Name of this fallacy and how to reach conclusion

Question

While handling some demographic data, I stuck in a position where (I did not disclose the actual data set and whom it is concerning, therefore I replace it with hypothetical data) I could not reach a conclusion.

• Suppose we have 2 class of people; A and B.
• We have two ideologies/ political opinions, Alpha and Beta.

The number of individual members identify or affiliate as follows:

$$\begin{array}{c|c|c|} & \text{Ideology- Alpha} & \text{ Ideology Beta} \\ \hline \text{Group-A} & 90 & 10 \\ \hline \text{Group-B} & 70 & 30 \\ \hline \end{array}$$

In this situation; News source 1 interpret the data as

"Only 30% members of group-B stands for ideology-Beta. Members of group-B does not want ideology-Beta"

And another News Source 2 interpret the data as

"About 75% members affiliated to the Ideology Beta, are supported by group-B, therefore Ideology beta embraces the rights and demands of people belonging to societal category group-B" .

Now, my question is; which one source to trust? using the exact same opinion poll, two sources are generating very much opposite narrative, which are incredibly confusing. Both the calculations are technically correct, and may be partly showing different facets of the truth. But how would I reconstruct the actual story or draw a conclusion from those data? Which one source is biased? or both representation is biased? And what is the name or documentation for this kind of bias?

Bonus question:

What should be the appropriate mode to represent such kind of data?

Courtesy: I have copied the code to format table from here https://math.meta.stackexchange.com/questions/4240/how-do-i-insert-a-table-when-asking-a-question

Answer 1

News source 1 gives the distribution of ideology conditional on membership of Group B. News source 2 gives the distribution of group membership conditional on Ideology beta. These statements each invert the argumet and conditioning variables in the other statement an both are correct in a purely quantitative sense.

The problem here arises in the latter source with the inference that "... therefore Ideology beta embraces the rights and demands of people belonging to societal category group-B". The assertion about the content of Ideology beta in this statement may or may not be true, but either way it does not follow as a sensible inference merely from the fact that there is a higher representation of Group B amongst the group committed to ideology beta. All that follows from this conditional distribution is the fact that Ideology beta is relatively more popular in Group B than in Group A. Thus, what can be said as an overall summary, which is reconcilable on the basis of these being reversed conditioning statements is that Ideology beta is not popular overall with Group B, but it is relatively more popular in that group than it is in Group A.

Answer 2

You can consider this as cherry picking. It is a narrow selective representation of the data that favors only a particular viewpoint.

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At the same time both news sources make also fallacious statements in their second sentences (but as explained below, I don't believe that these are the reason for the different viewpoints and path to resolve a paradox)

"Only 30% members of group-B stands for ideology-Beta. Members of group-B does not want ideology-Beta"

"About 75% members affiliated to the Ideology Beta, are supported by group-B, therefore Ideology beta embraces the rights and demands of people belonging to societal category group-B"

Based on the data you can't argue that these conclusions are true. It is not neccesarily true that members of group-B don't want ideology beta. And it is not neccesarily true that ideology beta embraces the rights and of group-B. (for example imagine votes in a mock election where the response of people is not aligned with what people want or what ideologies support).

But it is not that the conclusions are contradictions or and that they are neccesarily false, or that they are false because of the fallacious reasoning. It can be true that ideology $\beta$ favours group B yet most of group B don't want ideology $\beta$. The data would be a likely outcome for that situation and the data does increase the likelihood of the theories/statements. The fallacies here may be considered a fallacy like jumping to conclusions or inference-observation confusion.

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These fallacies are not the core reason for the different narratives (The news sources just exaggerate their views of the data as proof for stronger statements, but their views remain different).

When we strip the two news sources from the exaggerated language. Then we still get two different viewpoints

"Only 30% members of group-B stands for ideology-Beta."

"About 75% members affiliated to the Ideology Beta, are supported by group-B"

We can describe multiple effects in the data. There are main effects and there are interactions. This can be seen in the following diagram.

News source 1 stresses the main effect that ideology alpha is overall more popular.

News source 2 stresses the interaction effect that there is a difference in affiliation to ideology alpha and beta when we consider the group membership A and B.

Answer 3

I broadly agree with many aspects mentioned in previous answers.

I just want to state that the core of both statements in the question is true and correct, namely

Ideology-beta is only supported by a minority within Group-B,

and

Group-B members are the majority among Ideology-beta supporters.

There is no contradiction here, and looking at the data there is no reason for confusion. Now you have worded both statements in a way that they look contradictory by somehow suggesting that Group-B is generally against Ideology-beta in one case and that there is some "overall match" between Group-B and Ideology-beta in the other case. Neither of these is backed up by the data (let alone the made up and rather nonsensical pseudo-causal explanations), and keeping in mind that it is well possible that a majority within a minority class held by a certain group can be a minority within that group should remove all confusion. Of course the problem that makes this question interesting remains, which is that this situation can be used by manipulators to promote contradictory points. But this is just as it is. Data hold limited information and can be used for manipulation by selecting and misinterpreting certain aspects of the data.

What should be the appropriate mode to represent such kind of data?

