Statistics & Educational Measurement

Nominal, Ordinal, Interval, Ratio

The NOIR scales, central tendency under skewness, the empirical rule, and standard-score transformations.

Four Core Concepts

Concept 01

The Scales of Measurement (NOIR)

Core principle/rule: Every piece of educational data belongs to one of four hierarchical scales—Nominal, Ordinal, Interval or Ratio. The scale determines which mathematical operations can legally be performed.

NominalNames or categorises data with no quantitative value or inherent order—for example, student gender, blood group and roll numbers.

OrdinalRanks data in a specific order, but distances between ranks are unknown or unequal. The gap between 1st and 2nd may be 10, while 2nd to 3rd may be 2.

IntervalOrdered data with exact, equal intervals but no true absolute zero—for example, Celsius temperature, standardised IQ scores and a mathematics score out of 100.

RatioAll interval characteristics plus a true absolute zero indicating total absence—for example, student height, weight and time taken to finish a race.

Concept 02

Measures of Central Tendency and Skewness

Core principle/rule: Central tendency describes the “centre” of a dataset. When test scores are not perfectly symmetrical, skewness pulls the mean away from the centre.

Key facts

Mean: Arithmetic average; highly sensitive and easily pulled away by extreme outlier scores.
Median: Exact middle value when scores are ordered; completely unaffected by extreme outliers.

Normal DistributionPerfectly symmetrical:
Mean = Median = Mode

Positive / Right SkewMost scores very low—a very hard test; tail stretches right to a few high outliers.
Mean > Median > Mode

Negative / Left SkewMost scores very high—a very easy test; tail stretches left to a few low outliers.
Mean < Median < Mode

Concept 03

Standard Deviation and the Empirical Rule

Core principle/rule: Standard Deviation measures dispersion or spread of scores around the mean. The Empirical Rule states exactly how data fall within a Normal Distribution Curve.

Key facts

Small SD: Scores cluster tightly around the mean; class performance is similar.
Large SD: Scores are widely spread; student performance shows high disparity.
68–95–99.7 Rule: In a perfect normal curve, 68% of scores lie within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD of the mean.

Concept 04

Standard Scores — Z, T and Stanine

Core principle/rule: Raw scores—for example, 45/100 in Mathematics and 30/50 in English—cannot be directly compared across tests with different difficulty and variance. Standard scores convert them to universal scales.

Z-Score

Z = (X − μ) / σ

Mean = 0; SD = 1. Z-scores can be negative.

T-Score

T = 10Z + 50

Mean = 50; SD = 10. Removes negative values and decimals.

Stanine · Standard Nine

1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9

Mean = 5; SD ≈ 2.

Quick Reference

Table A · NOIR Measurement Scales

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Scale	Core Characteristic	Permissible Mathematical Operations	Educational Example
Nominal	Categorisation only.	Counting (frequencies), mode.	Student ID numbers, gender.
Ordinal	Ranks, but unequal gaps.	Median, percentiles.	Class ranks—1st, 2nd, 3rd.
Interval	Equal gaps; no true zero.	Addition, subtraction, mean, SD.	Standardised test scores, IQ.
Ratio	Equal gaps; true absolute zero.	Multiplication, division, all operations.	Time taken to complete a test.

Table B · Standard Score Profiles

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Score Type	Universal Mean	Universal SD	Primary Administrative Benefit
Z-Score	0	1	The foundational statistical standard.
T-Score	50	10	Eliminates all negative scores for clean reporting.
Stanine	5	≈ 2	Groups scores into nine broad, easily digestible bands.