Norm scores: z, T, IQ, stanine, percentile rank

In brief

Norm scores convert a person's raw score (for example points in a reading test) to a scale on which the person's position can be read off: z score, T score, IQ scale, stanine or percentile rank. One new column is created for each chosen type. The reference is your own sample (also per group) or a norm whose mean and standard deviation (SD) are entered from the test manual. Norm scores from your own sample do not replace test norms.

Where

On the "Columns" page under "Norm scores as a new column"; the numeric column is chosen there (group and ID columns excluded). The preview shows three people; "Undo" takes the new columns back.

The scales

• z score: z = (x − M) / SD, mean 0, SD 1. The basis of the other scores.
• T score: 50 + 10 · z (mean 50, SD 10). 40 to 60 is within one SD of the mean.
• IQ scale: 100 + 15 · z (mean 100, SD 15). 85 to 115 is within one SD.
• Stanine: nine levels with mean 5 and SD 2; 4 to 6 is the average range.
• Percentile rank (PR): share of comparison persons who score lower; ties count half. PR 84 means about 84% of the comparison group are below.

Reference

• Own data: mean, SD and rank order come from the column itself. Optionally separate by a group column (for example grade); all scores then apply within each group.
• Enter norm values: the mean and SD of the norm (matching age or grade) are copied from the test manual. The percentile rank follows from the normal distribution and assumes an approximately normal norm.

Interpretation

• Scores from your own sample describe only the position within this group: a T score of 60 in a high-achieving class means something different than in a norm sample. For diagnostics use the norm table of the test.
• z, T and IQ scores are linear transformations and keep the shape of the distribution; for skewed distributions the percentile ranks are more informative.
• Missing values, "no answer", missing group values and groups with only one person or without spread stay blank.

The columns are called z_column, T_column, IQ_column, Stanine_column and PR_column (with a number appended on a name clash).

For specialists

• Mean and SD (n − 1) as in R scale() and DESCRIPTIVES /SAVE.
• Empirical percentile rank: PR = 100 · (R − 0.5) / n with R the mean rank. SPSS RANK /PERCENT computes R / n · 100 and is 50 / n percentage points higher.
• Stanines follow from the empirical percentile rank (level shares 4, 7, 12, 17, 20, 17, 12, 7, 4%).
• With an entered norm: PR = 100 · Φ(z); stanine = 2z + 5, rounded and limited to 1 to 9.