Download 4.4 Statistical Paradoxes

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 113
4.4 Statistical Paradoxes
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How do we investigate variation?
 Study all of the raw data…shape and skew?
 Range…
 Quartiles…
 Five-number summary (BOXPLOT or BOX-and-WHISKER)…
 Interquartile range…
 Semi-quartile range…
 Percentiles…
 MAD…
 Variance
& Standard Deviation…
4.4 Statistical Paradoxes
Algebraic Form

Mathematical
Notation
x
x

 ( x  w)
w
x  each value; w  weight
Definition
Sum of all values
x
sum of all values
total number of values
Mean of an entire
population.
Weighted Mean
4.4 Statistical Paradoxes
Algebraic Form

Mathematical
Notation
s
2
Sx
x
Definition
Variance
Sample standard
deviation
Population standard
deviation
Number of data
n
4.4 Statistical Paradoxes
4.4 Statistical Paradoxes
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True Positives: The true positive rate is the
proportion of positive instances that were justly reported
as positive.
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False Positives:

True Negatives:
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False Negatives:
The false positive rate is the
proportion of negative instances that were erroneously
reported as being positive. (Type 1 error)
The true negative rate is the
proportion of negative instances that were justly reported
as negative.
The false negative rate is the
proportion of positive instances that were erroneously
reported as negative. (Type 2 error)
4.4 Statistical Paradoxes
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Simpson’s Paradox…Before
(Has nothing to do with the Simpsons)
A statistical paradox wherein the successes of
groups seem reversed when the groups are
combined. This result is often encountered in
social and medical science statistics, and occurs
when frequency data are hastily given causal
interpretation; the paradox disappears when
causal relations are derived systematically,
through formal analysis.
4.4 Statistical Paradoxes
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
Simpson’s Paradox…After
Example:
A common example of the paradox involves batting averages in
baseball: it is possible for one player to hit for a higher batting
average than another player during a given year, and to do so
again during the next year, but to have a lower batting average
when the two years are combined. This phenomenon, which
occurs when there are large differences in the number of at-bats
between years.
1995
1996
Combined
Derek Jeter
12/48=0.250
183/582=0.314
195/630=0.310
David Justice
104/411=0.253
45/140=0.321
149/551=0.270
4.4 Statistical Paradoxes
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HOMEWORK:
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Pg 182 # 1-10