INF 397C Introduction to Research in Library and Information Science Fall, 2003 Day 3

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INF 397CIntroduction to Research in Library and
Information ScienceFall, 2003Day 3
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Calculating percentiles

• From Runyon et al. (2000)

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Standard Deviation

• s SQRT(S(X - µ)2/N)

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Measures of Dispersion

• Range
• Semi-interquartile range
• Standard deviation
• s (sigma)

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Range

• Like the mode . . .
• Easy to calculate
• Potentially misleading
• Doesnt take EVERY score into account.
• What we need to do is calculate one number that
  will capture HOW spread out our numbers are from
  that Central Tendency.
• Standard Deviation

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Hmmm . . .
Mode Range
Median ?????
Mean Standard Deviation
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We need . . .

• A measure of spread that is NOT sensitive to
  every little score, just as median is not.
• SIQR Semi-interquartile range.
• (Q3 Q1)/2

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To summarize
Mode Range Easy to calculate. Maybe be misleading.
Median SIQR Capture the center. Not influenced by extreme scores.
Mean (µ) SD (s) Take every score into account. Allow later manipulations.
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A quick, real-time, example

• How many pets have you ever owned?
• Order
• Freq. dist. (cumu. freq., rel. freq. cumu. rel.
  freq.)
• Histogram
• Measures of central tendency
• Measures of spread

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Graphs

• Graphs/tables/charts do a good job (done well) of
  depicting all the data.
• But they cannot be manipulated mathematically.
• Plus it can be ROUGH when you have LOTS of data.
• Lets look at your examples.

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Your Charts/Graphs/Tables

• http//www.cnn.com/SPECIALS/2003/back.to.school/co
  llege/
• http//www.austin.isd.tenet.edu/k12/docs/ratings/2
  001-2002/227901006.pdf
• http//www.denverpost.com/Stories/0,1413,36257E53
  257E1609345,00.html
• http//www.usatoday.com/money/advertising/adtrack/
  2003-08-24-viagra_x.htm
• http//money.cnn.com/2003/08/07/pf/college/bestcol
  legedegrees/index.htm
• http//www.economist.com/agenda/displayStory.cfm?s
  tory_id2034869
• http//www.usatoday.com/money/perfi/housing/2003-0
  9-07-mym_x.htm
• http//www.bts.gov/publications/national_transport
  ation_statistics/2002/html/table_01_35.html
• Phone numbers memorability vs. dialability!!
• http//www.understandingusa.com/chaptercc12cs26
  0.html

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Some rules . . .

• . . . For building graphs/tables/charts
• Label axes.
• Divide up the axes evenly.
• Indicate when theres a break in the rhythm!
• Keep the aspect ratio reasonable.
• Histogram, bar chart, line graph, pie chart,
  stacked bar chart, which when?
• Keep the user in mind.

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The Normal Distribution(From Jaisingh 2000)
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So far . . .

• . . . weve talked of summarizing ONE
  distribution of scores.
• By ordering the scores.
• By organizing them in graphs/tables/charts.
• By calculating a measure of central tendency and
  a measure of dispersion.
• What happens when we want to compare TWO
  distributions of scores?

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Now, why would I want to do that?

• Is your child taller or heavier?
• Is this months SAT test any easier or harder
  than last months?
• Is my 91 in my Research Methods class better than
  my 95 in my Digital Libraries class?
• Is the new library lay-out better than the old
  one?
• Can more employees sign up, more quickly, for
  benefits with our new intranet site than with our
  old one?
• Did my class perform better on the TAKS test than
  they did on the TAAS test?

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Well?

• COULD it be the case that your 91 in your
  Research Methods class is better than your 95 in
  your Digital Libraries class?
• How?

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What if . . .

