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Commit 36f07062 authored by Robert Ricci's avatar Robert Ricci

Expand further on notes

parent dcb5006d
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\1 Today: Ratio games
\2 Primarily: explain why comparing ratios is not okay
\2 Understand the difference between rates and ratios
\1 Primary source for today: ``When Can You Meaningfully Add Ratios and Fractions?'' by Mochon, For the Learning of Mathematics, 1993
\2 \url{http://www.jstor.org/stable/40248090}
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the ratio
\2 \textit{When is this assumption broken?}
\1 Standard method for adding fractions:
\1 Definitions: rates and ratios:
\2 Ratio: Relationship (quotient) between two values
\2 Rate: Ratio reduced to a unit value (eg. per 1 hour, not $N$ hours)
\1 Standard method for adding fractions (Method \#1)
\2 $\frac{a}{b} + \frac{c}{d} = \frac{ad+cb}{cd}$
\1 Example A: Basketball player
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\2 $\frac{3}{12} + \frac{2}{12}$ when added as fraction does \textbf{not} get the right answer
\2 Method \#2: Add both ``tops and bottoms''
\3 $\frac{a}{b} + \frac{c}{d} = \frac{a + c}{b + d}$
\3 This is the vector math way of adding\ldots
\2 Simplifying fractions to $\frac{1}{4}$ and $\frac{1}{6}$ breaks this strategy
\1 Example B: Gas mileage
......@@ -81,9 +87,32 @@
is one quarter black, one quarter white. What percentage of
the paint in the combined can is white? (unanswerable)
\1 General form: $\frac{A}{B}$
\2 $\frac{A_1 + A_2}{B_1 + B_2}$
\2 $\frac{A_1 + A_2}{\frac{A_1}{A_1+A_2}B_1 + \frac{A_2}{A_1 + A_2}B_2}$
\1 So: When adding ratios, you have to determine which of these methods is appropriate
\2 ``Just adding'' is not often not the correct strategy
\2 \ldots and averaging using adding, of course
\2 And this is why ratio games work
\2 \textit{So, where can this get you into trouble?}
\3 Adding / averaging ratios where the amount of work done is different
\3 Intentionally weighting the amount of work done
\3 Averaging speedups
\3 Using the wrong one as the base
\3 Incomparable base---can't compare speedups
\3 Combining ratios or rates with different units
\3 Comparing percentages of different numbers
\1 Rules according to Jain
\2 If one is better on all benchmarks, not possible to reverse conclusion
\2 \ldots but relative performance can be overstated by picking the wrong
base
\2 If there are inconsistent results, contradictory conclusions can often
be drawn
\3 If metric is LB, use your metric as base
\3 If metric is HB, use opponent
\3 Lengthen benchmarks that are helpful to you
\1 Derive conditions using Table 11.10
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