Psychological Statistics
      Measurement
Psychology 207B
Erwin Segal

Homework Page

Sets, Subsets, Populations and Samples
• Statistics begins with a set of things of the same kind. These things are elements (individuals, members) of the set. A set is a collection of individuals.

Examples of sets include

– students in this class
– people who smoke
– families in the USA
– flips of a coin
– nightly sleep habits
– even numbers
• Two sets A and B are the same if they contain exactly the same individuals.
• Set A is a subset of set B if all of the elements in Set A are elements of Set B.
• Sets can be classified by their cardinality, the number of members that they contain. This may be large or small, even infinite or theoretical.
• Populations and samples are sets and subsets used in statistics.

Measurement identifies a well defined procedure that assigns a measure (symbol or number) to each member of a set. Every individual receives exactly one value of the measure. Here are different measures assigned to the above pictured sets. The property that takes on different values over the set of measures is called a variable.

The Sets of Measures are bona fide sets in their own right as well as representatives of the individuals that they measure.

Scales of Measurement
• Nominal--categories or names, measures on individuals are either equal, = ,or not equal, <>, (examples are sex, jersey number, hometown)

• Ordinal--nominal plus relative order (transitivity),  greater ,>, or less,<. (rank in class, mineral hardness, percentile)

Quantitative Scales: Measures of the individuals have some properties of numbers. Computations on the numbers are reliable and can be meaningful.
• Interval--ordinal operations of plus an underlying scale, (calendar date, Farenheit temperature)--equal differences are equal, e.g. 8-3=10-5=5.
• Ratio--interval plus a true zero, (height in inches, weight in pounds, time to run 100 meters). All number operations hold.

On numerical measures
• Discrete variables--counting individuals
        – Exact values
        – five people, 36 families
• Continuous variables--assumes an underlying dimension is measured,
        – measures can be continuously refined with more precise measures.
        – length, weight, I.Q., anxiety

– values of measures can be mapped onto points on a line
– The line represents the variable or dimension of measure
• Sometimes different but similar values may be categorized together. At times being too precise hides meaningful similarities. For example, one might wish to group all IQ scores between 95 and 105 in the same category. In this case all the scores in the category are considered mathematically equivalent. We then have a way to easily assess how many scores are essentially average.