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Statistics: First Steps Andrew Martin PS 372 University of Kentucky
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Variance Variance is a measure of dispersion of data points about the mean for interval- and ratio-level data. Variance is a fundamental concept that social scientists seek to explain in the dependent variable.
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Standard Deviation Standard deviation is a measure of dispersion of data points about the mean for interval- and ratio-level data. Like the mean, standard deviation is sensitive to extreme values. Standard deviation is calculated as the square root of the variance.
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Example Calculate the ID and IQV for this PS 372 class grades using the following frequencies or proportions: Grade Freq. Prop. A 4 (.12) B 7 (.21) C 4 (.12) D 7 (.21) E 12 (.34)
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Index of Diversity ID = 1 – ( p 2 a + p 2 b + p 2 c + p 2 d + p 2 e ) ID = 1 - (.12 2 + .21 2 + .12 2 + .21 2 + .34 2 ) ID = 1 - (.0144 + .0441 + .0144 + .0441 + .1156) ID = 1 - (.2326) ID = .7674
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Index of Qualitative Variation 1 – ( p 2 a + p 2 b + p 2 c + p 2 d + p 2 e ) 1 - (1/K)
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Index of Qualitative Variation .7674 (1 – 1/5) .9592
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Data Matrix A data matrix is an array of rows and columns that stores the values of a set of variables for all the cases in a data set. This is frequently referred to as a dataset.
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Properties of Good Graphs Should answer several of the following questions: (JRM 384) 1. Where does the center of the distribution lie? 2. How spread out or bunched up are the observations? 3. Does it have a single peak or more than one? 4. Approximately what proportion of observations in in the ends of the distributions?
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Properties of Good Graphs 5. Do observations tend to pile up at one end of the measurement scale, with relatively few observations at the other end? 6. Are there values that, compared with most, seem very large or very small? 7. How does one distribution compare to another in terms of shape, spread, and central tendency? 8. Do values of one variable seem related to another variable?
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Statistical Concepts Let's quickly review some concepts.
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Population A population refers to any well-defined set of objects such as people, countries, states, organizations, and so on. The term doesn't simply mean the population of the United States or some other geographical area.
Samples are drawn in some known manner and each case is chosen independently of the other.
From here on out, when the book uses the term sample, random sample or simple random sample, it's making reference to the same concept, which is a sample chosen at random.
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Hypothesis Testing Many claims can be translated into specific statements about a population that can be confirmed or disconfirmed with the aid of probability theory. Ex: There is no ideological difference evident in the voting patterns of Republican and Democrat justices on the U.S. Supreme Court.
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Point and Interval Estimation The goal here is to estimate unknown population parameters from samples and to surround those estimates with confidence intervals. Confidence intervals suggest the estimate's reliability or precision.
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Hypothesis Testing Start with a specific verbal claim or proposition. Ex: The chances of getting heads or tails when flipping the coin is are roughly the same. Ex: The chances of the United States electing a Republican or Democrat president are roughly the same.
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Hypothesis Testing Next, the researcher constructs a null hypothesis. A null hypothesis is a statement that a population parameter equals a specific value.
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Hypothesis Testing Following up on the coin example, the null hypothesis would equal .5. Stated more formally: H 0 : P = .5 Where P stands for the probability that the coin will be heads when tossed. H 0 is typically used to denote a null hypothesis.
An alternative hypothesis is a statement about the value or values of a population parameter. It is proposed as an alternative to the null hypothesis.
An alternative hypothesis can merely state that the population does not equal the null hypothesis, or is greater than or less than the null hypothesis.
Suppose you believe the coin is unfair, but have no intuition about whether it is too prone to come up heads or tails.
Stated formally, the alternative hypothesis is:
H A : P ≠ .5
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Hypothesis Testing Perhaps you believe the coin is more likely to come up heads than tails. You would formulate the following alternative hypothesis: H A : P > .5 Conversely, if you believe the coin is less likely to come up heads than tails, you would formulate the alternative hypothesis in the opposite direction: H A : P < .5
Make a decision rule based on some criterion of probability or likelihood.
In social sciences, a result that occurs with a probability of .05 (that is, 1 chance in 20) is considered unusual and consequently is grounds for rejecting a null hypothesis.
Other common thresholds (.01, .001) are also common.
In light of the decision rule, define a critical region. The critical region consists of those outcomes so unlikely to occur that one has cause to reject the null hypothesis should they occur.
So there are areas of “rejection” (critical areas) and non-rejection.
Collect a random sample and calculate the sample estimator.
Calculate the observed test statistic. A test statistic converts the sample result into a number that can be compared with the critical values specified by your decision rule and critical values.
Examine the observed test statistic to see if it falls in the critical region.
Make practical or theoretical interpretation of the findings.
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