# How do I go about finding a plateau within a set of numbers?April 23, 2010 10:21 AM   Subscribe

How do I go about finding a plateau within a set of numbers?

Okay, so, let's say I have a shoe store which sells designer shoe laces and I have a bunch of competitors in the area. We're each given a single pair of designer laces per store, and I know how much the other stores are selling their laces for, and here are those numbers:

.96
.99
.99
.99
1.97
1.99
4.90
16.48

The shoe laces will sell over time, and these numbers will stay the same. I'd like to sell my laces somewhat soon so that I have more money to buy additional laces from the supplier, but I also don't want to sell them just for the least amount. In fact, in that set, I'd want to sell my laces for \$4.89, right next to that \$4.90 number.

How do I go about figuring out where that sort of plateau is within a set of numbers? Here's another set:

8.75
9.00
9.07
9.31
10.00
14.95
18.39
19.93
19.95
23.91

and in that set i'd want to sell my laces for 14.95 or 18.38.

I know nothing about statistics (is this statistics?) but imagine there is some sort of formula or algorithm or procedure that has existed since ancient Greece.
posted by SamuelF to Science & Nature (21 answers total) 1 user marked this as a favorite

Plateau would mean something else to me, but it sounds like you want an arithmetic mean.
posted by Blazecock Pileon at 10:35 AM on April 23, 2010

I'm not sure I understand exactly what you're asking, but it seems to me you could simply measure the percentage increase each price is above the next lower price. E.g.,

8.75
9.00 (2.8% increase over 8.75)
9.07 (0.8% increase over 9.00)
9.31 (2.6%)
10.00 (7.4%)
14.95 (49.5%)
18.39 (23.0%)
19.93 (8.4%)
19.95 (0.1%)
23.91 (19.8%)

And just note the largest percentage increases over the next lower price.
posted by DevilsAdvocate at 10:36 AM on April 23, 2010

You might round the mean up or down to the nearest figure, minus a cent. For example, the mean of the first set of numbers is \$3.66. The closest value to this mean is \$4.90. The second closest is \$1.99. Etc.
posted by Blazecock Pileon at 10:38 AM on April 23, 2010

DevilsAdvocate, you're on the right track. How would you deal with a set like this:

3.63
3.63
7.45
7.45
9.98
10.90
10.95
48.14

In this case, the 48.14 price is unreasonable and the laces would never sell, so it would probably make sense to price between 3.63 and 7.45. I suppose we're not only looking for percentages, but also for avoiding outliers as well.

Thanks for the help so far, everyone!
posted by SamuelF at 10:46 AM on April 23, 2010

You want to look at something like the cumulative distribution of prices. The cumulative distribution is what you get if you make a graph of "y many laces sell for less than x price' where x is, say, all prices between the lowest and the highest, in small increments (ten cents, say). What you are calling a plateau will be a region of this graph where it goes up sharply, since small increases in price will include many more laces. You'd be looking, then, for places where the graph moves from being very flat to very steep, since those correspond to where there is a large gap in the price.

If you plot the difference between consecutive points of that cumulative price distribution, then you can look for places where it goes from having very flat slope (increase near zero) to very high slope. If you were to write this as an algorithm, the details of what you want to consider "flat" and "high" would need to depend on your exact data, but if you just want to use the plots and then find the exact places by eye, you can do that without much calibration.
posted by Schismatic at 10:48 AM on April 23, 2010

Your comparison is meaningless without knowing how many shoelaces the competitors sell. If the expensive stores sell theirs almost as rapidly as the cheaper stores, then you should charge more. If the cheapest option is selling their shoelaces much more rapidly than the expensive store, then you'd want to price yours lower. You want to optimize your profit, number sold*(price per unit - cost per unit) - fixed costs. Anything else is numerology. I have no idea how you decided that \$4.89 was the best price in that first example. You want to sell your laces for five times what four different competitors are charging? Bad idea.

In other words, setting the optimal price depends on many more factors than just the list of competitors prices.
posted by kiltedtaco at 10:59 AM on April 23, 2010 [3 favorites]

In this case, the 48.14 price is unreasonable and the laces would never sell, so it would probably make sense to price between 3.63 and 7.45.

