Nominal and Ordinal Scales in Statistics

Scales of Measurement

• Nominal scales: simplest form of measurement, involve classification or categorization based on one or more distinguishing characteristics, mutually exclusive and exhaustive categories.
• Ordinal scales: permit classification and rank ordering on some characteristic, data is ordinal in nature, no absolute zero point.
• Interval scales: contain equal intervals between numbers, each unit on the scale is exactly equal to any other unit, no absolute zero point.

Descriptive Statistics

• Variance: equal to the arithmetic mean of the squares of the differences between the scores in a distribution and their mean.
• Formula for variance (s2): deviation scores.
• Standard deviation: square root of the variance.
• Skewness: nature and extent to which symmetry is absent in a distribution.
  • Positive skew: relatively few scores fall at the high end of the distribution, may indicate the test was too difficult.
  • Negative skew: relatively few scores fall at the low end of the distribution, may indicate the test was too easy.
• Kurtosis: steepness of a distribution in its center, platykurtic (relatively flat), leptokurtic (relatively peaked), or mesokurtic.

The Normal Curve

• Development of the concept of a normal curve began in the 18th century with Abraham Demurer and the Marquis de Laplace.
• Karl Friedrich Gauss made substantial contributions, referred to as the “Laplace-Gaussian curve”.
• Karl Pearson coined the term “normal curve”, used to refer to behavior that is usual, average, normal, standard, expected, or typical.

Norms

• Norms: test performance data of a particular group of test takers, designed for use as a reference when evaluating or interpreting individual test scores.
• Normative sample: group of people whose performance on a particular test is analyzed for reference.
• Norming: process of deriving norms, can be modified to describe a particular type of norm derivation, e.g. race norming.
• User norms or program norms: descriptive statistics based on a group of test takers in a given period of time, rather than norms obtained by formal sampling methods.

Sampling to Develop Norms

• Standardization or test standardization: process of administering a test to a representative sample of test takers to establish norms.
• Population: complete universe or set of individuals with at least one common, observable characteristic.
• Sampling: process of selecting a portion of the universe deemed to be representative of the whole population.
  • Stratified sampling: process of selecting a sample to prevent sampling bias.
  • Purposive sample: sample selected because it is believed to be representative of the population.
  • Incidental sample or convenience sample: sample that is convenient or available for use, generalization of findings must be made with caution.

Types of Norms

• Percentiles: dividing a distribution of scores into 100 equal parts.
  • Xth percentile is equal to the score at or below which x% of scores fall.

Validity

• Criterion-related validity: judgment of how adequately a test score can be used to infer an individual’s most probable standing on some measure of interest.
• Concurrent validity: degree to which a test score is related to some criterion measure obtained at the same time.
• Predictive validity: degree to which a test score predicts some criterion measure.
• Criterion: standard against which a test or a test score is evaluated.
  • Characteristics of a criterion:
    • Relevance: pertinent or applicable to the matter at hand.
    • Validity: evidence exists that the criterion is valid for the purpose for which it is being used.
    • Lack of contamination: criterion measure is not based on predictor measures.