A phenomenon which can be accounted for though measurement

Think of a relationship – what are the variables?

How can we measure those variables?

What variables are easy to measure? Which are more challenging?

Is the variable independent?

Is the sample obtained randomly? Is it a population?

Variables

Types of studies:

Cross-sectional

Time series

Experimental

Case Study

Decide on the type of variable based on the type of study

Operationalizing and measuring variables are crucial to conducting a valid study

Variables and Your Study

Variables should include observations that are random if from a sample

Observations that are not random suffer from non-random error

Example of non-random error in a variable

Sample space includes all possible values

Discrete (nominal, ordinal, interval) or continuous (ratio)

Probability and Variables

Use when something is or is not

Example: How many times has Apu been shot? How many times has Apu died?

Analysis of this type of data is different since the range is only 0 and 1

**Dichotomized Variables**

Although not inferring causation, you consider what may affect the other

What causes the effect is the independent variable

What is affected is the dependent

But be careful! Simultaneity may exist…

crime -> urban decay OR urban decay ->crime ?

Confounding/Spurious variables

Harvested corn volume is inversely related to the deaths of elderly… does this make sense?

Independent and Dependent

Measured variables should contain variation

That variation corresponds to the probability of something occurring

To make this simple, we use dice, coins, cards, ect. to explain probability because tests can be replicated easily and the range is fixed

Variables and Variation

Coins:

S={H,T}

Probability of Heads?

Dice:

S={1,2,3,4,5,6}

Probability of 1?

Four coins/Four tosses:

S={1,2,3,4}

Probability of having 2 heads in the series

Probability

Two dice:

S={2,3,4,5,6,7,8,9,10,11,12}

Probability of 6?

Murders per capita:

S={0,.01,.02,.03…..∞}

Probability of 5 murders per capita?

As the n increases toward the N so does the Xbar decreases to the true µ

**Law of Large Numbers**

Used often to describe something we can count.

Average number of students in a stats course – how can you measure this?

Average number of pies eaten at a fair – how can you measure this

Averages can be deceiving

The average of a dice is 4.2, but should you bet on 4?

**Average**

X is a random variable and a and b are fixed numbers, then μa + bX = a + bμX

If X and Y are random variables, then

μX + Y = μX + μY

This is true if the variable(s) are truly random!

Rules of Means

X1(p1)+X2(p2)….Xi(pi)=μ

Var= ((x1-μ)^2)p1+ ((x2-μ)^2)p2…. ((xi-μ)^2)pi

Standard deviation is the square root of variance

Standard deviation puts the variance in respect to the values of the variable

Mean & Variance of Discrete Variables

X1(p1)+X2(p2)….Xi(pi)=μ

Var= (x1-μ)2p1+ (x2-μ)2p2…. (xi-μ)2pi

Standard deviation is the square root of variance

Standard deviation puts the variance in respect to the values of the variable

Mean & Variance of Discrete Variables

Standard deviation of a Normal Distribution

Operationalization

How can we measure (quantify) a phenomenon?

What are some ways to measure religiosity?

Try these subjects:

Coffee consumption

Wealth

Criminality

Happiness

Political Rights

Freedom

Standard Deviation of Discrete Data

First calculate the variance of the sample, then square root for the standard deviation

If you have a population divide by N instead of N

The (n-1) accounts for the greater variance inherent in samples