How do you know the laces would sell at 7.45, or even 3.63? In your prior example, there's no way to know based on the numbers alone whether 23.90 or 18.38 or 14.94 is too high. kiltedtaco is right—it's easy to find large jumps in the range of prices, but the optimal price can't be determined from the list of competitors' prices alone.
posted by DevilsAdvocate at 11:09 AM on April 23, 2010

kiltedtaco, you're absolutely right. Let's assume the cheapest ones sell fastest/first. I'm trying to figure out the optimal place to position the laces, selling for as much as possible in a reasonable timeframe.
posted by SamuelF at 11:12 AM on April 23, 2010

It's still going to depend on other factors, such as how quickly the supplier replaces the item one it's sold, and how much demand for the item there is, and how willing customers are to wait for a better price. You say that each store has a single pair at a time. Once the laces are sold, how long until the store gets another pair in? Two minutes? Two days? Two months? How many competing stores are there, and how many people want the laces? If all the stores selling laces at less than \$2 are out, are the laces something that someone has to have RIGHT NOW, or are they something that consumers are willing to wait for in order to save \$3?

Frankly, I'm having a hard time wrapping my head around this question as I'm having difficulty thinking of an example of a good that's sold in the manner you describe, but even if there were, the above questions would all affect the optimal price.
posted by DevilsAdvocate at 11:21 AM on April 23, 2010

kiltedtaco, you're absolutely right. Let's assume the cheapest ones sell fastest/first. I'm trying to figure out the optimal place to position the laces, selling for as much as possible in a reasonable timeframe.

As kiltedtaco says, you can't determine that with the data given. Unless you assume that the sales speed is perfectly (inversely) proportional to price (ie laces at \$1 sell for twice as fast as \$2 laces and 4x as fast as \$4 laces) - in which case its largely irrelevant how you price them, over time everyone gets the same income from the laces (although not profit. If the laces cost 50c to buy from the supplier. The person selling 20 for \$1 each would make \$10, whereas someone who sold 10 for \$2 each would make \$15)

Without knowing how fast they sell at each price (or at least some of the prices) and what a 'reasonable timeframe' is, its impossible to determine the optimal selling price
posted by missmagenta at 11:29 AM on April 23, 2010

sorry - didn't mean inversely there since I said speed not time taken to sell, but you get my meaning, I'm sure.
posted by missmagenta at 11:30 AM on April 23, 2010

I still don't know what "plateau" means here. I think of a high, wide, flat area, but your first example has only one value there; there' s nothing wide about it. The mode is .99, so why didn't you choose .98?

Maybe if you answer some more samples, we can come up with some algorithm. What about these? What should be the answer for these inputs?

[10, 20, 30, 40, 50]

[10, 10, 10, 10, 50]

[10, 10, 10, 50, 50, 50]

[10, 50, 50, 50]

[10, 30, 30, 30, 50]

[10, 30, 60, 100]
posted by cmiller at 11:34 AM on April 23, 2010

Maybe if you answer some more samples, we can come up with some algorithm.

...or explained how you chose the figures that you chose. What criteria did you use/decisions did you make to come to those conclusions? I can't see any pattern or logic in the numbers you've chosen?
posted by missmagenta at 11:39 AM on April 23, 2010

[10, 20, 30, 40, 50]

[10, 10, 10, 10, 50]

[10, 10, 10, 50, 50, 50]

[10, 50, 50, 50]

[10, 30, 30, 30, 50]

[10, 30, 60, 100]

The general idea is, as the cheaper ones sell they won't necessarily be replenished. In this example, we'll assume they won't be. The default for pricing would be the lowest (rank 1). In the third example, I can sell the laces for \$40 more, and only lose 1 rank. What I'm looking for is a formula I can tweak to adjust the sensitivity and price/rank exchange.

Thank you all, again, for your help!
posted by SamuelF at 11:58 AM on April 23, 2010

[10, 20, 30, 40, 50]

Why did you choose 30 here rather than any of the others?

[10, 30, 30, 30, 50]

Why did you choose 30 here rather than 50?

[10, 30, 60, 100]

Why did you choose 60 here instead of 100? What would you have chosen if it had been [0.10, 0.30, 0.60, 1.00]?

