Statistics Overview

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Statistics Overview
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Two Categories of Statistics

• Descriptive Statistics
• Typically used to describe the sample used in a
  study.
• At a minimum, sample statistics should include
  the a measure of central tendency and a measure
  of variability for each group studied.
• Inferential Statistics
• Used to make inferences to a target population.

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Descriptive Statistics

• The type of statistic used depends upon the type
  of scale used to measure data.
• There are four types of measurement scales
• Nominal
• Ordinal
• Interval
• Ratio
• The seminal work on scales of measurement is
  Stevens (1946), available in the Directory of
  Journal Articles.

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Measurement

• First, what is measurement?
• Stevens (1946) defined measurement as the
  assignment of numbers to aspects of objects or
  events according to rules (p. 677).
• Measurement maps a set of objects onto a set of
  numbers such that there is a one-to-one
  correspondence between the objects and the
  numbers assigned to them.

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Nominal Scale (Categorical Data)

• Assigns numbers as labels to objects, or classes
  of objects.
• Used for measuring categorical data
• 1 if have a computer at home 2 if does not have
  a computer at home.
• Voted as a (1) democrat, (2) republican, (3)
  libertarian, (4)independent, or (5) other in last
  election.
• Numerical values are assigned arbitrarily.
• Rules used for classification are important.

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Descriptive Statistics Used with Categorical Data

• Frequencies (counts) and percentages.
• How many cases are in each category.

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Ordinal Data (Ranks)

• The assignment of numbers to persons or objects
  in such a way that the numbers reflect a
  rank-ordering on the attribute in question.
• Example include
• Class rank,
• Beauty pageant scoring,
• Percentiles.

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Descriptive Statistics Used with Ordinal Data

• Measure of Central Tendency Median
• Middle score in a distribution of scores.
• Score at the 50th percentile.
• Measure of Variability the Range
• Difference between highest score and lowest score
  (Xhigh-Xlow).
• Inter-quartile range (XQ3-XQ1).

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Interval and Ratio Scales

• Numbers are assigned so that, in addition to
  satisfying the requirements of the ordinal scale,
  differences between numbers can be meaningfully
  interpreted.
• Suppose we have four individuals measured on an
  IQ scale 60, 80, 100, and 120.
• Here the individual with an IQ of 100 is 20
  points higher in IQ than an individual with an IQ
  of 80. The same can be said for the individual
  with an IQ of 120 as compared to the one with an
  IQ of 100.
• It cannot be said that the individual with an IQ
  of 120 has twice the IQ as the person with an IQ
  of 60. (This statement would require a ratio
  scale).

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Descriptive Statistics used with Interval and
Ratio Data

• Measure of Central Tendency the mean.
• Arithmetic average.
• Measure of variability the Variance (s2) or
  Standard Deviation (s)

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Definition in Words.

• The standard deviation tells how far, on average,
  a score drawn, at random, from a distribution
  deviates from the mean of that distribution.
• The variance is the average squared deviation of
  scores from the mean of the distribution i.e.
  the mean squared deviation.

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The Correlation as a Descriptive Statistic

• The correlation coefficient, r, quantifies the
  linear relationship between two variables.
• r varies between -1, through 0 to 1.
• An r of -1 (or close to -1) implies a perfect (or
  nearly perfect) inverse relationship between two
  variables.
• An r of 0 (or close to 0) implies NO relationship
  between the two variables.
• An r of 1 (or close to 1) implies a perfect (or
  nearly perfect) direct relationship between to
  variables.

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Inferential Statistics

• Allow inferences about population parameters
  based on sample statistics.
• Rarely would we be interested in the value of
  sample statistics.
• Rather we are interested in using the sample
  statistics as estimators of the population
  parameters.
• i.e., we use to estimate µ.

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Statistical Tests

• Many of the most-used statistical tests yield
  inferential statistics.
• Tests of null hypotheses, H0.
• t test, to test the null hypothesis that the
  means of two samples were drawn from the same
  population (i.e., H0 µ1 µ2).
• F test, to test the null hypothesis that the
  means of several samples were all drawn from the
  same population (i.e., H0 µ1 µ2 µk.

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Type I and Type II Errors
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Probabilities of Type I andType II Errors

• Alpha (a) gives the probability of a Type I
  error.
• This is the level of significance set by the
  researcher.
• a .05 or a .01 are, typical by convention.
• Should be distinguished from p (the probability
  of an outcome under the assumption that the null
  hypothesis is TRUE.)

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Probabilities of Type I andType II Errors

• Beta (ß) gives the probability of a TYPE II
  error.
• Usually not known.
• Its compliment (1 - ß), known as power is of
  great importance.
• Power is the probability of REJECTING the null
  hypothesis when it is FALSE.
• Researchers usually want to maximize power.

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Maximizing Power

• There are four main ways to enhance the power of
  a quantitative study
• Increase the sample size.
• Relax the probability of a Type I error (i.e.,
  use a larger a level).
• Use a one-tailed test.
• Reduce the error variance within groups.
• Use treatments that yield a larger effect size.

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Effect Size

• Calculating an effect size can aid in
  interpreting statistical results.
• ? gives the distance, in standard deviation
  units, between the experimental group and the
  control group.
• See the next slide

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Picture of a Normal Curve
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Confidence Intervals

• According to Hays (1988) a confidence interval
  gives and estimated range of values with a
  givenprobability of covering the parameter
  (p. 206).

• A set of 95 Confidence Intervals around ? (i.e.,
  ? or 1.96 standard errors of ?)
• ----------?----
• ------?--------
• ------------?--
• ---?-----------
• ? ---------------

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Additional Topics in Statistics

• Reliability of measurements.
• The problem of missing data.
• What is the unit of analysis.
• Multi-level analysis.
• Parametric vs Nonparametric statistical
  procedures.

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The End