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Events that we cannot predict wirh certainty. "Unpredictable"
Example is flipping a coin. 


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Processes in which all outcomes have the same probability.
Example: fair coin, fair dice 


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If two events are independent, it means that knowing one outcome does not allow you to predict the other outcome.
Example: Just because you got 5 heads in a row during a coin toss does not mean you are "due" for a tails. There is still a 5050 chance of getting either a heads or tails. 


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A random process with only two outcomes.
Can have the feature of equal probability or not. 


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The plot of the two outcomes (discrete). N= number of trials, p= probability of success and each event is independent. 


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When a process has a countable number of possible outcomes (more than 2). Can have equal probabilities for each outcome or not. 


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Every number in a given range has an equal chance of showing up. 


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Shows the likelihood that various values of the variable will occur. Shows us what kinds of outcomes are common and what kinds are rare. 


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Knowing all the possible results and there are a countable number of them. 


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 Used to describe random processes
 Describing data
 Help us to understand how large the random component is in the things we observe



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Normal (Gaussian) Distribution 

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Is a continuous probability distribution. A frequency distribution in which there are many possible values with a probability of occurring. *data tends to be around a central value with no bias left or right, creates what we know as the "bell curve" 


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Avergage.
The sum of a set of numbers divided by how many numbers there are.
2+4+8+10=22
24/4=6 (mean)
Equation: EXi/N 


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The extent to which the sample does not reflect the population. The sample may no be representing a certain portion of the population accurately. 


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Sample average=population mean + sampling error 


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During an experiment, the random assignment of subjects to experimental control groups may randomly produce differences between the two groups (control and experimental), resulting in an error 


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Difference between the actual value of a quantity and the value obtained by a measurement. Repeating the measurement will reduce the random error.



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E= the effects of every variable, (the random component) on the outcome.
X= the independent vairable
Y= outcome or the dependent variable (the thing we are trying to explain) 


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The mean of the sampling distribution 


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A probability distribution that shows what happens when we repeat the smapling process many times. 


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Number used to describe the variation in a variable's distribution
*the square root of the variance



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The standard deviation of the sampling distribution. 


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When values are transformed in a way that allows for comparisons between variables measured on different scales. 


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When the sum or the mean of large random smaples are plotted, the shape is of a bell (normal distribution).
Tells us what is a good guess of the mean. 


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E(X^{1}X)^{2}/N1
X^{1}= each number in the sample
X=mean
N= number of samples
Ex: E(1, 6, 8, 9)
Mean = 1 + 6 + 8 + 9/4 = 6
(16)^{2}+ (61)^{2} +(81)^{2}+(91)^{2}/41 = 163/3 =54.33



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Is a normal distribution but with longer tails. Sample size is small and the sample falls further away from the mean. 


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Occurs if the mean of the sampling distribution for the estimator (expected value) is equal to the population value being estimated.
Bias = population value  expected value of estimator 


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The field of studey that involves the collection and analysis of numerical facts or data. 


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Things differ from person to person 


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Every member of the population we are studying has the same chance of appearing in the sample. 


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Set of participants who receive the "manipulation" 


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Set of participants in an experiment who do not receive the "manipulation" 


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Observations are assigned to categories with no ordering among categories. Can only be assigned to one category.
Ex: Married, widow, divorced, single, seperated. 


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A nomial measurement with only two possible categories.
Ex: male or female. 


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Number represent rank ordering but we do not know how far aprt the ranks are.
Ex: strongly agree, agree; neither agree or disagree; disagree; and strongly disagree. 


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Have regular numbers that indicate how far apart observations are.
Ex: number of years of education, income. 


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Level of measurement in which observations are assigned numbers and the numbers indicate how far apart observations are. Different form interval because have a natural zero point. 


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Qualitative vs. Quantitative 

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Qualitative: nominal variables (discrete)
Quantitative: ordinal and interval measurements (continuous vairables) 


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Studying one kind of thing and making conclusions about another unit of analysis. 


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Used to study the occurrence of events. 


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Analysis of data over time. Can help us solve the problem of casual ordering. 


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The collection of data on many examples at one point in time. Cannot determine causation. 


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Observe untis at several points in time. Provides a large sample size and time odering to help determine causation. 


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Is a subset of all the units (population) on which we would like to have data. 


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Measures of Central Tendency 

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What is typical in a data set and the extent to which observations differ from one another (mean, median & mode). Is an attempt to summarize the data set with one number that is "typical" of all numbers in the data set. 


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How different observations are from one another. 


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The middle.
Order numbers from highest to lowest and it is the number found in the middle. If there is an even number, take the two numbers in the middle and divide by two. 


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The score that occurs most frequently in the data.
Ex: 1, 3, 4, 3, 6, 7, 4, 3, 9, 3
Mode: 3 


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Exceptional data points. Impacts the mean the most.
Ex: 1, 2, 3, 4, 26
26 is the outlier. 


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Allows us to make estimates of values in the population while taking into account that our estimates are never accurate (idicates reliability).
Mean + Z(S.e/standard error)



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+/ 1 > 68%
+/2 > 95%
+/3 > 99% 


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A measure of whether the data are peaked or flat relative to a normal distribution 


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Is a measure of symmetry or the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. 


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Have a distinct peak near the mean, decline rather rapidly, and have heavy tails 


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Have a flat top near the mean rather than a sharp peak and little/thin tails. A uniform distribution would be the extreme case. 


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A graphical technique for showing both the skewness and kurtosis of data set. 


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3 Moment Generating Functions 

Definition
 Location (mean, median & mode).
 Variation (range, standard deviation & variance)
 Symmetry (skewness, kurtosis)



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The thing on which data was collected (organizations, countries, people) 


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Variable that is used to try to explain/predict a dependent variable. Known as "X" in the equation. 


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The variable that we are tring to explain/predict. Known as "Y" in the equation. 


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In a linear regression model in which that errors have the expectation of zero and are uncorrelated, and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator 


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One factor (X) leads to another factor (Y) to occur 


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Tells us where the numbers are coming from. Likelihood of an event occurring. 


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The collection and analysis of data so that we are able to infer to a population. Interested in the causal realtionship as politcial scientists. 


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Information that we break into categories (gender, race, religion, etc) then assign it a number (nominal). 


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Cumulative Distribution Function (CDF) 

Definition
 The proportion of population with value less than X
 The probability of having a value less than x



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Probability Density Function 

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Describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region 


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Counts the number of event and the number of times these events occurred in a given interval of time. 


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Any number on the number line. 

