Saturday, 26 April 2014

Anson Belt & Buckle Review: Part 2

It's been a long time, but I'm finally following up on my initial review of my Anson Belt & Buckle purchase. For the benefit of those too lazy to read all the details, here's the important information:
  • These belts have stood up to my abuse and exceeded my expectations.
  • My opinion of them is very positive. I would recommend Anson Belt & Buckle to anyone and I would consider buying from them again if I need a belt in the future for myself or for a gift to somebody else.
  • Anson Belt & Buckle reports that they've changed suppliers and improved their straps since the time these belts were purchased, meaning the products they currently sell aren't quite the same as what I've reviewed. 

And now for the full analysis. First, I'll mention that the co-founder, David Ferree, commented on my first post. Evidently my review gets enough traffic to warrant attention from the company. David expressed concern that my previous review contains negative elements that might discourage potential customers and he points out some relevant errors and omissions I made. He also offered a full refund if I was dissatisfied and returned the belts, so good on them for making customer satisfaction a high priority. That said, I will not delete or substantially revise my original post (though I have added a brief note at the beginning of it). At the time that I wrote the first part of the review it was accurate to the best of my knowledge and an honest account of my opinions and experiences. Nevertheless, I think David's comments and concerns (I've quoted him verbatim in italics) are valid, so I'll address them here:
  1. Our individual straps and buckles actually sell for $24.95 and and [sic] a complete belt for $49.90.
    I checked the website recently (which has been redesigned since my previous visit) and confirmed that individual buckles and straps are currently priced at $24.95 USD, so a complete belt is just under $50 before taxes, shipping, and any other applicable fees. They've also added several more straps and buckles to choose from, so there's a lot more selection available than when I bought my belts.
  2. We also have a Lifetime Guarantee which I don't think you mentioned.
    I didn't mention the guarantee in the first post because I was unaware of any guarantee. But in my recent visit I found that it is plainly visible on the splash page that Anson Belt & Buckle offers some kind of lifetime guarantee. That definitely offers some reassurance to potential new customers. However, I can't find any further information regarding the guarantee. There is usually some information on these things written in 'legalese' to mitigate against frivolous claims or people generally abusing the policy.
  3. I think that overall your feeling and perception of our product was a positive one...Indeed, my opinion of the product was generally positive. I had some reservations about the quality and durability and described them honestly. 
In a separate email to me, David stated that they've changed suppliers since I made my order and that their straps have been improved upon. I can't verify that statement without buying and testing another belt, but based on all the changes I saw on the website, I'm not surprised that the supplier has changed too. It also means that the belts I have are at least slightly different from the products Anson Belt & Buckle is currently selling.

On to the actual belts. I have been wearing the dark brown belt about 3 to 4 times a week for the past year. Here are a few recent photos of it to show you what it looks like today:

Brown leather belt looks warped after wearing it for a year.
The kink formed where the belt's pinched by my belt loop when I sit.

Close-up of the outer side of the belt at the kink.

Close-up of the top finished edge of the belt at the kink.
The edge finishing has started to separate from the strap.

Close-up of the top finished edge of the belt at the tongue.

The belt buckle I wear most often. A little scratched but nothing obvious from afar.

I think the kink in the belt is normal if you sit a lot, so in and of itself the kink didn't concern me. The edge finishing has taken some damage, but the damage hasn't propagated. It looks like the strap's edge finishing is mediocre while the rest of the belt of good quality. I may be critical, but I'm not unreasonable; the flaws I've highlighted are practically invisible and I'm generally quite pleased with the belt.

In addition to wearing a belt for a year, I also subjected the cut end (the one I didn't dissect) to a harsh test of my own design to simulate long-term use and severe abuse. I tested the durability of the buckle's lever, buckle's surface finish, and the leather strap, spread out over a year so that I wouldn't lose my mind doing the experiment. I've reported my findings below:

Buckle Lever Durability
Test description: I opened and closed the lever a total of 3000 times to simulate taking off or adjusting the belt. I was a little rough too, pressing the latch to its limit in both directions using firm pressure with my thumb. This was to make sure it could handle repeated forces beyond what you need to just make it work.

Result: The buckle still works like it was new.

Buckle Finish Durability
Test description: I scratched the buckle (in an inconspicuous area so I could still use it afterward) a total of 100 times to simulate accidental abrasion from carrying awkward large objects or brushing up against a wall. I alternated between using a knife blade and a rock to do the scratching (so 50 scratches each).

