Psychology 412 Applied Data Analysis

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Psychology 412Applied Data Analysis

• Instructor Adam Kramer
• Week 1

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Science

• A method for testing theories or answering
  questions
• Seek evidence which supports
• Seek evidence which contradicts
• The key seek evidence.
• The result Bad theories fall away, good theories
  persist

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Sciences

• Physics, chemistry, psychology, sociology,
  political science all do this
• Q How much does an atom weigh?
• A Put one on a scale and see!
• Q What happens if you mix baking soda and
  vinegar?
• A Try it and see!
• Q Does chocolate make people happy?
• A What do you mean by people

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Psychology as Science

• Atoms, chemicals, etc., are nearly identical on
  the relevant dimensions
• They weigh the same, react the same
• People are nearly identical on some dimensions
• They all die in extreme temperatures, react
  similarly to some drugs
• But psychology asks different questions.

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People are different.

• But also similar psychology looks for maxims or
  similarities despite the differences among people
• Or, what are people like on average?

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A sea of variability!

• What can so much variation tell us?
• What is the trend despite the variation, and how
  sure are we that its real?
• This is the task of statistics in social science
• Detecting trends despite variation
• Explaining variation in some way

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An example Satisfaction

• I am satisfied with my life.
• 7 - Strongly agree
• 6 - Agree
• 5 - Slightly agree
• 4 - Neither agree nor disagree
• 3 - Slightly disagree
• 2 - Disagree
• 1 - Strongly disagree

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So what does it mean?

• So, now weve got a variablewhats in it? What
  does it tell us?

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The mean

• The average is the sum of the items divided by
  the number of the items.
• µ or m are symbols for mean, x represents a
  variable, i is an index to an observation in the
  variable, n is the symbol for the number of data
  points, ? means sum.
• Mean and average are used interchangeably.
• Each observation contributes equally.

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Median, Mode

• The median is the value in the middle of your
  variable (or between the two middle values).
• 50 of observations above, 50 below
• The mode is the value most frequently observed
• Each observation does not contribute equally

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Mean median mode which when?

• All have different definitions and meanings, but
  science is about understanding and description.
• Use the summary statistic that is most
  interesting or central to your point!

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Example SWL

• Todays SWL If I could live my life over, I
  would change almost nothing
• 1 - Strongly disagree, disagree, slightly
  disagree, neither agree nor disagree, agree,
  strongly agree - 7
• How close are mean, median, mode?

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None is enough
Lecture 1 ended here

• We need some notion of variability, too.
• Variability around some point
• Use the mean!
• Difference from average makes sense when the
  mean is meaningful
• Each point contributed to the mean, so has an
  effect on its deviation

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Mean deviations

• Summed deviations
• Always equals zero
• So square them
• Then the signs go away
• Adding absolute values is messy
• Average squared deviation
• n-1 because weve already used the mean, which
  makes n overestimate.
• This is the variance.

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Mean deviations

• Standard deviation
• Un-square the sum of squared deviations
• An average of deviations from the mean
• Quantifies how much variability there is in the
  data set
• In the same UNIT (satisfactions) as the original
  variable
• The unsquared average squared deviation

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The Z Score

• Each score xi has a Z score
• Z is how far from the mean x is, in standard
  deviation units
• In other words, it is the standardized score
• Z is distribution-free because the variability
  has been controlled for

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Z scores

• Our scores µ6.11, sd0.6
• adam arron brent jason killian
  whitney dylan hal lauren
• 1, 6.0000 6.0000 6.0000 6.0000 6.0000
  6.0000 7.000 5.000 7.000
• 2, -0.1849 -0.1849 -0.1849 -0.1849 -0.1849
  -0.1849 1.479 -1.849 1.479
• Im 0.18 sd below the mean
• Means less satisfied, but not by much

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The Normal Distribution

• Whats an SD good for?
• For normal distributions, 68 of observations
  within 1sd, 95 within 2
• In other words, a Z score tells us how ODD a
  score is

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How big is big?