See the two correct statements above. (I'm not saying that this is generally how it should be done as it will always depend on the audience - the data themselves are so simple that one may well let them speak for themselves, but if people are just interested in majorities it's what I said.)

By the way I think a fallacy (maybe not the only one) behind the confusion/manipulated interpretations is the idea that the majority within a group should be considered as "everything"/"dominating"/"the only part worth taking into consideration". Not sure how that's called, but basically the first statement sounds as if the 70% of Group-B supporting Ideology-alpha were actually 100% (or 100% of "what counts" within the group), and the second statement gives the 75% a general authority among Ideology-beta. And if it were indeed 100%, the two statements were indeed technically contradictory (unless we allow to talk about 100% out of zero in a situation with no support for one of the ideologies whatsoever).

PS: I'd probably call this something like "The-majority-is-all-there-is fallacy". Looking a bit around, I find a few somewhat related fallacies that are not quite the same, closest maybe the Fallacy of Composition.

Answer 4

Update: To be very precise; this is already known paradox in probability theory; which is called the confusion of the inverse.

See the Marble problem discussed below. Where probability of a cubic marble to be pink; is asymmetrical to a pink marble to be cubic.

$P(A|B) ≠ P(B|A)$ ..... (Not necessarily equals to, except the values may coincide)

This is a very common fallacy in conditional probability.

Using Bayesian theorem;

$$P(\text{Pink}|\text{Cube})= \frac{P(\text{Cube}|\text{Pink})P(\text{Pink})}{P(\text{Cube})}$$ $$=\frac{75\% . 20\%}{50\%}=30\%$$

So using Bayesian theorem, the 2 representations of the data are consistent, just they are answer to two different questions.

As user Scortchi - Reinstate Monica suggested, this can be described as a subset of affirming the consequent fallacy, a fallacy related to cause and effect.

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I would call it a generalization (that may lead to stereotype) or misattribution (esp. culture bias)

I would interpret is this way.

Suppose I have 2 kind of crop plants : Rice and Wheat

Now I have a virus, which affect only 10% of the rice plant and 30% of the wheat plants.

The statement "among wheat plants, most are are somewhat immune to the virus, only 30% of them are susceptible"- this is an answer to a different question.

The statement "among wheat plants and rice plants, wheat plants are 3 more susceptible to the virus"- this is also true, and an answer to a separate question.

Now, using one of the answer to nullify the other, is not correct, because they are answers to different question.

Giving more mundane example.

Suppose we have some marbles; which can be categorized according to colour (blue and pink); or by shape (round or cubic).

$$\begin{array}{c|c|c|} & \text{BLUE} & \text{PINK} \\ \hline \text{ROUND} & 90 & 10 \\ \hline \text{CUBIC} & 70 & 30 \\ \hline \end{array}$$

As an overall, if a marble is a cube, it is not very much likely to be pink. But if a marble is pink, it is very much likely to be a cube.

Both information can be useful while making government policy etc.

Suppose Group-1 is a non-vegetarian eater. Group-2 prefers vegetarian food only.

Most members of both group prefer a lecture to be delivered in English, but some members from both group prefer a lecture be delivered in Hindi.

$$\begin{array}{c|c|c|} & \text{English} & \text{Hindi} \\ \hline \text{Non-veg} & 90 & 10 \\ \hline \text{Veg} & 70 & 30 \\ \hline \end{array}$$

From this given data, it would be aweful to generalize that vegetarian eaters are necessarily Hindi speakers; but it can be said if you arrange a comedy show only for the Hindi speakers from the people in the given sample, you may have to arrange more food coupons for vegetarian people.

Answer 5

I see in this problem a symptom that is rarely mentioned in textbooks, for the simple reason that it’s not really about statistics per sé, but more about scientific method and philosophy of science.

I’m answering this on my phone, I’ll see if I can find a nice paper that I vaguely remember reading about this, which explains this in detail when I have my computer after the weekend. I’ll summarize the general concept here.

The problem is not in the statistics - both analyses are mathematically correct.

The problem lies in the real-world inference. What textbooks describe is typically: Statistical Hypothesis > Statistical Test or Analysis > Statistical conclusion.

But what we are doing in real-world work is: Substantive Hypothesis > Statistical Model > Statistical Hypothesis > Statistical Test or Analysis > Statistical Conclusion > Substantive Conclusion.

In some work in real life, especially exploratory work, we may skip the first step, and sometimes we replace the last step by a more detailed investigation. And different people may make their own modifications for their own applications. For this answer, none of that is relevant: In any case, and with any modification, there has to be some translation from statistics to real-world.

That’s what’s going on here. Both reports have chosen a different substantive hypothesis, the same model, but a different analysis (as dictated by their respective substantial hypotheses) and therefore end up at a different substantive conclusion. No amount of math can correct for this difference: they answer different questions.

These conclusions are not even mutually exclusive, and neither is exhaustive (I.e. they make no mention of the “other” viewpoint, so that information is left unextracted from the available evidence).

The above suggests the following substantive hypothesis: both analyses were either performed by an incompetent analyst, or deliberately designed to support a certain bias, or a combination of these. I’ll let you all decide for yourself on the answer to that ;)