Buckle after the first day of testing.
Buckle at the conclusion of the tests.
Aside: I'm a much better photographer now than I was a year ago.

Result: The finish looks like hell where I tested it, but there's no flaking. The weird bubble-like thing on one of the buckles (shown in my first post) is probably a minor cosmetic flaw related to the manufacture of the whole part, not with some cheap finish. The finish on the buckle I tested was tough; definitely tough enough for long-term normal use. Just keep in mind that if you do a lot of hard work where your belt buckle gets abraded, don't expect these buckles to come out completely unscathed.

Belt Strap Durability
Test description: I folded, twisted, and contorted the strap in various ways at one location. I subjected the strap to a total of 5000 contortions to simulate situations like being stressed by a belt loop while sitting or being squashed in your luggage during travel. I also dragged the belt back and forth over different surfaces to simulate accidental abrasion from things like carrying stuff close to the body or brushing up against walls.
Inner side of the specimen after the first day of tests.
Outer side of the specimen after the first day of tests.
Outer side of the test specimen at the conclusion of testing.
A little wrinkled from the stresses and weathered from abrasions, but still in pretty decent shape.

Top edge of the test specimen at the conclusion of testing.
The edge finish has separated from the belt.

Inner side of the test specimen at the conclusion of testing.
Wrinkled from all the stresses I subjected it to, but otherwise in good shape.

Bottom edge of the test specimen at the conclusion of testing.
Edge finish separated from the belt.

Result: The belt has clearly taken on a worn and weathered appearance, but considering how harsh my test was and the low expectations I had when I discovered it was Chinese laminated leather, the belt has fared extremely well. The edge finish wasn't as tough as the rest of the belt and started cracking and peeling off early on. However, the rest of the belt held together and you can see that the damage remained minimal. I wouldn't wear a belt with my Sunday best in that worn condition, but I think it would still be okay to wear for casual, everyday use.

Clamped End Durability
Test description: I yanked and twisted on the clamped end of the belt strap 1500 times. I also removed and then reseated the buckle 100 times, to simulate switching around the strap/buckle combinations.

Clamped-end of the test specimen at the conclusion of testing.
Results: The tooth marks left by the clamping mechanism have fretted a little, but the strap and the clamp mechanism stood up to the abuse.

Overall, I've been impressed. The buckles are working just as well as when I first took them out of the box. The part of the buckle I scratched simply looks scratched up. The finish is reasonably tough and the damage is limited only to the scratches; there was no flaking or peeling. The strap on the real belt is slightly worn and a little bent out of shape, as would be reasonably expected of any belt after wearing 3 or 4 times a week for a year. The test specimen has certainly seen better days, but considering the severity of the test, fared quite well. In my opinion, the straps are tough enough for long-term, regular daily use. However, if you do a lot of manual work that makes your belt take a lot of abuse, I think it'd be optimistic of you to not expect the belt to start coming apart eventually. Then again, the outward appearance of the belt doesn't exactly say "I'm perfect for farm chores!", so I'm not sure many people will benefit from that word of caution anyway. In conclusion, I think Anson's belts are a good buy and I would recommend them to others.

Friday, 25 April 2014

More Fun with Probability and Birthdays

In my previous post I examined the probability of there being a shared birthday in a random group of n people. Let's take this analysis a little further and look at the probability of there being birthdays within a certain proximity to each other, say for example within 4 days time. After all, it is common to hold birthday celebrations on a weekend when they're more convenient, so two people with a birthday within 4 days of each other could conceivably choose to celebrate their birthdays on the same day. This problem gets a little more complicated than the same-day birthday. The probability of two randomly selected people having a birthday within 4 days of each other is 9/365 (about 2.47%), because B could have his birthday on the same day as A, or in any of the four days before, or in any of the four days after. Just as before, it is simpler to calculate the probability of there being no shared birthday as we increase our group size and then subtract from 100%. When we add a third person we now have two possibilities. One possibility is that A and have their birthdays at least 9 days apart, in which case there are 18 days where C can't have his birthday. A second possibility is that A and have their birthday more than 4 days apart but less than 9 days apart, in which case there is some overlap and there could be as few as 14 days eliminated for C. This brackets the probability of there being no birthdays close together in a group of three people to somewhere between 6.20% and 7.28% (exact solution is 7.26%; formula is given a few lines down). Adding a fourth similarly means there are as few as 19 and as many as 27 days eliminated, bracketing the probability of there being no birthdays close together in a group of four people to between 11.09% and 14.13% (exact solution is 14.08%).

