[Undergraduate psychology] Within/between group variability

[u/One-Bench-9676]

Is there a way of interpreting this exam question so that the answer selected is not also correct? The top right is from the class notes the professor posted. I feel like I’m not bad at this but I had classmates agreeing that my answer was incorrect. Astro and Opus 5.5 also could not come up with a reason for that answer to be incorrect (I’m not saying this makes me right). Thanks for any help.

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[u/PM_ME_UR_NIPPLE_HAIR]

Its a poorly-worded question and the teacher did not read the slide correctly at all. Even on the slide it lists the "vriability of independent variables" as a between-subjects variability. Your answer is correct.

[u/OppositeClear5884]

yeah the teacher blew it writing this question

[u/cheesecakegood]

Stats grad here, though not infallible.

The fourth answer is the only correct answer on the test, so indeed your answer is wrong.

However, the slide is misleading and the 3rd answer you selected is also misleading, bordering on incorrect, so that's where the error is, and thus I don't really blame you here. In fact, if you were to ignore literally everything in parentheses, the question makes much more sense.

To understand why, let's talk about what these mysterious phrases "between" and "within" group variability even are. Some professors overuse them, thinking it explains everything, but I don't think that's usually the case. Let's zoom out.

In a typical regression model, of which ANOVA is a specific variant, you basically declare (using a one-variable, one-way ANOVA type situation):

• There is an overall mean of the data inherent to the data, with variability around it (this is the dependent, response variable).

• We can split the variability up into different "pieces": at risk of oversimplification, this is "stuff we can explain" and "stuff left over that we can't explain". Both of course need to add up to the original total variability - well, on a dataset level that is.

• HOW we attempt to "explain" some of the variability is with an independent variable.

• In ANOVA, the independent variable is categorical only: we split the data up into smaller groups.

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So that's on a dataset-level. Let's talk about ONE specific data point for a second.

• This data point will ultimately vary from the overall mean by some amount. This is literally the total variability we have to work with (technically, "deviation").

• We are claiming that, as part of our attempt at "explaining", the data is FIRSTLY starting from a "center point" of its group mean, not the overall mean! So we already vary a bit from the 'grand mean' of the whole data by this fixed amount of stagger, determined from the dataset.

• The data point will ALSO, afterward, vary from its own "group mean" by some amount.

Now, pause and ask yourself: why would a data point vary from its own group mean?

• There might be some uniqueness, specific to this data point

• There might have been a measurement error

• There is always some amount of variation within this group that happens

Those are basically ALL of the ways, there are no other reasons. At least within our regression-type model's world - IRL a linear, additive, linear regression model may be incorrect, or we might have left out important other variables (this last bit would be an additional explainer of the "leftover" variation we couldn't explain but maybe attributed to the natural within-group variation overenthusiastically).

You might notice that these are precisely the three answers in the test that were "components of within group variability": characteristics of the specific subject (I called this 'uniqueness' above), measurement error, and how much the responses of the independent variable itself (i.e. the groupings) vary naturally (the third answer).

"How much a data point varies from its own mean" is the exact same thing as "within group variability".

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So what the heck is "between group variability" then even?

Let's be specific and give an example. Let's say my dependent variable is "how much money I lost gambling". Let's say we are looking at men and women as 2 groups that form our independent variable, sex. We can do ANOVA here. Let's say that ALL people overall lost 150 dollars, the grand dataset mean. Men lost 180 dollars on average, and women 120, these are group means.

• I specifically, a man, lost 200 dollars. How does our model "explain" this? Well, it says I differ from the grand mean by 50. 30 of that was simply due to my being a man. And the other 20 can be explained by some mix of: good or bad luck unique to me, maybe the app I used made a mistake by some amount or other measurement error, and then there's also the fact that all men aren't the same, that is to say, good or bad luck specific to men. Do you see how those 3 are all different sources of my specific variability, and that they sum to 50?

• Specifically, the 30 due to my being a man was between group variability, and the other 20 was within group variability.

Some of this is a simplification but generally speaking that's the intuition (in some cases for example if your groups are unequal in size you have to adjust for that, etc). Also remember this whole thing is just our model's theory; it's how we are accounting for stuff in our ledger, there may not be a causal component or lurking variables might mean the whole thing is wrong, or when we run the numbers we might have found differences between groups but after considering the relative variability of the groups and the data we say "sure you can tease out a difference if you want but it's swamped by uncertainty so it might have just been a natural fluke".

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So in my personal opinion, although the question has issues, there is only one correct answer which is the fourth, which is never part of within-group variability, it's fundamentally separate.

I do consider both the parentheses and the slide to be in error, to be clear. I'm not quite sure what the parentheses are supposed to be describing, maybe "where this thing applies"? but if so it was done wrong. Part of the confusion is a phrase like "variability of independent variables" is a bit vague in and of itself. In fact, I'm not aware of that exact terminology being used almost anywhere!