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Statistics: First Steps Andrew Martin PS 372 University of Kentucky
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.
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.
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)
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
Index of Qualitative Variation 1 – ( p 2 a + p 2 b + p 2 c + p 2 d + p 2 e ) 1 - (1/K)
Index of Qualitative Variation .7674 (1 – 1/5) .9592
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?
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?
Statistical Concepts Let's quickly review some concepts.
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.
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.
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.
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.
Hypothesis Testing Next, the researcher constructs a null hypothesis. A null hypothesis is a statement that a population parameter equals a specific value.
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.
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
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