Statistics Overview

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Statistics Overview

• Some New, Some Old Some to come

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Science of Statistics

• Descriptive Statistics methods of summarizing
  or describing a set of data tables, graphs,
  numerical summaries 
• Inferential Statistics methods of making
  inference about a population based on the
  information in a sample

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Levels of Measurement

• Nominal The numerical values just "name" the
  attribute uniquely no ordering of the cases is
  implied.
• Ordinal Attributes can be rank-ordered here,
  distances between attributes do not have any
  meaning.
• Interval The distance between attributes does
  have meaning.
• Ratio There is always an absolute zero that is
  meaningful this means that you can construct a
  meaningful ratio.
• It's important to recognize that there is a
  hierarchy implied in the level of measurement
  idea. At each level up the hierarchy, the current
  level includes all of the qualities of the one
  below it and adds something new. In general, it
  is desirable to have a higher level of
  measurement.

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Variables

• Individuals are the objects described by a set of
  data may be people, animals or things
• Variable is any characteristic of an individual
• Categorical variable places an individual into
  one of several groups or categories
• Quantitative variable takes numerical values for
  which arithmetic operations make sense
• Distribution of a variable tells us what values
  it takes and how often it takes these values

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Correlation

• Correlation can be used to summarize the amount
  of linear association between two continuous
  variables x and y.
• A positive association between the x and y
  variables is indicated by an increase in x
  accompanied by an increase in y.
• A negative association is indicated by an
  increase in x accompanied by a decrease in y.
• For more information see http//www.anu.edu.au/nce
  ph/surfstat/surfstat-home/1-4-2.html

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Chi-square

• A chi square statistic is used to investigate
  whether distributions of categorical variables
  differ from one another.
• The chi square distribution, like the t
  distributions, form a family described by a
  single parameter, degrees of freedom.
• df (r 1) X (c 1)
• For a detailed example, see http//math.hws.edu/ja
  vamath/ryan/ChiSquare.html

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Hypothesis Testing

• Hypothesis testing in science is a lot like the
  criminal court system in the United States
  consider How do we decide guilt?
• Assume innocence until proven'' guilty.
• Proof has to be beyond a reasonable doubt.''
• Two possible decisions guilty or not guilty
• Jury cannot declare someone innocent

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Statistical Hypotheses

• Statistical Hypotheses are statements about
  population parameters.
• Hypotheses are not necessarily true.
• The hypothesis that we want to prove is called
  the alternative hypothesis, Ha.
• Hypothesis formed which contradicts Ha is called
  the null hypothesis, Ho.
• After taking the sample, we must either Reject
  Ho and believe Ha or Fail to Reject Ho because
  there was not sufficient evidence to reject it.

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Type I and II Error

• Consider the jury trial
• If a person is really innocent, but the jury
  decides (s)he's guilty, then they've sent an
  innocent person to jail.
• Type I error.
• If a person is really guilty, but the jury finds
  him/her not guilty, a criminal is walking free on
  the streets.
• Type II error.
• In our criminal court system, a Type I error is
  considered more important than a Type II error,
  so we protect against a Type I error to the
  detriment of a Type II error. This is typically
  the same in statistics.

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P-value

• The choice of alpha is subjective.  
• The smaller alpha is, the smaller the critical
  region. Thus, the harder it is to Reject Ho.  
• The p-value of a hypothesis test is the smallest
  value of alpha such that Ho would have been
  rejected.
• If P-value is less than or equal to alpha, reject
  Ho.
• If P-value is greater than alpha, do not reject
  Ho.

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Confidence Intervals

• Statisticians prefer interval estimates.  
• Point Estimate /- Critical Value Standard
  Error
• The degree of certainty that we are correct is
  known as the level of confidence.
• Common levels are 90, 95, and 99.
• Increasing the level of confidence,
• Decreases the probability of error
• increases the critical point
• widens the interval
• Increasing n, decreases the width of the interval

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Gamma

• This is a statistics utilized in cross-tabulation
  tables.
• Typically viewed as a nonparametric statistic.
• The Gamma statistic is preferable to Spearman R
  or Kendall tau when the data contain many tied
  observations. Gamma is a probability
  specifically, it is computed as the difference
  between the probability that the rank ordering of
  the two variables agree minus the probability
  that they disagree, divided by 1 minus the
  probability of ties.
• It is basically equivalent to Kendall tau, except
  that ties are explicitly taken into account.
• Detailed discussions of the Gamma statistic can
  be found in Goodman and Kruskal (1954, 1959,
  1963, 1972), Siegel (1956), and Siegel and
  Castellan (1988).

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Gamma

• This statistic also tells us about the strength
  of a relationship.
• Can be used with ordinal or higher level of data.
• For a more detailed discussion of Lambda, Gamma
  and Tau, see http//72.14.209.104/search?qcache8
  ZS4_FvVqrgJms.cc.sunysb.edu/mlebo/_private/Class
  es/POL501/Lecture252012.pdfgammaANDlambdaAND
  tauANDstatisticshlenglusctclnkcd39

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Considering Bias

• A sample is expected to mirror the population
  from which it comes, however, there is no
  guarantee that any sample will be precisely
  representative of the population from which it
  comes. The difference between the sample and the
  population is referred to as bias.
• Sampling BiasA tendency to favor selecting
  people that have a particular characteristic or
  set of characteristics. Sampling bias is usually
  the result of a poor sampling plan. The most
  notable is the bias of non response when people
  of specific characteristics have no chance of
  appearing in the sample.
• Non-Sampling ErrorIn surveys of personal
  characteristics, unintended errors may result
  from
• The manner in which the response is elicted
• The social desirability of the persons surveyed
• The purpose of the study
• The personal biases of the interviewer or survey
  writer

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Enjoy the exploration!

• Questions or comments