4.5 Explains ANOVA
This essay was written by ChatGPT (4.5), based on a rather detailed prompt (and follow-up conversations). The problem being discussed is taken from a book published by Microsoft, authored by Wayne Winston. I have made minor edits for clarity, but I am not the author of this essay. I also asked ChatGPT to write (in italics) about how to start to think about writing good prompts.
The point of this essay is to try and show you how LLM's can generate lessons that help you understand a topic better. Used as a complement to a textbook, your own efforts and what you learn in your classrooms, LLM's can become a great learning companion - but that's not an excuse to kick back and chill! Quite the contrary, it is a way to be able to study more, and better.
The essay follows below:
Imagine yourself as a statistics student, comfortably seated in a quaint coffee shop, enjoying a steaming cup of tea. Just as you ponder the complexities of life (and statistics!), a bookstore owner from next door walks in, a puzzled expression on her face. Recognizing your stats textbook, she approaches with a hopeful smile.
“I've been told statistics students are quite helpful!” she begins warmly. “Perhaps you can help me make sense of my sales data? You see, I sell the same book at the front, middle, and back of my bookstore, and I have data from two different stores. My question is simple: where should I place my books to maximize sales?”
She hands you two neat sets of sales figures:
Store 1 Sales Data:
PositionSalesAverageFront7, 20, 8, 8, 29.0Middle3, 19, 1111.0Back2, 16, 25, 1314.0
Store 2 Sales Data:
PositionSalesAverageFront7, 10, 8, 9, 119.0Middle10, 11, 1211.0Back12, 13, 15, 1614.0
Interestingly, averages alone don't fully explain the story. Not only do they not explain the story, they in fact make the story weirder in this case. Why weirder? Because both sets of averages are exactly the same.
But hey, ask yourself this: with store one, how confident are you that sales next week will be around 9 for the front of the store? They have been as low as 2, and as high as 20! And how confident are you about the front shelves of store 2? Now do you see the problem?
While averages clearly represent central tendencies, they don't reveal how consistently these averages reflect actual performance. Consistency—or lack thereof—is captured by variance:
Variation typeStore 1Store 2Within-group variation574.022.0Across-group variation55.755.7
Store 1: Notice how high the within-group variation is (574.0) compared to the across-group variation (55.7). This indicates substantial inconsistency in sales within each placement. Despite having distinct averages, this large within-group variation means we cannot confidently conclude that the placement significantly impacts sales.
Store 2: Here, the within-group variation (22.0) is considerably lower than the across-group variation (55.7), clearly indicating that differences between positions are meaningful. ANOVA confirms this intuition, making a strong statistical case for placing books at the back. Now, here it is easy to see that 55.7 is dramatically lower than 574. But what if the difference was less clear? What if the within-group variation was 200 instead of 574? Or even closer—perhaps 66.7? At what stage do we say the difference is substantial enough or insignificant enough to compare averages confidently?
That's precisely where an objective statistical test like ANOVA becomes invaluable. ANOVA helps us "outsource" this decision-making to computed statistics, determining objectively whether variations justify comparisons of averages. Remember, the essence of ANOVA is to guide us clearly and objectively on whether comparing averages across more than two groups makes statistical sense—even if, to the casual observer, the two sets of averages seem identical, the underlying datasets might differ greatly to a statistician!
This brings us to an important point: why ANOVA—Analysis of Variance—and not ANOME—Analysis of Means? ANOVA specifically analyzes whether the differences among group averages are meaningful by comparing them to the internal consistency (within-group variance) of the groups. Averages alone are signals, clearly indicating central tendencies, but variance reveals the reliability of those signals.
Now let's step back and become a little 'meta' about learning and writing: notice how this explanation unfolded? It began with a relatable story, clearly defined the problem, explicitly presented numerical evidence, and seamlessly transitioned into the core statistical explanation. This structured yet conversational approach didn't happen by accident—it came from a thoughtfully crafted prompt.
So, what exactly is a "prompt" in this context? It's simply the clear, focused instruction you provide—detailing exactly what knowledge you seek, how you prefer it presented, and what outcomes you want. A well-written prompt explicitly outlines the style (informal, engaging, intuitive), the content (clearly defined goals and subtopics), and desired outcomes (increased conceptual clarity and practical application). Notice how the clarity of the bookstore owner's question and your own internal question guided the explanation, ensuring the answer precisely matched the learning goal.
Returning to our bookstore story, the owner walks away content and informed. You, the statistics student, sip your drink and reflect—not just on ANOVA, but on how good questions unlock meaningful answers.
Would you appreciate a follow-up essay exploring statistical significance or deeper techniques for writing powerful prompts? If yes, please, put me out of a job and go have a conversation with ChatGPT (or your LLM of choice) about the topic, what about it you are not clear about, and where specifically you would like help. Customize it to your requirement by specifying as much detail as you need about who you are, what your interests are, what you like studying and where you struggle.
An LLM absolutely can do a better job of teaching you, especially as a complement to all of what you already do to learn. But hey, realize that chatting with an LLM doesn't make your job as a student easier - quite the contrary. It makes it harder!
Why? Because a) you have no excuses left and b) from here on in, you can only blame yourself if you're not using LLM's well enough.
Learn better!