• The mean in Research Methods was 50, and the mean
  in Digital Libraries was 99?
• (What, besides the fact that everyone else is
  trying to drop the Research class!)
• So

You Mean
Res. Meth. 91 50
Dig. Lib. 95 99
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The Point

• As I said last week, you need to know BOTH a
  measure of central tendency AND a measure of
  spread to understand a distribution.
• BUT STILL, this can be convoluted . . .
• Well, daughter, how are you doing in grad school
  this semester?

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Well, Mom . . .

• . . . I have a 91 in Research Methods but the
  mean is 50 and the standard deviation is 12. But
  I only have a 95 in Digital Libraries, whereas
  the mean in that class is 99 with a standard
  deviation of 1.
• Of course, your moms reaction will be, Just
  call home more often, dear.

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Wouldnt it be nice . . .

• . . . if there could be one score we could use
  for BOTH classes, for BOTH the TAKS test and the
  TAAS test, for BOTH your childs height and
  weight?
• There is and its called the standard score,
  or z score. (Get ready for another headache.)

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Standard Score

• z (X - µ)/s
• Hunh?
• Each score can be expressed as the number of
  standard deviations it is from the mean of its
  own distribution.
• Hunh?
• (X - µ) This is how far the score is from the
  mean. (Note Could be negative! No squaring,
  this time.)
• Then divide by the SD to figure out how many SDs
  you are from the mean.

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Z scores (contd.)

• z (X - µ)/s
• Notice, if your score (X) equals the mean, then z
  is, what?
• If your score equals the mean PLUS one standard
  deviation, then z is, what?
• If your score equals the mean MINUS one standard
  deviation, then z is, what?

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An example
Test 1 Test 2
Kris 76 76
Robin 52 86
Marty 58 80
Terry 58 90
SX 244 332
µ
Mode, median?
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Lets calculate s Test 1
X X-µ (X-µ)2
Kris 76 15 225
Robin 52 -9 81
Marty 58 -3 9
Terry 58 -3 9
S 244 0 324
/N 61 81
s 9
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Lets calculate s Test 2
X X-µ (X-µ)2
Kris 76 -7 49
Robin 86 3 9
Marty 80 -3 9
Terry 90 7 49
S 332 0 116
/N 83 29
s 5.4
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So . . . z (X - µ)/s

• Kris had a 76 on both tests.
• Test 1 - µ 61, s 9
• So her z score was (76-61)/9 or 15/9 or 1.67. So
  we say that Kriss score was 1.67 standard
  deviations above the mean.
• Test 2 - µ 83, s 5.4
• So her z score was (76-83)/5.4 or -7/5.4 or 1.3.
  So we say that Kriss score was 1.3 standard
  deviations BELOW the mean.
• Given what I said last week about two-thirds of
  the scores being within one standard deviation of
  the mean . . . .

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z (X - µ)/s

• If I tell you that the average IQ score is 100,
  and that the SD of IQ scores is 16, and that
  Bobs IQ score is 2 SD above the mean, whats
  Bobs IQ?
• If I tell you that your 75 was 1.5 standard
  deviations below the mean of a test that had a
  mean score of 90, what was the SD of that test?

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Notice . . .

• The mean of all z scores (for a particular
  distribution) will be zero, as will be their sum.
• With z scores, we transform raw scores into
  standard scores.
• These standard scores are RELATIVE distances from
  their (respective) means.
• All are expressed in units of s.

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Practice Questions
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Probability

• Remember all those decisions we talked about,
  last week.
• VERY little of life is certain.
• It is PROBABILISTIC. (That is, something might
  happen, or it might not.)

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Prob. (contd.)

• Lifes a gamble!
• Just about every decision is based on a probable
  outcomes.
• None of you raised your hands last week when I
  asked for statistical wizards. Yet every one
  of you does a pretty good job of navigating an
  uncertain world.
• None of you touched a hot stove (on purpose.)
• All of you made it to class.

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Probabilities

• Always between one and zero.
• Something with a probability of one will
  happen. (e.g., Death, Taxes).
• Something with a probability of zero will not
  happen. (e.g., My becoming a Major League
  Baseball player).
• Something thats unlikely has a small, but still
  positive, probability. (e.g., probability of
  someone else having the same birthday as you is
  1/365 .0027, or .27.)

