean, not that the client got 16% of items correct. Translating a percentile into its SD location is exactly the bridge the exam wants you to build.
A client earns a Wechsler Full Scale IQ of 130. Using mean = 100 and SD = 15, this score corresponds to approximately which z-score and percentile?
A study reports a correlation of r = .50 between a counseling aptitude test and later supervisor ratings. What proportion of the variance in supervisor ratings is accounted for by the test?