Quantitative Data Analysis: Statistics

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Quantitative Data Analysis Statistics
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Sherlock Holmes

• "... while man is an insoluble puzzle, in the
  aggregate he becomes a mathematical certainty.
  You can, for example, never foretell what any one
  man will do, but you can say with precision what
  an average number will be up to. Individuals
  vary, but percentages remain constant. So says
  the statistician"

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Overview

• General Statistics
• The Normal Distribution
• Z-Tests
• Confidence Intervals
• T-Tests

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General Statistics

•  
• THE GOLDEN RULE
• Statistics NEVER replace
• the judgment of the expert.

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Approach to Statistical Research

• Formulate a Hypothesis
• State predictions of the hypothesis
• Perform experiments or observations
• Interpret experiments or observations
• Evaluate results with respect to hypothesis
• Refine hypothesis and start again
• (Basically the same as all other research)

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Hypothesis Testing

• H0 Null Hypothesis, status quo
• HA Alternative Hypothesis, research question
• So, either
• "The data does not support H0"
• or
• "We fail to reject H0"

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Types of Data

• Continuous
• height, age, time
• Discrete
• of days worked this week, leaves on a tree
• Ordinal
• Good, O.K., Bad
• Nominal
• Yes/No, Teacher/Chemist/Haberdasher

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Picturing The Data

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Pie Charts

• Nominal/Ordinal
• Only suitable for data that adds up to 1
• Hard to compare values in the chart

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Bar Charts

• Nominal/Ordinal
• Easier to compare values than pie chart
• Suitable for a wider range of data

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Dot Plots

• Nominal/Ordinal
• Represents all the data
• Difficult to read

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Box Plots

• Nominal/Ordinal
• 1IQR, 3IQR
• Outliers

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Scatter Plots

• Excellent for examining association between two
  variables

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Histograms

• Continuous Data
• Divide Data into ranges

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Time-Series Plots

• Time related Data
• e.g. Stock Prices

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Question 1

• In a telephone survey of 68 households, when
  asked do they have pets, the following were the
  responses
• 16 No Pets
• 28 Dogs
• 32 Cats
• Draw the appropriate graphic to illustrate the
  results !!

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Question 1 - Solution

• Total number surveyed 68
• Number with no pets 16
• gtTotal with pets (68 - 16) 52
• But total 28 dogs 32 cats 60
• gt So some people have both cats and dogs

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Question 1 - Solution

• How many? It must be (60 - 52) 8 people
• No pets 16
• Dogs 20
• Cats 24
• Both 8
• -------------------------
• Total 68

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Question 1 - Solution

• Graphic Pie Chart or Bar Chart

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The Literary Digest Poll

• 1936 US Presidential Election
• Alf Landon (R) vs. Franklin D. Roosevelt (D)

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The Literary Digest Poll

• Literary Digest had been conducting successful
  presidential election polls since 1916
• They had correctly predicted the outcomes of the
  1916, 1920, 1924, 1928, and 1932 elections by
  conducting polls.
• These polls were a lucrative venture for the
  magazine readers liked them newspapers played
  them up and each ballot included a
  subscription blank.

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The Literary Digest Poll

• They sent out 10 million ballots to two groups of
  people
• prospective subscribers, who were chiefly upper-
  and middle-income people
• a list designed to "correct for bias" from the
  first list, consisting of names selected from
  telephone books and motor vehicle registries

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The Literary Digest Poll

• Response rate approximately 25, or 2,376,523
  responses
• Result Landon in a landslide (predicted 57 of
  the vote, Roosevelt predicted 40)
• Election result Roosevelt received approximately
  60 of the vote

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The Literary Digest Poll

• POSSIBLE CAUSES OF ERROR
• Selection Bias By taking names and addresses
  from telephone directories, survey systematically
  excluded poor voters.
• Republicans were markedly overrepresented
• in 1936, Democrats did not have as many phones, 
  not as likely to drive cars, and did not read the
  Literary Digest
• Sampling Frame is the actual population of
  individuals from which a sample is drawn
  Selection bias results when sampling frame is not
  representative of the population of interest

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The Literary Digest Poll

• POSSIBLE CAUSES OF ERROR
• Non-response Bias Because only 20 of 10 million
  people returned surveys, non-respondents may have
  different preferences from respondents
• Indeed, respondents favored Landon
• Greater response rates reduce the odds of biased
  samples

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Terminology

• Population is a set of entities concerning
  which statistical inferences are to be drawn.
• Sample a number of independent observations from
  the same probability distribution
• Parameter the distribution of a random variable
  as belonging to a family of probability
  distributions, distinguished from each other by
  the values of a finite number of parameters
• Bias a factor that causes a statistical sample
  of a population to have some examples of the
  population less represented than others.

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Outliers (and their treatment)

• An "outlier" is an observation that does not fit
  the pattern in the rest of the data
• Check the data
• Check with the measurer
• If reason to believe it is NOT real, change it if
  possible, otherwise leave it out (but note).
• If reason to believe it is real, leave it out and
  note.

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The Mean

• The Mean (Arithmetic)
• The mean is defined as the sum of all the
  elements, divided by the number of elements.
• The statistical mean of a set of observations is
  the average of the measurements in a set of data

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The Variance

• But there can be a lot of variance in individual
  elements,
• e.g. teacher salaries
• Average 22,000
• Lowest 12,000
• Difference 12,000 - 22,000 -10,000

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The Variance

• Sum of (Sample - Average) 0, thus we need to
  define variance.
• The variance of a set of data is a cumulative
  measure of the squares of the difference of all
  the data values from the mean divided by sample
  size minus one.

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Standard Deviation

• The standard deviation of a set of data is the
  positive square root of the variance.

- 1
- 1
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Question 2

• Find the mean and variance of the following
  sample values
• 36, 41, 43, 44, 46

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Question 2

• Mean (36 41 43 44 46)/5 42
• Variance
• Difference Square
• 36 42 -6 36
• 41 42 -1 1
• 43 42 1 1
• 44 42 2 4
• 46 42 4 16
• ----------------------------------------
• 58
• 58 / (5 -1) 58 / 4 14.5

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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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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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One Tail / Two Tail

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

• 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

• Does FF-PD-G have more of the popular vote than
  FG-L ?
• In a random sample of 721 respondents
• 382 FF-PD-G
• 339 FG-L
• Can we conclude that FF-PD-G has 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-PD-G 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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One Tail / Two Tail
T-test
Z-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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T-Tests

• The t-test is often called the Student's t-test.
  It was created by a chief brewer named William S.
  Gossett who worked for the Guinness Brewery. He
  discovered this statistic as part of his work in
  the brewery to compare the different brewing
  processes for changing raw materials into beer.
• 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-Test

•  
• 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