• If Z is big, it means someones score is many
  sds from the mean
• 2 is big for normal distributions
• That person is odd or doesnt belong if
  theyre further than we would expect by chance.
• GIVEN the variability, is this person odd?
• Z CONTROLS for variability, but still

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The Shape of Distributions

• One SD means different things for different
  distributions
• So its hard to say a PERSON is weird

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The Central Limit Theorem

• Regardless of the distribution of items, the
  distribution of MEANS is normal (if you have
  enough)

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The Z test

• So GIVEN how many observations went into the mean
  (here, 10000), we can actually quantify HOW ODD a
  mean is
• But only because we can place it on the normal
  distribution

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The Null Hypothesis

• Assuming that your sample is pulled from a
  certain population, how likely is the mean of the
  sample?
• How far is your samples mean from the null
  hypothesis?

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Back to science

• Testing a hypothesis means
• Finding evidence to support a theory
• Finding evidence to disprove a theory
• In psychology, we hypothesize relationships among
  variables
• But we have a lot of variability
• Are observed relationships accidental?

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The Null Hypothesis

• Is the relationship strong enough to overcome the
  variability?
• Or, how bizarre or unlikely would our observed
  results be if there wasnt really a relationship
  there?
• Formally
• Assume there is no relationship (H0)
• Quantify how unlikely the data are
• The task for the term!

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Our data

• We can ask two questions using this technique
• Is our measure DIFFERENT from some point (like
  0)?
• Are two measures DIFFERENT from each other?
• Are two measures RELATED to each other?

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Our data

• We can ask two questions using this technique
• Is our measure DIFFERENT from some point (like
  0)? Z T-test
• Are two measures DIFFERENT from each other?
  T-test
• Are two measures RELATED to each other?
  Correlation

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Satisfied?

• Compared to UO students
• UO students show µ5.05, sd1.48
• So, are we, on average higher?
• Average means mean!
• Is our mean greater than 5.05?
• Well, duh, its 6.11.
• But is that MEANINGFUL?

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Null hypothesis

• 6.11 is meaningfully higher than 5.05 if it is
  really unlikely that wed have randomly gotten a
  number that high GIVEN that we are just uo
  students.
• So whats a lot of deviation?
• We have a measure of the deviationbut how do we
  know if it matters?

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The Central Limit Theorem

• If the population mean and sd are known, then OUR
  mean is part of a NORMAL distribution for means
  of size 9 around the population mean.

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The Z-Test

• If we know the mean and standard deviation of the
  population, we can quantify how far above it we
  are.
• We are 0.717 standard deviations above the mean

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The Z-Test

• But SD isnt good enough, because were using our
  mean, which is produced by 9 observations.
• The standard error is the sd over sqrt(n)
• We are 2.15 standard errors above the mean

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The normal distribution

• 2.15 standard errors above the mean

98.4
1.6
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Are we normal?

• so theres a 1.4 chance that a null effect (us
  just being random UO students) would produce a
  result as extreme as 6.11
• Z2.15, p.014

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Are we normal?

• Our standardized average, of 1.39, is higher than
  91.8 of averages which a true null hypothesis
  would produce.
• Is that suitably unlikely?
• So wed say p.082
• but where did that mean and sd come from

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Not really parameters.

• The mean and sd for the population was actually
  from a study with n273.
• Z-tests only work with parameters.
• T-tests allow for uncertainty on both sides!

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Independent-samples t

• complicated.
• Take the difference in means, over the standard
  error of the difference
• which is estimated from the standard errors of
  the components, weighted by their sample sizes.

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IST

• t is larger than z!
• But all ts were not created the same.

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Degrees of Freedom

• When we compute the mean, we use all the numbers
• The set of numbers has some value pulled out of
  it.
• So, every time a mean is computed and used, we
  lose a degree of freedom.
• In this example, we compare two means, so we lose
  two df
• Df 8272-2 278

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t distributions
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Our t distribution

• Now only a 0.9 chance of a null hypothesis
  producing results as extreme as ours!

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Other kinds of t

• The t-distribution can be used for other things
  as well
• One-sample t Compare a samples mean to any set
  number (are we satisfied µgt4?)
• Dependent samples t Compare paired datanext
  week.