Communicating Quantitative Information

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Communicating Quantitative Information

• Diagrams
• Sampling issue
• Risk Communicating Risk
• Smoking. Hormone Replacement Therapy
• Dimension
• Homework Prepare/Design diagram/chart. Postings

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General comment

• ASK QUESTIONS
• There often is more than way to
• explain a problem
• do a problem

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Fitting together puzzle

• 400 freshmen, 250 taking math

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More information

• 400 freshmen, 250 taking math
• 100 taking neither math nor science so out of
  the not taking math part
• Note this part is 400-250 is 150

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out of not math part

• Dark blue represents 100 taking neither

50 taking science!
100
250
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next information

• Of the ones taking math, 40 are taking science
• .40 times 250 is 100
• Divide the freshmen into those taking math and
  those not
• Final answer 100 50 taking science

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Special Survey

• Will come back to topic of Sampling
• Accuracy (confidence, error) of tests, surveys
  depends on
• quality of sample
• size of sample
• My sister asked me can young people identify
  Einstein?
• Use my students / sample of my students
• Last Spring
• students responding to request to take survey in
  both my classes
• Last Spring quantity was pretty small (14 11)

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Preview

• Margin of error
• Claim actual result (for whole population) is
  within certain limits (answer plus/minus MoE)
• Confidence
• confidence that this particular sample is not so
  unusual as to make results wrong where wrong
  means the actual result is outside the margin of
  error.
• Generally, the means (averages) of samples are
  distributed normally around the mean of the whole
  population and the SD of this distribution is
  smaller (tighter) than the SD of the distribution
  for this quantity for the whole population.

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Thought experiment.

• Want to get average height of people in the
  class.
• Claim impossible to measure everyone, so use a
  sample.
• Only time to measure 6 people
• 62 72 68 66 69 71 for a sample mean of 68
• Statistics say (IF the sample is random) then we
  can be 95 confident that the mean of the whole
  class is between61.4 and 74.6
• will spend some more time on this

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Thought experiment

• Want to know favorable rating of the President
  from whole USA population.
• Ask a sample
• p (proportion) of sample view President
  favorably.
• Want to make a statement with 99 confidence
• Formula for the margin of error, call it E we
  can be 95 sure that the proportion of the whole
  population favorable is within p-E and pE
• If we want to be 99 sure, then formula will give
  a bigger margin of error, call this F we can be
  99 sure that the proportion of the whole
  population favorable is within p-F and pF.

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Another way

• Random sample of size N means that each person in
  the whole population equally likely to be in the
  sample
• There are many samples of size N
• The results of a sample of size N vary, but
• Some samples of size N are very different from
  the whole population, but most arent
• In most cases, the sample result will be close to
  the result for the whole population
• What do I mean by close? Within the margin of
  error
• What do I mean by most? This refers to the
  confidence interval (19/20, 99/100)

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Formally

• The averages of samples of size N are normally
  distributed
• The average (mean) is the mean of the whole
  population
• The standard deviation is smaller by a factor of
  square root of N
• Think of narrow mountain
• To half (reduce to ½ what it was) the required
  margin of error, you need to quadruple (4) the
  sample size

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Note

• The size of the whole population does not enter
  into these calculations!

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Back to special survey

• Objective to find out if college students know
  about Einstein
• Extra credit used as motivation
• (didnt work that well for you?)
• Is this factor independent of what is being
  tested?????

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Quality of sample

• Does not mean how good you arein any way.
• Does mean how representative of general college
  population.
• The former class was practically all seniors.
  That class and all since tend to be journalism,
  history, political science majors
• Students who don't have specific required
  mathscience courses
• These factors mean sample is not representative
  of college population!

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Quality of sample

• Opportunity
• subjects available to me in my classes. Are
  they/you typical of 'young people'?(My sister
  thought yes.)
• Response bias
• students who took up offer. Are they/you more
  likely to 'know Einstein' than those who didn't.
• higher level of general curiosity
• diligence at obtaining extra credit

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Tester reliability

• I was generous in categorizing answers as
  correct.
• Two questions considered separately
• 23 out of 26
• 24 out of 26

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Reporting

• Confidence at level alpha that actual proportion
  is within error of tested proportion
• More confidant at larger interval
• I am 95 confident (chances are only 1/20 that
  this is wrong) that the actual proportion is at
  least 83 that knows Einstein.
• I am 99.5 confident (chances are 1/200 that this
  is wrong) that the actual proportion is at least
  78

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Formulas

• Based on the finding that means of samples are
  close to normally distributed with standard
  deviation function of tested proportion and size
  of sample.
• One-tailed test (just checking one side because
  tested proportion close to 1)

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Correlation (again)

• Two variables
• common examples
• height weight
• mortality set of health risks factors (e.g.,
  smoking history)
• Are the two correlated? Does value of one predict
  some of value of the other?

