Partitioning and Measurability

Partitioning and Measurability:
what makes a variable measureable?
©1999 Edward G. Rozycki

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edited 4/20/14

Variables and values.

A variable is a class of sets called values.

Examples:

Cat = {Tabby, Manx, Siamese, ...}

Height = {..., 1', 2', ...,5'5",...}

Appetizers = {clams casino, onion rings, crudité}

A value is a set of items or instances or occurrences.

Examples:

Tabby = {my cat, your cat, this cat, the cat I ran over, ...}

Onion rings = {this onion ring, the one you just ate, ...}

5'5" = {Sam's height, the height of the window, ...}

Imagine a large set, called S, comprised of items, instances and occurrences, where each individual item, instance or occurrence is mentioned only once. We will call such a set a group of individuals. For example,

S = {Sam's height, that building, the onion ring you just ate, your Aunt Emmy's oldest cat, the sound of your doorbell when it last rang, your youngest sister's left eyebrow,...}.

The question we want to address is this: Is S a measurable variable?

Partitioning

S is measurable if and only if it can be partitioned. That is, S can be measured if and only if we can find a set of categories to be the values of S, such that

a. All of the items, instance and occurrences in S can be categorized by a value of S, and

b. No item falls in more than one value of S.

This set of categories, values of S, constitute a partition of S. There may be many partitions of S available. Or there may be none.

Exercise: For each of the sets given below, check to see if the set is a group of individuals, if not, adjust it. Then check to see of any of the possible partitions actually partitions it, if not, adjust it.

Sets

A = {your left foot, your nose, your chair, your father, your car}
B = {your head, Kirkbride Hall, Widener University, Penn State, the last piece of pizza you ate, the last time you saw your car}

C = {your right foot, my right foot, LBJ's left hand, LBJ's right foot, your driver's license, the Empire State Building, the Taj Mahal, the Empire State Building, your driver's license}

D. = {Ronald Reagan, Mel Gibson, Dan Quayle, Margaret Thatcher, Meryl Streep, George Gipp, Bush's VP}

Possible Partitions: Which sets, A through D, are partitioned by which of P1 through P5?

Your Things

Vehicles

Parents

Body Parts

Vehicles

Parents

Actors

Politicians

Women

Body Parts

Documents

Buildings

Males

Females

See, also, Measurability & Educational Concerns

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