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INF 397C
Introduction to Research in Library and
Information Science
Fall, 2003
Day 4
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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4 things today
1.
2.
3.
4.
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NEW equation for σ
Practice exercises
z scores and “area under the curve”
Start to look at experimental design
(maybe)
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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NEW equation for σ
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• σ = SQRT(Σ(X - µ)2/N)
– HARD to calculate when you have a LOT of
scores. Gotta do that subtraction with every
one!
• New, “computational” equation
– σ = SQRT((Σ(X2) – (ΣX)2/N)/N)
– Let’s convince ourselves it gives us the
same answer.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Practice Questions
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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z scores – table values
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• z = (X - µ)/σ
• It is often the case that we want to know
“What percentage of the scores are
above (or below) a certain other score”?
• Asked another way, “What is the area
under the curve, beyond a certain point”?
• THIS is why we calculate a z score, and
the way we do it is with the z table, on p.
306 of Hinton.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Going into the table
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• You need to remember a few things:
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–
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–
–
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We’re ASSUMING a normal distribution.
The total area under the curve is = 1.00
Percentage is just a probability x 100.
50% of the curve is above the mean.
z scores can be negative!
z scores are expressed in terms of (WHAT – this is
a tough one to remember!)
– USUALLY it’ll help you to draw a picture.
• So, with that, let’s try some exercises.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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z table practice
1.
2.
3.
4.
5.
6.
7.
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What percentage of scores fall above a z score of
1.0?
What percentage of scores fall between the mean
and one standard deviation above the mean?
What percentage of scores fall within two standard
deviations of the mean?
My z score is .1. How many scores did I “beat”?
My z score is .01. How many scores did I “beat”?
My score was higher than only 3% of the class. (I
suck.) What was my z score.
Oooh, get this. My score was higher than only 3%
of the class. The mean was 50 and the standard
deviation was 10. What was my raw score?
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Probability
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• 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.)
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Prob. (cont’d.)
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• Life’s 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.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Probabilities
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• 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 that’s 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%.)
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Just because . . .
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• . . . There are two possible outcomes,
doesn’t mean there’s 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 I’ve defined them.) Not equally likely.
• But the odds of a flipped coin being
“heads,” . . . .
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Let’s talk about socks
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Prob (cont’d.)
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• Probability of something happening is
–
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–
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# 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) = ?
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Probabilities – and & or
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• 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 . . .
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Prob (II)
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• 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.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Prob (II)
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• http://www.midcoast.com.au/~turfacts/maths.ht
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– 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."
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Let’s try with Venn diagrams
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Additional Resources
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• 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/tutorials.
html
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–
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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.html
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Think this through.
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• What are the odds (“what are the
chances”) (“what is the probability”) of
getting two “heads” in a row?
• Three heads in a row?
• Six heads in a row?
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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So then . . .
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• 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 b’day) = 1.00
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Happy B’day to Us
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• But we had 43.
• Probability that the first person has a
birthday: 1.00.
• Prob of the second person having the
same b’day: 1/365
• Prob of the third person having the same
b’day as Person 1 and Person 2 is 1/365
+ 1/365 – the chances of all three of
them having the same birthday.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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Sooooo . . .
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• http://www.people.virginia.edu/~rjh9u/birt
hday.html
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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• http://highered.mcgrawhill.com/sites/0072494468/student_view0
/statistics_primer.html
• Click on Statistics Primer.
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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More practice problems
R. G. Bias | School of Information | SZB 562BB | Phone: 512 471 7046 | [email protected]
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