Quantitative Data Analysis: Statistics

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Quantitative Data Analysis Statistics Part 2
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Overview

• Part 1
• Picturing the Data
• Pitfalls of Surveys
• Averages
• Variance and Standard Deviation
• Part 2
• The Normal Distribution
• Z-Tests
• Confidence Intervals
• T-Tests

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The Normal Distribution

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The Normal Distribution

• Abraham de Moivre, the 18th century statistician
  and consultant to gamblers was often called upon
  to make lengthy computations about coin flips. de
  Moivre noted that when the number of events (coin
  flips) increased, the shape of the binomial
  distribution approached a very smooth curve.
• In 1809 Carl Gauss developed the formula for the
  normal distribution and showed that the
  distribution of many natural phenomena are at
  least approximately normally distributed.

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Abraham de Moivre

• Born 26 May 1667
• Died 27 November 1754
• Born in Champagne, France
• wrote a textbook on probability theory, "The
  Doctrine of Chances a method of calculating the
  probabilities of events in play". This book came
  out in four editions, 1711 in Latin, and 1718,
  1738 and 1756 in English.
• In the later editions of his book, de Moivre
  gives the first statement of the formula for the
  normal distribution curve.

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Carl Friedrich Gauss

• Born 30 April 1777
• Died 23 February 1855
• Born in Lower Saxony, Germany
• In 1809 Gauss published the monograph Theoria
  motus corporum coelestium in sectionibus conicis
  solem ambientium where among other things he
  introduces and describes several important
  statistical concepts, such as the method of least
  squares, the method of maximum likelihood, and
  the normal distribution.

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The Normal Distribution
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The Normal Distribution

• Age of students in a class
• Body temperature
• Pulse rate
• Shoe size
• IQ score
• Diameter of trees
• Height?

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The Normal Distribution
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The Normal Distribution
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Density Curves Properties
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The Normal Distribution

• The graph has a single peak at the center, this
  peak occurs at the mean
• The graph is symmetrical about the mean
• The graph never touches the horizontal axis
• The area under the graph is equal to 1

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Characterization

• A normal distribution is bell-shaped and
  symmetric.
• The distribution is determined by the mean mu, m,
  and the standard deviation sigma, s.
• The mean mu controls the center and sigma
  controls the spread.

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Same Mean, Different Standard Deviation
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1
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Different Mean, Different Standard Deviation
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1
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Different Mean, Same Standard Deviation
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1
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The Normal Distribution

• If a variable is normally distributed, then
• within one standard deviation of the mean there
  will be approximately 68 of the data
• within two standard deviations of the mean there
  will be approximately 95 of the data
• within three standard deviations of the mean
  there will be approximately 99.7 of the data

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The Normal Distribution
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Why?

• One reason the normal distribution is important
  is that many psychological and organsational
  variables are distributed approximately normally.
  Measures of reading ability, introversion, job
  satisfaction, and memory are among the many
  psychological variables approximately normally
  distributed. Although the distributions are only
  approximately normal, they are usually quite
  close.

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Why?

• A second reason the normal distribution is so
  important is that it is easy for mathematical
  statisticians to work with. This means that many
  kinds of statistical tests can be derived for
  normal distributions. Almost all statistical
  tests discussed in this text assume normal
  distributions. Fortunately, these tests work very
  well even if the distribution is only
  approximately normally distributed. Some tests
  work well even with very wide deviations from
  normality.

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So what?

• Imagine we undertook an experiment where we
  measured staff productivity before and after we
  introduced a computer system to help record
  solutions to common issues of work
• Average productivity before 6.4
• Average productivity after 9.2

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So what?
After 9.2
0
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Before 6.4
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So what?
Is this a significant difference?
After 9.2
10
0
Before 6.4
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So what?
or is it more likely a sampling variation?
After 9.2
10
0
Before 6.4
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So what?
After 9.2
10
0
Before 6.4
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So what?
After 9.2
10
0
Before 6.4
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So what?
How many standard devaitions from the mean is
this?
After 9.2
10
0
Before 6.4
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So what?
How many standard devaitions from the mean is
this?
and is it statistically significant?
After 9.2
10
0
Before 6.4
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So what?
s
s
s
After 9.2
10
0
Before 6.4
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One Tail / Two Tail

• One-Tailed
• H0 m1 gt m2
• HA m1 lt m2
• Two-Tailed
• H0 m1 m2
• HA m1 ltgtm2

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STANDARD NORMAL DISTRIBUTION

• Normal Distribution is defined as
• N(mean, (Std dev)2)
• Standard Normal Distribution is defined as
• N(0, (1)2)

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STANDARD NORMAL DISTRIBUTION

• Using the following formula
• will convert a normal table into a standard
  normal table.

