Numbers and Measurement

• Measurement: the process of assigning numerals or numbers to observations

• We enhance our ability to measure attributes by defining the basic size of the unit of measurement for a test -- the use of real numbers

• Real numbers are also continuous because they represent any quantity along a number line

Properties of Measurement in Relation to Numbers

• The score from a psychological test is based on the properties of numbers and how we treat 0

• The level of measurement of a particular score can be determined by the presence or absence of 4 properties

Nominal

• number of classes (classification), mode

• Qualitative: sex or hair color, distinguishing labels or categories

Ordinal

• median, percentiles, order statistics

• Quantitative: class rank, hardness of minerals, order finish in a competitive running race

Interval

• equality of intervals of scores along the score continuum

• Quantitative: temperature (Celsius), standardized test scores

Ratio

• equality of ratio

• Quantitative: temperature (Kelvin)

Scaling

• the mathematical techniques used for determining what numbers should be used to represent different amounts of a property or attribute being measured

Statistical Foundations for Psychometrics

Descriptive Statistical Techniques

• used to organize and describe data

• order and group scores into distributions and describe the scores

• calculate a single number that summarizes a set of scores

• represent scores graphically

• applied to samples and populations

Inferential Statistical Techniques

• used to help make educated guesses about populations based on the samples

Variables, Frequency Distributions, and Scores

• measurements acquired on a variable/s are part of the data collection process -- refers to a property whereby members of a group differ from one another (i.e., measurements change from one person to another)

Constant

• refers to property whereby members of a group do not differ from one another (e.g., all persons in a study or taking an examination are female; thus, biological sex is constant)

• variables can be qualitative or quantitative; quantitative variables can be discrete or continuous

Discrete

• values can only be whole numbers

Continuous

• can assume any value

Frequency Distribution

• a tabulation of the number of occurrences of each score value

Shape, Central Tendency, and Variability of Score Distributions

• The shape of a distribution is defined as symmetric, positively skewed, and negatively skewed; can also be unimodal or bimodal

Central Tendency

• the score value at the center (position at the center of the x-axis) that marks the center of the distribution of scores -- mean, median, and mode

• Since a measure of central tendency is a single number, it is a concise way to provide an initial view of the set of scores.

• Measures of central tendency can quickly and easily be compared.

• Many inferential statistical techniques use a measure of central tendency to test hypotheses of various types.

• Measures of variability for a set of scores or measurements provide a value of how spread or dispersed the individual scores are in a distribution -- variance and standard deviation

Variance

• the degree to which measurements or scores differ from the mean of the population or sample

Standard Deviation

• the square root of the variance

Percentiles

• used to provide an index of the relative standing for a person with a particular score relative to the other scores (persons) in a distribution

Z-Score

• A way to rescale or standardize the value of 0 so that it means the same thing in every distribution. We can then directly compare scores from different tests with different distributions.

T-Score

• a mean of 50 and a standard deviation of 10

Stanine Cycle

• All raw scores are converted to a single-digit system of scores ranging from 1 to 9. The mean of stanine scores is always 5 and the standard deviation is 2.