In the third example, I can sell the laces for \$40 more, and only lose 1 rank.

In the final example, you could also sell the laces for \$40 more and only lose 1 rank, yet you chose not to. What's the difference between the third example (or perhaps even more closely, the second example) and the last?

I'm asking because we need some kind of consistent principle to make a mathematical model that does what you want it to do, and I can't discern any consistent principle based on the examples so far. That doesn't necessarily mean there isn't one, it might just mean we don't have enough examples yet to see what it is.
posted by DevilsAdvocate at 12:10 PM on April 23, 2010

I think you want a histogram.

It'll tell you how much sells in a given price range. E.g.

0.90 - 0.99 - 15
1.00 - 1.09 - 12
1.10 - 1.19 - 23

etc. and then by looking at the graph you can figure out where the price 'plateaus'

Send me your data & I'll send you back some information on it - graphs, and if there is a formula to be extracted, a formula.
posted by MesoFilter at 7:02 PM on April 23, 2010

[10, 10, 10, 10, 50]

[10, 30, 30, 30, 50]

Oh, wait, so if I take the first one, and find someone willing to sell for "1", your answer changes to 10? Adding an outlier changes it? Or, is there something more about the magnitude of the numbers?

[1, 10, 10, 10, 10, 50]
posted by cmiller at 8:17 PM on April 23, 2010

4x10 = 40
1x50 = 50

50 wins.

10 x 1 = 10
30x3 = 90
50 x 1 = 50

90 wins.

1x10
1x20
1x30
1x40
1x50

50 should win, I don't know why he picked 30.

1x10
1x30
1x60
1x100

100 should win, I suspect he meant to make it 2x60.

10x3 = 30
50x3 = 150

50 wins.
posted by MesoFilter at 8:37 PM on April 23, 2010

Here's an example histogram based on the data given in the example in the OP.

price range x sales = approximate revenue (upper range price x number sold)

0.00 - 1.00 x 4 = ~4.00
1.01 - 2.00 x 2 = ~4.00
2.01 - 5.00x 1= ~5.00
5.01 - 10.00 x 5 = ~50.00
10.01 - 15.00 x 1 = ~15.00
15.01 - 20.00 x 4 = ~80.00
21.01 - 25.00 x 1 = ~25.00

You want to charge between \$15 and \$25 to maximize revenue, that's what we call the maxima.

If you want to sell quickly, you may want to price between \$5 and \$10, since those sell the fastest.
posted by MesoFilter at 9:00 PM on April 23, 2010

I guess what I am looking for is this:

Within a set of numbers, I'm more willing to trade early places than I am willing to trade later places. So, here's an example:

[10,20,30,40,50]

To go from rank 1 (9.99) to rank 2 (19.99) I gain 10 but lose 1 rank. To go from rank 4 (39.99) to rank 5 (49.99) I also gain 10 but lose 1 rank, but that last rank is more valuable, and it would be preferable to have a variable that could be adjusted so that the default behavior can be adjusted to be more aggressive (trade less) or more greedy (trade more).

The original question was more in this sense:

If I have a set of numbers

[8.75, 9.00, 9.07, 9.31, 10.00, 14.95, 18.39, 19.93, 19.95, 23.91]

is there any way to identify those price clusters within it? That is, [8.75 … 10.00] or even [18.39 … 19.95]? It seems like I'd be looking for a density function of some sort.

The cumulative distribution of prices also sounds promising, and this is something I'll explore as well. I'm okay with having to dig into some math for this and stretch my brain.
posted by SamuelF at 12:19 PM on April 24, 2010

I've taken all the numbers you've posted to this thread and did some calculations.
``` Price    # Sold Revenue
\$0.96    1      \$0.96
\$10.40   15     \$155.94
\$19.83   5      \$99.16
\$29.27   3      \$87.80
\$38.70   0      \$0
\$48.14   1      \$48.14 ```
The data is rather limited, but the conclusion is pretty clear. Price it around \$10 if you want to move it. It won't hurt to have a few \$20 and \$30 options in stock too.
posted by MesoFilter at 5:19 PM on April 24, 2010

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