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# Statistics: Module 2 Variables and Probability

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## Jibey Asthappan

on 15 December 2015

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#### Transcript of Statistics: Module 2 Variables and Probability

Variables and Probability
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 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}
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
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