The Beauty of Bell Curves, with applications to perfume

As of today, September 1, 2013, at 10:34 pm Berlin time, the Parfumo database lists 31,748 perfumes. For the past few months, I have been attempting to pull all of my perfume reviews together in a searchable archive. I'm not quite finished, but as of right now, there are 2,114 reviews. 

The goal of Parfumo appears to be to list all perfumes in existence, whether discontinued or readily available. My goal? Just to travel around the olfactory universe a bit. Do I have any intention of reviewing another 28K perfumes? No, of course not. Maybe I'll call it quits when I reach 3K, but one thing is clear: I have no desire to sniff the vast majority of perfumes in existence. Why? The answer, my fragrant friends, lies in the beautiful bell curve:

Bell curves depict a "normal" or Gaussian distribution of some quality or thing covering a fixed range. They are perfect for evaluations of things which come in all sorts of varieties and specifically when we choose to rate those things using simple numerical scales. I rate perfumes on a scale from 1 to 10, with 1 being rock bottom scrubber, and 10 being incredibly wonderful, even transcendent. In a normal distribution of things along the x axis, the number of items (shown on the y axis) at the lowest end will be matched by the number of items at the highest end. The ratings eventually drift off to nothingness at both ends when continuous and not discrete measures are used.

Bell curves are useful for handling lots of things in real life, believe it or not. Take people, for example, but let's also stick with perfume. Of the people you happen to encounter in your life, how many of them are completely anosmic? Probably not very many. If we were to graph that trait, I imagine that only a tiny number would be completely anosmic, and it might well match the number of people who are hyperosmic to the point of throwing a fit whenever anyone wears a scent in their presence. Most people fall somewhere in between. 

There are lots of statistical nuances between mean and median, and so forth, but for our purposes, to think about what we should expect when we set out to test perfumes, this simple bell curve is good enough.

The numbers didactically displayed on this particular version of the bell curve indicate that 68% of things fall right down the middle of the range, and 96% of things fill the broad underbelly of the curve. Most things, including perfumes, are average. Some are above average; some are below average. The extreme outliers are the top 2% and the bottom 2%. 

Let's think about those numbers for a moment. If you test 100 perfumes, how many, realistically speaking, should be masterpieces? Well, if perfume ratings are plotted along a normal distribution, then you should expect the number of scrubbers to pretty much equal the number of masterpieces. This is not to say that any two perfume wearers will agree on which ones those are. 

Every single trait, every sensitivity to every scent (and ingredient) included in a perfume, and every single taste can also be understood in terms of a bell curve distribution. Consequently, we should not expect to see all that much convergence in opinions about perfumes, it seems to me. What we should expect, however, in every case, is that to any given perfume wearer, most perfumes should be average. This might even be a tautology.

I have often wondered how I could explain my innate disdain for people who throw temper tantrums about perfumes which they happen not to like. Yes, they are childish, of course, but how and why exactly? The answer, my fragrant friends, lies here in the bell curve. We should rationally expect the worst 2% of all perfumes to be just that. It would be crazy to expect more than 2% of all perfumes to be masterworks, would it not?

Not so fast, sherapop. It's all going to turn on the reviewer. If someone actually expects every perfume to be a masterpiece, then he will be disappointed 98% of the time. On the other hand, if a person has no powers of discrimination, then everything will smell great. Will it not? But what, you may by now be wondering, does the actual perfume rating distribution look like for a real live perfumista?

Inspired by Undina, the reigning Queen of Perfume Stats, I set out this afternoon to plot my very own ratings and determine whether or not I evaluate perfumes numerically in accordance with a normal bell curve distribution. Here's what I found:

As you can clearly see, my ratings lean to the right, relative to a normal distribution. Although my numbers of absolute scrubbers and masterpieces are very similar, I am more generous in bestowing ratings on the wearable perfumes in the broad underbelly of the curve. I give an unexpectedly large number of 6's and 7's, which is probably because my approach is to award ratings based on an "all things considered" system. I take into account the cost of perfumes which strike me as an exceptionally good value, and as a result, I'll give a Molinard a 7 which I might have given only a 5 or a 6, had it not cost me nearly nothing.

All of this makes me wonder whether I should try to not do "all things considered" ratings. Until I remember that people may read my reviews, and they may be looking for some useful advice, especially if they already know that we have similar tastes. Keeping those people in mind, I feel that I should continue on with my "all things considered" ratings. So if a perfume costs $800 and smells like an average designer launch, then I may give it only a 4 instead of a 5, even though it is completely wearable. In this way I am basically expressing my opinion that it is not a perfume which I would recommend purchasing. Why? Because I don't see the point in spending $800 for a perfume very similar to a perfume which costs $80.

There is another possible explanation for the right-side heaviness of my curve: I do not seek out perfumes which I am fairly sure a priori will not smell good. I tried a couple of the Coty drugstore scents, and they were so horrible that I simply decided to avoid liquids in that general territory. This means that I am not really sampling a normal distribution of perfumes. I am not randomly spritzing in the dark. I decide to sample some perfumes and not others, and that selection process weeds out more of the perfumes that might have shown up in the 2, 3, and 4 rating region of the curve. 

I also test a lot of niche perfumes. Whether or not I happen to think that they are great, they are often very nice, and because I care very much about high quality ingredients, even a solid niche perfume which does not break any new ground may receive a 7 from me, just because it smells so nice. 

Now I'd like to open up the floor. What does your ratings distribution look like, and why? Can you think of other explanations for the leaning to the right of mine? Of course, I could just be indiscriminate, but that does not explain why the far left and the far right do appear to be normal. Or are they? I need to do a quick calculation. 

The total number of reviews is 2114, so 2% is about 42! This means that my extreme termini are unexpectedly low--at both ends of the graph. 

So it's really true, after all: Most perfumes are average! Or at least I believe that they are...