*In 2014-2015 I wrote an education column called "There Are No Dumb Questions" for the website "MuseHack". As that site has evolved, I have decided to republish those columns here (updating the index page as I go) every Wednesday. This ninth column originally appeared on Thursday, November 27, 2014.*

#### Why don’t more prices end in round numbers?

In large part due to the “Left Digit Effect” - but as a bonus, I’ll also mention “Benford’s Law”, and the pricing of your own items. It seems a sensible topic to tackle as we head into the season of Holiday Shopping, right? Not to mention Black Friday/Cyber Monday for the Americans.

No one (that I’ve been able to discover) knows where the practice of ending a price in a “9” or “99” began. It’s been suggested that doing so would force the cashier to open the till to make change, so that the sale would become a matter of record. But perhaps business owners simply started noticing that pricing at xx.99 was good for sales. Because it is. This has been shown experimentally. Even going back to the 1960s, when a liquor store in Southern California found that pricing their wine UP to 99 cents (from 79 and 89 cents) increased the number sold. How could this be?

One issue is the Western manner of reading, which involves scanning from left to right. So, upon seeing the price $39.99, the left most digit is seen first, and is thus given greater weight - even though by the time we get to the end, the price might as well be $40. (Particularly in countries which no longer mint pennies!) Consider, does it immediately register to you that a $5.99 item is actually double the cost of a $2.99 item? The term “Left Digit Effect” is used to describe how consumers reading $5.xx will interpret “$5 and change”, even if the cents given mean the cost is “Almost $6”. Which, granted, doesn’t quite explain why raising a price would result in better sales, but there’s an element of psychological pricing involved too - if you DO see the price as “Almost $6”, you may get the false impression that the item is somehow on sale. Even if $5.99 is the regular price.

All that said, there’s one other mathematical aspect in play, involving percentages.

#### Thirty Percent Chance

It turns out that not all leading (“left”) digits are created equal. While truly random numbers (like the lottery) will be evenly spread out across all digits, and truly constrained numbers (like ones which actively eliminate digits) are not subject to the following effect, a set of random measurements (for instance, house addresses) will tend to start with a “1” more often than a “2”, a “2” more often than a “3”, and so on. In fact, the left-most digit in most data sets turns out to be a “1” fully 30% of the time! That’s not even close to one ninth! The mathematics behind it is referred to as Benford’s Law, which describes the probability of the first digit (“d”) using logarithms. This law is even “scale invariant”, meaning it works regardless of whether you measure in metric, in imperial, in dollars or in euros. Why is this useful? Well, for one thing, when the expected first digit pattern is MISSING, we can identify voting anomalies, or catch those committing tax fraud. Yet how does this connect back to shopping?

At first, it seems like a complete contradiction - shouldn’t we see more “1”s, not “9”s? But remember, Benford’s law talks about the leftmost digit. The second digit does not follow the trend to the same extent, and by the time you reach the fifth digit, number choice is fairly uniform from 0-9 (all other things being equal). Why? Let’s consider the percentages. If an item is valued at $10, to move that first digit to $20, you need to double the price - a 100% increase. But for an item at $20, moving it to $30 merely requires a 50% increase - even though both cases involved an additional $10. And moving a $90 item to $100 is trivial - only a bit over a 10% increase in price. (At which point changing the $100 item to the next initial digit, $200, is again fully double.) Such is the nature of logarithms. So why not leave the price at $99? There's not much percentage to be gained by changing it.

Consider also discounts. If there is a 50% discount on any item (under $100), it will end up starting with a “1” so long as the regular price was anywhere between $20 and $40 - yet with the leading digit now being a “1”, it might appear to be an especially good deal. If we increase the discount to 60%, an item at $19.99 would have been less than $50 anyway ($49.98) - yet we may not stop to consider that actual drop in price. We may also perceive a $100 item being marked down to $89.99 as being a much better deal than seeing a $116 item priced down to $105.99, because of the change in place value - even though the price differences are the same.

So, can we relate some of this to pricing your own items for sale? Well, while the “Left Digit Effect” might be in play, a study last year suggested that customers

*prefer*to pay in round numbers. Because really, when was the last time you were at the gas pump, trying to hit a total that ended in .99? In fact, given a “pay-what-you-want” download plan for the video game “The World of Goo”, this study found that 57% of consumers chose to pay round dollar amounts. (I’ve also noticed Kickstarter pledges tend to go in round numbers - is that built into their system?) Some stores will even use round number prices to create the impression of added value or quality. But before we disregard the psychology entirely, there are applications outside of shopping. The link (below) about game prices uses the “Left Digit Effect” as a reason to award 3000 experience points - rather than only 2950 - when you’re coding up your game. After all, the percentage increase from 2950 to 3000 is below 2%.

Will you now pull out your calculator when doing your shopping? Probably not, so your best take away here is to avoid making spur of the moment decisions, particularly when looking at what, on the surface, seems to be a “great deal”. Oh, and you should also check snopes.com before following any other advice about “secret codes” used in pricing.

__For further viewing__:

1. Why Game Prices End in .99

2. Benford’s Law (with graphs)

3. GoodQuestion on WCCO News (3.8 min* video)

* - see what I did there?

**Got an idea or a question for a future TANDQ column? Let me know in the comments, or through email!**
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