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Just because . . .

• . . . There are two possible outcomes, doesnt
  mean theres a 50/50 chance of each happening.
• When driving to school today, I could have
  arrived alive, or been killed in a fiery car
  crash. (Two possible outcomes, as Ive defined
  them.) Not equally likely.
• But the odds of a flipped coin being heads, . .
  . .

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Lets talk about socks
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Prob (contd.)

• Probability of something happening is
• of successes / of all events
• P(one flip of a coin landing heads) ½ .5
• P(one die landing as a 2) 1/6 .167
• P(some score in a distribution of scores is
  greater than the median) ½ .5
• P(some score in a normal distribution of scores
  is greater than the mean but has a z score of 1
  or less is . . . ?
• P(drawing a diamond from a complete deck of
  cards) ?

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Probabilities and or

• From Runyon
• Addition Rule The probability of selecting a
  sample that contains one or more elements is the
  sum of the individual probabilities for each
  element less the joint probability. When A and B
  are mutually exclusive,
• p(A and B) 0.
• p(A or B) p(A) p(B) p(A and B)
• Multiplication Rule The probability of
  obtaining a specific sequence of independent
  events is the product of the probability of each
  event.
• p(A and B and . . .) p(A) x p(B) x . . .

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Prob (II)

• From Slavin
• Addition Rule If X and Y are mutually exclusive
  events, the probability of obtaining either of
  them is equal to the probability of X plus the
  probability of Y.
• Multiplication Rule The probability of the
  simultaneous or successive occurrence of two
  events is the product of the separate
  probabilities of each event.

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Prob (II)

• http//www.midcoast.com.au/turfacts/maths.html
• The product or multiplication rule. "If two
  chances are mutually exclusive the chances of
  getting both together, or one immediately after
  the other, is the product of their respective
  probabilities.
• the addition rule. "If two or more chances are
  mutually exclusive, the probability of making ONE
  OR OTHER of them is the sum of their separate
  probabilities."

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Lets try with Venn diagrams
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Additional Resources

• Phil Doty, from the ISchool, has taught this
  class before. He has welcomed us to use his
  online video tutorials, available at
  http//www.gslis.utexas.edu/lis397pd/fa2002/tutor
  ials.html
• Frequency Distributions
• z scores
• Intro to the normal curve
• Area under the normal curve
• Percentile ranks, z-scores, and area under the
  normal curve
• Pretty good discussion of probability
• http//ucsub.colorado.edu/maybin/mtop/ms16/exp.ht
  ml

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Think this through.

• What are the odds (what are the chances) (what
  is the probability) of getting two heads in a
  row?
• Three heads in a row?
• Three flips the same (heads or tails) in a row?

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So then . . .

• WHY were the odds in favor of having two people
  in our class with the same birthday?
• Think about the problem!
• What if there were 367 people in the class.
• P(2 people with same bday) 1.00

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Happy Bday to Us

• But we had 43.
• Probability that the first person has a birthday
  1.00.
• Prob of the second person having the same bday
  1/365
• Prob of the third person having the same bday as
  Person 1 and Person 2 is 1/365 1/365 the
  chances of all three of them having the same
  birthday.

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Sooooo . . .

• http//www.people.virginia.edu/rjh9u/birthday.htm
  l

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• http//highered.mcgraw-hill.com/sites/0072494468/s
  tudent_view0/statistics_primer.html
• Click on Statistics Primer.

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Who wants to guess . . .

• . . . What I think is the most important sentence
  in S, Z, Z (2003), Chapter 2?

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p. 19

• Penultimate paragraph, first sentence
• If differences in the dependent variable are to
  be interpreted unambiguously as a result of the
  different independent variable conditions, proper
  control techniques must be used.

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Homework

• Keep reading.
• See you next week.