Proportions and Percentages Help

[episcopoandrew]

Hi all,
I've noticed that of the questions I've been missing on my preptests recently, a lot of them have stimuli talking about percentages and proportions. Does anyone have any resources or drills and/or strategies to improve on these questions? I remember reading a chapter on this subject in the Powerscore LR Bible. The version of the Manhattan LR book I used did not have a chapter on percentages and proportions. Maybe past versions had a chapter on it? Either way, I need tips, help, any of the above on these questions. I do know that when the stimulus mentions the proportion going up, it does not necessarily mean that there are more of one thing because it's possible that one of the two groups decreased, thus increasing the proportion of the other group.

Any help would be appreciated.

[ohthatpatrick]

Here is an excerpt from a draft I wrote for the most recent edition of the book (but it didn't get used in that edition).

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Proportions vs. Actual Numbers
The author presents proportional information about groups but makes a numerical conclusion, failing to consider how the different sizes of those groups could invalidate the numerical conclusion.

EXAMPLE:
100% of bags with bombs will trigger an alert.
1% of bags without bombs will trigger an alert.
From this, an author erroneously concludes, "99% of the time you hear an alert, there’s a bomb.

Think about the real numbers of the groups being discussed. How many bags will have bombs? Let’s say 10 per day, which is still horrifyingly high. How many bags will not have bombs in them? Let’s say 1000 per day.

With these numbers, we’ll get 10 alerts with actual bombs in the bag (100% of 10 = 10).
And we’ll get 10 alerts with no bombs in the bag (1% of 1000 = 10).

So there will be 20 alerts, but for only 10 of them will a bomb be present. That’s 50% of alerts, not 99%.

EXAMPLE
AN author concludes that left-handed people are not more accident prone than right-handed people. After all, more household accidents are caused by right-handed people.

We could make a similarly fallacious argument by saying that “Tesla’s are not more accident prone than Honda’s. After all, more accidents involve Honda’s than Tesla’s.”

If you know that Honda’s are the most common car on the road and that Tesla’s are new and incredibly rare, it’s a pretty obvious fact of life that more accidents will involve Honda’s. That doesn’t mean that a Honda is more accident prone.

That term indicates the idea of relative frequency. What percent of Tesla drivers get into accidents? What percent of Honda drivers do? That’s the data we would need to evaluate whether one type of car (or driver, depending on how we interpret causality) is more prone to accidents.

Similar to Honda’s and Tesla’s, there are way more right handed people in the world than left handed people. (Congratulations, left handed people, you come off looking pretty cool in this analogy) So it’s not that interesting to say that more household accidents are caused by right-handed people.

If it turns out that 16% of left handed people, as opposed to 8% of right handed people, have had a household accident in the last month, then it looks like left handed people are twice as prone to accidents.

EXAMPLE
At Morris High School, ¼ of part-time teachers quit during the first year. 1/3 of full-time teachers do. So can we conclude, as an author did, that more full-time than part-time teachers at Morris now quit during their first year?

Make sure you register the switch from proportion (fractions) to real numbers (more teachers).

Even though 1/3 (33.3%) is a bigger fraction than ¼ (25%), we don’t know the size of the groups being compared: 33% of 100 is not bigger than 25% of 1000.


Percent Change
LSAT authors who present a percent change almost always assume a change to the numerator, and fail to consider a change to the denominator.

Let’s say that in a room, there are 4 girls and 6 boys. (40% female, 60% male)
A few minutes later, the room is 50% female, 50% male.
What changed?

The short, real answer is who knows?! There are so many ways we could end up with a room that’s ½ female, ½ male.

LSAT authors, however, will present the facts a little differently and force their thinking into one possibility.

Last year, the clientele at our furniture store were 40% female. Since the hiring of our new CEO, our clientele has been 50% female. Clearly, women are increasingly drawn to our brand by the public persona of our new CEO.

There are actually several different possible flaws with this argument. Hopefully, you immediately became skeptical when you heard the author’s conclusion explain the reason for the percent change in his conclusion.

What are the two questions we ask ourselves to address the pressure points of a causal explanation?
1. Some OTHER WAY to explain the curious fact (premise)?
2. PLAUSIBILITY of AUTHOR'S WAY (conclusion)?

We could definitely attack the plausibility of the author’s story by calling out assumptions such as, “female clientele are aware there is a new CEO”. And we could present alternative stories for why the clientele has shifted demographics, such as, “It’s not the new CEO, it’s the popular commercials that recently ran that were targeted more at a female audience.”

But a third option here is to call into question the implicit assumption that there are more females shopping at this store this year.

Remember the room with 4 women, 6 men:
We could make it a 50/50 gender split by adding 2 females, making it 6 women, 6 men.
But we could have also gotten the 50/50 split by subtracting two males, making it 4 women, 4 men.

The percent of women comes from the fraction of (# of women) / (total people)

In the first scenario, we went from 40%, (4 women)/(10 total), to 50% (6 women)/(12 total) by increasing the numerator, which indirectly increases the denominator.

In the second scenario, we went from 40%, (4 women)/(10 total), to 50% (4 women)/(8 total) by decreasing the denominator, after we removed two males.

It is this second scenario that LSAT authors always fail to consider. When the percentage of Total X that is Part A rises, LSAT authors assumes that we have more Part A to thank. But it’s possible that the higher % of Part A is just caused by a lower % of everything else.


Gambler’s Fallacy
Since the last five flips of this coin have landed on heads, the next flip is highly likely to be tails.

Sounds pretty reasonable, because we feel that we’re due for tails at this point. But a flip of a coin is an isolated event. Each side of the coin has a ½ probability (neither likely nor unlikely). The history of previous flips in no way affects the likelihood of the current flip.

In the train industry, there is an average of one major accident per five years. So if a certain train line has just had a major accident, you’re guaranteed at least a few years of a safe window for travel on that train line.

This author is just interpreting the idea of average frequency way too literally. An average of one accident per five years still allows for two accidents in the same year. Given an average rate, we can’t treat that rate as a guaranteed interval.

Proving Overlap
All A’s are B’s. Some B’s are C’s. Thus, some A’s are C’s.

Remember Venn diagrams? They would show you members of two separate groups with an overlapping region that contains people in both groups. We’ll discuss this math in a later chapter, but we’ll briefly plant the seed right here.

All US Presidents have been male. Some males have landed on the moon. Thus, some US Presidents have landed on the moon.

The feel of these arguments is that there is some shared quality in each of the two premises. In one premise the shared quality relates to A. In the other premise, it relates to C. The conclusion attempts to prove that at some point, A and C must both occur.

Males are related to US Presidents. Males are related to moon landings. Hence, there must be some US President – Moon Landing overlap.

There are very narrow situations in which it’s actually fair to draw this sort of mandatory overlap conclusion.
"Most A's are B" + "Most A's are C" = Some overlap between B and C
or
"All A's are B" + "any type of fact involving A and C" = Some overlap between B and C

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Here’s a quick recap of the big statistical flaws mentioned:
Proportion vs. Number – bigger proportion doesn’t necessarily mean a bigger number (need group sizes)
Percent Change – a percent increase/decrease for a Part doesn’t necessarily mean there’s more/less of that Part (could just be the result of other parts of the pie shrinking or growing)
Gambler’s Fallacy – do not apply ratios, averages, probabilities in a literal fashion or consider them to be influenced by prior events
Proving Overlap – just because two things are both associated with a third thing doesn’t mean you can prove that the two things have an association with each other. Know your legal overlap inferences so that you can guard against the phony ones.

[episcopoandrew]

That was super helpful. Thank you!