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Exercise

• If the average IQ in a given population is 100,
  and the standard deviation is 15, what percentage
  of the population has an IQ of 145 or higher ?

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Answer

• P(X gt 145)
• P(Z gt ((145 - 100)/15))
• P(Z gt 3)
• From tables 99.87 are less than 3
• gt 0.13 of population

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Trends in Statistical Tests used in Research
Papers
Historically
Currently
Quoting P-Values
Confidence Intervals
Hypothesis Tests
Results in Accept/Reject
Results in p-Value
Results in Approx. Mean
Testing
Estimation
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Confidence Intervals 

• A confidence interval is used to express the
  uncertainty in a quantity being estimated. There
  is uncertainty because inferences are based on a
  random sample of finite size from a population or
  process of interest. To judge the statistical
  procedure we can ask what would happen if we were
  to repeat the same study, over and over, getting
  different data (and thus different confidence
  intervals) each time.

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Confidence Intervals 
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Jerzy Neyman

• Born April 16, 1894
• Died August 5, 1981
• Born in Bessarabia, Imperial Russia
• statistician who spent most of his professional
  career at the University of California, Berkeley.
• Developed modern scientific sampling (random
  samples) in 1934, the Neyman-Pearson lemma in
  1933 and the confidence interval in 1937.

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Egon Pearson

• Born 11 August 1895
• Died 12 June 1980
• Born in Hampstead, London
• Son of Karl Pearson
• Leading British statistician
• Developed the Neyman-Pearson lemma in 1933.

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• Neyman and Pearson's joint work formally started
  in the spring of 1927.
• From 1928 to 1934, they published several
  important papers on the theory of testing
  statistical hypotheses.
• In developing their theory, Neyman and Pearson
  recognized the need to include alternative
  hypotheses and they perceived the errors in
  testing hypotheses concerning unknown population
  values based on sample observations that are
  subject to variation.
• They called the error of rejecting a true
  hypothesis the first kind of error and the error
  of accepting a false hypothesis the second kind
  of error.
• They called a hypothesis that completely
  specifies a probability distribution a simple
  hypothesis. A hypothesis that is not a simple
  hypothesis is a composite hypothesis.
• Their joint work lead to Neyman developing the
  idea of confidence interval estimation, published
  in 1937.

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Confidence Intervals 

• Neyman, J. (1937) "Outline of a theory of
  statistical estimation based on the classical
  theory of probability" Philos. Trans. Roy. Soc.
  London. Ser. A. , Vol. 236 pp. 333380.

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Confidence Intervals 

• If we know the true population mean and sample n
  individuals, we know that if the data is normally
  distributed, Average mean of these n samples has
  a 95 chance of falling into the interval.

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Confidence Intervals 

• where the standard error for a 95 CI may be
  calculated as follows

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Example 1
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Example 1

• Did FF have more of the popular vote than FG-L ?
• In a random sample of 721 respondents
• 382 FF
• 339 FG-L
• Can we conclude that FF had more than 50 of the
  popular vote ?

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Example 1 - Solution

• Sample proportion p 382/721 0.53
• Sample size n 721
• Standard Error (SqRt((p(1-p)/n))) 0.02
• 95 Confidence Interval
• 0.53 /- 1.96 (0.02)
• 0.53 /- 0.04
• 0.49, 0.57
• Thus, we cannot conclude that FF had more of the
  popular vote, since this interval spans 50. So,
  we say "the data are consistent with the
  hypothesis that there is no difference" 

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Example 2
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Example 2

• Did Obama have more of the popular vote than
  McCain ?
• In a random sample of 1000 respondents
• 532 Obama
• 468 McCain
• Can we conclude that Obama had more than 50 of
  the popular vote ?

