Norms and Standardized Scores

PSYC 520

Norm and Criterion Referencing

  • Norm referencing
    • How well one has done relative to the norm group
    • E.g., Weschler tests, GRE
  • Criterion referencing
    • How well one has done in a more absolute sense
    • E.g., qualifying examinations for doctors, airline pilots, etc

Norms

  • Norm group: a reference group providing a context for interpreting scores

  • Norms: scores from the norm group

Standard 5.8 (p. 104)

Norms, if used, should refer to clearly described populations. These populations should include individuals or groups with whom test users will ordinarily wish to compare their own examinees.

Standard 5.9 (p. 104)

Reports of norming studies should include precise specification of the population that was sampled, sampling procedures and participation rates, any weighting of the sample, the dates of testing, and descriptive statistics. Technical documentation should indicate the precision of the norms themselves.

The “Three R’s”

  • Recency
  • Representativeness
  • Relevance

Norm-Referenced Scores

  • Percentile Ranks

  • Standardized scores

    • \(z\)-scores
    • \(T\)-scores

  • Normalized scores

    • Normalized \(z\)-scores
    • Stanines
    • Normal curve equivalents

Percentile Ranks

f cf % c% PR
25 3 3 5.769231 5.769231 2.884615
26 8 11 15.384615 21.153846 13.461538
27 7 18 13.461538 34.615385 27.884615
28 5 23 9.615385 44.230769 39.423077
29 5 28 9.615385 53.846154 49.038461
30 7 35 13.461538 67.307692 60.576923
31 3 38 5.769231 73.076923 70.192308
32 2 40 3.846154 76.923077 75.000000
33 2 42 3.846154 80.769231 78.846154
34 6 48 11.538462 92.307692 86.538461
35 2 50 3.846154 96.153846 94.230769
36 2 52 3.846154 100.000000 98.076923

Notes on Percentile Ranks (PRs)

\[ \begin{aligned} \mathit{PR} & = \frac{\mathit{cf} - 0.5f}{n} \times 100 \\ & = ((c\%) - 0.5 (\%)) \times 100 \end{aligned} \]

  • Percentile rank is not intervally scaled
    • Averaging is not meaningful
  • Percentile ranks vs. percentile points
    • E.g., PR31 = 70; P70 = 31

Standardized Scores

  • \(z\) scores
    • \(z = \frac{X - \bar X}{\mathit{SD}_X}\)
    • Mean = 0, SD = 1
  • \(T\) scores
    • \(T = z \times SD_\text{new} + M_\text{new}\)
    • Typically, mean = 50, SD = 10

Normalized Scores

Should be used very cautiously, as it may lead to non-meaningful results

Stanines

  • Convert raw scores to nine possible values, each corresponding to a section of a normal curve

Normal curve equivalents

  • Convert normalized \(z\)-scores to \(T\)-scores with mean = 50 and SD = 21.06

Developmental-Level Scores

  • Grade-equivalent (GE) scores
  • Age-equivalent (AE) scores

Obtained by using common/anchor items in adjacent grade/age levels

E.g., one can interpolate that a Grade 3 student has GE = 4.7.

Criterion-Referenced Testing

Standard 5.5 (p. 103)

When raw scores or scale scores are designed for criterion-referenced interpretation, including the classification of examinees into separate categories, the rationale for recommended score interpretations should be explained clearly.

  • E.g., Psychopathology, skills

Standard 5.21 (p. 107)

When proposed score interpretations involve one or more cut scores, the rationale and procedures used for establishing cut scores should be documented clearly.

Standard 5.23 (p. 108)

When feasible and appropriate, cut scores defining categories with distinct substantive interpretations should be informed by sound empirical data concerning the relation of test performance to the relevant criteria.