There is an exact solution that accounts for the probability of the overlaps in the spacing between birthdays, but the explanation of how that's accomplished would be a little math intensive and there's already more than enough math in this post. Just take my word for it that the total possible permutations of n people's birthdays spaced at least k days apart from each other is equal to:
which, in factorial notation, looks like this:

With this result we can move on to calculating the probability that no birthdays in a group of random people are within k days of each other. Subtract that from 1 and you get the the probability that there is at least one birthday in the group within k days of another person's. That equation looks like this:

It looks more complicated than the formula for probability of same-day birthdays, but the general form is the same. It converges to 1 pretty rapidly as n increases. And the larger k is, the faster it converges. If we take k = 0, the equation reduces to the earlier expression for birthdays on the same day. In the graph and table below you can see the results for yourself.

As you can see, among just 10 people there's a pretty good chance there are two or more birthdays in the same week. What I think is pretty cool is the huge difference between k = 0 and k = 1. For instance, in a group of 25 people, there's about 57% probability that two people have the same birthday, but almost 93% probability that two people have birthdays within one day of each other (i.e. on the same day or on consecutive days). Test it out among your co-workers or a sample of your Facebook friends and see for yourself. If you're clever, you might even be able to use this knowledge to make some bets and relieve a few naive people of their money.

Saturday, 19 April 2014

The Surprising Probability of Shared Birthdays

Have you ever wondered why, in a group of maybe a few dozen people (say a class of students or the staff in an office), it is fairly common for there to be a shared birthday? There are 365 days in a year but only 20 to 30 people in a typical elementary school classroom. Commonsense says shared birthdays in a small group ought to be rare, right?

When it comes to understanding probability and randomness, our commonsense often leads us astray. Our brains are better suited at comprehending patterns, structure, and order, so much so that when faced with chaos and randomness we tend to search for patterns and attempt to impose order. Our belief of what a random sample should look like is often not very random at all. So when we have a group of 20 random people, we'd like to believe that their birthdays should be evenly distributed throughout the year.

Let's analyze the shared birthday problem and find out how probable shared birthdays really are. First off, what are the chances that two randomly selected people have the same birthday? This is a rather straightforward problem, assuming that birthdays are evenly distributed among the 365 calendar days (let's neglect those leap-year birthdays). The probability is 1/365 (about 0.27%). Nothing counter-intuitive about that; our gut feeling should be that it's unusual for two random people to have the same birthday.

Where our intuition starts to lead us astray is when we start adding more random people to the sample. Let's go up to three people now (A, B, and C). There is a 1/365 chance that B has the same birthday as A. Equivalently, there's a 364/365 chance that B doesn't share a birthday with A. Having taken up two days of the year with A and B means that C has a 363/365 chance of not sharing a birthday with either A or B. The probability of there being no shared birthday in the group is therefore [364/365] * [363/365] (about 99.18%). To find the probability of there being a shared birthday, just subtract the probability of there being no shared birthday from 100%. In other words, there's about a 0.82% chance that there is a shared birthday among three randomly selected people. The probability is small, but keep in mind all we did was add a third person and we nearly tripled the probability of a shared birthday in the group. If we add a fourth person, we get a probability of 1 - {[364/365] * [363/365] * [362/365]}, which is about 1.64%. That's about double the chance of a shared birthday in the group of three.

We can calculate the probability of there being a shared birthday in any size of random sample using the following complicated-looking equation:
Probability of a shared birthday in a group of n randomly selected people.
where n is the number of randomly selected people in the group and the ! indicates the factorial function. The probability of a shared birthday rapidly approaches 100% because that 365^n factor in the denominator makes that whole fraction really small really fast as n increases. You can see for yourself if you plot the equation at different value of n like I have below:

Looking at it in a slightly different way, we can show what's the minimum size of group you need to have for a given chance of there being a shared birthday:

The results may be surprising. You might find them hard to believe. Intuitively, we know that a group of 366 or more people must have 100% probability of a shared birthday, and a group of 1 person has a 0% probability of a shared birthday. But beyond that, most folks' intuition is way out to lunch. You probably didn't guess that with just 23 people, it is more likely than not (50.73% chance) that there is a shared birthday in the group, or that in a group of 70 people there's only about a 1 in 1000 chance of there not being a shared birthday. But now that you've seen the analysis, hopefully it's no longer surprising that there were shared birthdays in your classes or workplace.