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Example 2 95 CI

• Sample proportion p 532/1000 0.532
• Sample size n 1000
• Standard Error (SqRt((p(1-p)/n))) 0.016
• 95 Confidence Interval
• 0.532 /- 1.96 (0.016)
• 0.532 /- 0.03136
• 0.5006, 0.56336
• Thus, we can conclude that Obama had more of the
  popular vote, since this interval does not span
  50. So, we say "the data are consistent with
  the hypothesis that there is a difference in a
  95 CI" 

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Example 2 99 CI

• Sample proportion p 532/1000 0.532
• Sample size n 1000
• Standard Error (SqRt((p(1-p)/n))) 0.016
• 99 Confidence Interval
• 0.532 /- 2.58 (0.016)
• 0.532 /- 0.041
• 0.491, 0.573
• Thus, we cannot conclude that Obama had more of
  the popular vote, since this interval does span
  50. So, we say "the data are consistent with
  the hypothesis that there is no difference in a
  99 CI" 

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Example 2 99.99 CI

• Sample proportion p 532/1000 0.532
• Sample size n 1000
• Standard Error (SqRt((p(1-p)/n))) 0.016
• 99.99 Confidence Interval
• 0.532 /- 3.87 (0.016)
• 0.532 /- 0.06
• 0.472, 0.592
• Thus, we cannot conclude that Obama had more of
  the popular vote, since this interval does span
  50. So, we say "the data are consistent with
  the hypothesis that there is no difference in a
  99.99 CI" 

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T-Tests

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William Sealy Gosset

• Born June 13, 1876
• Died October 16, 1937
• Born in Canterbury, England
• On graduating from Oxford in 1899, he joined the
  Dublin brewery of Arthur Guinness Son.
• Published significant paper in 1908 concerning
  the t-distribution

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• Gosset acquired his statistical knowledge by
  study, and he also spend two terms in 19061907
  in the biometric laboratory of Karl Pearson.
• Gosset applied his knowledge for Guinness both in
  the brewery and on the farm - to the selection of
  the best yielding varieties of barley, and to
  compare the different brewing processes for
  changing raw materials into beer.
• Gosset and Pearson had a good relationship and
  Pearson helped Gosset with the mathematics of his
  papers.
• Pearson helped with the 1908 paper but he had
  little appreciation of their importance.
• The papers addressed the brewer's concern with
  small samples, while the biometrician typically
  had hundreds of observations and saw no urgency
  in developing small-sample methods.

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T-Tests

• Student (1908), The Probable Error of a Mean
  Biometrika, Vol. 6, No. 1, pp.1-25.

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T-Tests

• Guinness did not allow its employees to publish
  results but the management decided to allow
  Gossett to publish it under a pseudonym -
  Student. Hence we have the Student's t-test.

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T-Tests

• powerful parametric test for calculating the
  significance of a small sample mean
• necessary for small samples because their
  distributions are not normal
• one first has to calculate the "degrees of
  freedom"

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•  
• THE GOLDEN RULE
• Use the t-Test when your
• sample size is less than 30

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T-Tests

• If the underlying population is normal
• If the underlying population is not skewed and
  reasonable to normal
• (n lt 15)
• If the underlying population is skewed and there
  are no major outliers
• (n gt 15)
• If the underlying population is skewed and some
  outliers
• (n gt 24)

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T-Tests

• Form of Confidence Interval with t-Value
• Mean /- tValue SE
• --------
  -------
• as before as
  before

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Two Sample T-Test Unpaired Sample

• Consider a questionnaire on computer use to final
  year undergraduates in year 2007 and the same
  questionnaire give to undergraduates in 2008. As
  there is no direct one-to-one correspondence
  between individual students (in fact, there may
  be different number of students in different
  classes), you have to sum up all the responses of
  a given year, obtain an average from that, down
  the same for the following year, and compare
  averages.

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Two Sample T-Test Paired Sample

• If you are doing a questionnaire that is testing
  the BEFORE/AFTER effect of parameter on the same
  population, then we can individually calculate
  differences between each sample and then average
  the differences. The paired test is a much strong
  (more powerful) statistical test.

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Choosing the right test

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Choosing a statistical test
http//www.graphpad.com/www/Book/Choose.htm
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Choosing a statistical test
http//www.graphpad.com/www/Book/Choose.htm