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Linear model

• Linear line.
• X and Y (standard names for two
  variablesvariables, values that vary!)
• Y a bX
• if a 0, bgt0 if agt0, bgt0

Note negative values of X and/or Y may or may
not be valid
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Linear Model

• agt0, blt0
• (This will be basis of negative correlation.
  Still a relationship, but in the negative. As X
  gets big, Y gets small.)

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Cab fare

• (Numbers are not right, but the idea is)
• 3 to get in
• 2 every ¼ of a mile
• Y is the fare/total cost (not including tip!) and
  X is distance, given in miles rounded up to the
  nearest quarter mile.
• Fare 3 2(miles 4)
• Example rode ½ a mile. Fare is 3 22 7

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(rough) graph of cab fare

• Points
• (0,3), (1,8)

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Aside

• Units miles versus quarter miles, miles versus
  feet versus kilometers versus need to be
  understood. Some stories/calculations/experiments
  succeed or fail based on getting the units
  right!
• space flight that failed due to
  misunderstanding/lack of agreement on units.

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Correlation

• Two variables, X and Y.
• Make a graph (computer program does not make a
  graphyou think about a graph)
• Process determine line that would be the best
  fit
• defined as minimizing sum of the squares of the
  distances from the line ('least squares')

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Excel example

• List two sets of numbers
• Graph using scatter plot
• Use
• correl(B2B8,C2C8) .96927

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Other models

• other relationship quadratic, log,
  exponential, etc.

Say you know deer population at two points in
time. Is/will the growth be linear or
exponential?????
Pop.
Time
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Caution

• Correlation is not cause
• coincidental
• both caused by other factor
• Cause is not.absolute determination.
• other factors

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Terminology (reprise)

• False positive wrongly say someone/something has
  condition.
• False negative wrongly say someone/something
  does NOT have condition when, if fact, he or she
  or it does
• Control group group in experiment that does not
  have treatment.
• treatment group group in experiment that does
  have treatment

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Double-blind study

• Randomly assign subjects to
• treatment
• control (may give placebo)
• Subject does not know which.
• Tester/evaluator does not know which
• See what happens. Time period may be long.
• Smoking cannot be studies using a double-blind
  study!

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Retrospective study

• Of the people who did/have X, ask how many did Y?
• Not as reliable.
• Also need to study group that do not have X.
• 85 of people with lung cancer report that they
  smoked.
• (How many George Burns are there?)

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Smoking and Lung Cancer

• Strong correlation
• more smoking increases chances of lung cancer
• smoking comes before the cancer
• many different studies
• Women's incidence of lung cancer went up when
  women started smoking
• Incidence going down in groups decreasing smoking
• Biological evidence
• nicotine experiments with animals
• lab study of lungs, blood, blood pressure, etc.

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What's likely to kill you

• http//www.reason.com/blog/show/128501.html

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Data dimension

• The data that is worth presenting in graphics
  form (as opposed to clear text) is generally
  complex multi-dimensional.
• Dimension measurement, extent, reach
• the degree of manifoldness time has 1 dimension,
  space has 3.
• Edward Tufte (and others)
• don't give data dimensions it doesn't have. Don't
  use 3D for bar graphs
• read (borrow) his books

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As with 3D bar charts when you only have points,
avoid rainbow, when the data is
one-dimensional(Note shades of blue chart
better for color-blind visitors.)
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Sign and dimension

• Previous example contrasted height on land with
  depth of the ocean.
• Next chart is problematic one dimension of
  shading used for negative and positive values.
• is this a problem?

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Dimension

• Identifying dimension is important and may not be
  obvious.
• Challenger disaster problems were associated
  with temperature.
• Values 'along' a dimension may be discrete or
  continuous or forced into discrete (quantized)
  categories or discrete but many values
• weight and height are continuous, but we round
  (down or up) to a standard unit
• registration is done by grouping credits earned

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Recent news

• Large study found that low-fat diet did not
  appear to have statistically significant effects
  on mortality due to cancer or heart disease
• Subjects were older women (post-menapausal women)
• Posting opportunity project I opportunity find
  more than one article and write good (better)
  report.
• Will cover more on this topic.

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Recent examples

• China/world water
• http//www.nytimes.com/interactive/2007/09/28/worl
  d/asia/choking_on_growth_2.htmlstory4
• Mortgage foreclosures/Atlanta
• http//www.nytimes.com/2007/07/09/business/09aucti
  ons.html
• Iraq neighborhoods
• http//www.nytimes.com/interactive/2007/09/06/worl
  d/middleeast/20070907_BUILDUP_MAIN_GRAPHIC.html

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Small multiples

• Several (many) graphs/diagrams of the same format
  

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Homework

• Identify complex topic (such as health risks,
  sports records, voting)
• multiple dimensions/factors multiple categories
  timeline?, geography?
• Find reputable source (more than one source even
  better)
• Determine critical findings
• determine audience
• Design/build diagram (chart, graph)
• Bring to class to present AND to turn in. Be
  professional!
• as appropriate, consider using 'small multiple'
  idea as done for 31 days
• as appropriate, consider charts presented on
  health risks
• This could be topic for your project I paper
  charts
• Continue postings.