Lego: Christmas gift without bricking the bank

Four years ago right around Christmas time, I had posted an analysis of how the prices of Lego sets were related to their number of pieces.

Checking a Lego Christmas catalog last week, I had the feeling that:

1. prices had gone up in general
2. yet the 10-pieces-per-dollar ratio I had estimated 4 years ago still appeared to be valid.

This conclusion would then naturally be that the number of pieces had also gone up.

I pulled all the prices and number of pieces by Lego theme (over 40 nowadays!), and revisited the analysis.

The data needed some clean-up again, especially regarding sets with with abnormal prices for the number of pieces: the Duplo sets (oversized blocks are much more expensive), the Mindstorms sets (the color sensor is over $40) and Powered-Up sets (the move hub is $80). Similarly to last time, I also split themes into those that are based on an external partnership (Star Wars, Harry Potter, Marvel, DC...) and those entirely developed by Lego (City, Friends, Creator Expert...).

Here's the plot of pieces versus price for all sets less than $200 (95% of sets), and removing the outliers mentioned in the previous paragraph:

It's quite tempting to look at the range for the pieces-to-price ratio across sets, splitting by theme:

Out of the top 10 themes with best median ratio, 9 are themes Lego developed in-house, and only one is through external partnership (car makers, deal might not have been as lucrative as for Marvel or Star Wars rights).

We can fit a linear model to fit the number of pieces against price, partnership, and their interaction:

The result is that for non-partnership sets, we get a pieces-to-price ratio of approximately 10.7, ratio that drops to just over 9 for the partnership sets. The difference is highly significant and confirms our previous observations. The intercepts are slightly different but not significantly so.

A question we might be interested in is whether we observe any diminishing or increasing returns as the prices go up. This can be done by observing the pieces-to-price ratio for various pice buckets:

As we might have expected, our pieces-to-price improves as we go into the more expensive sets. This makes sense as the pieces are only one of the costs for sets, there are many other fixed costs in terms of packaging and marketing that will make the smaller sets less attractive to the smart buyer.
Non-partnership sets appear to always outperform the partnership sets, but one should keep in mind that the sample size is pretty small in the $200+ bucket.
The previous result also throws some shade on our linear models, the slopes (pieces-to-price ratio) are actually not constant. If we wanted to improve the models, we would probably need to include a quadratic term for price.

So to recap, your best pieces per dollar will come from non-partnership and bigger sets. Of course it could be your child has a specific theme in mind, so to help out with your Christmas purchases, here's the list of the best sets for each theme:

Theme	Product Name	Pieces	Price	Ratio
Architecture	trafalgar-square	1197	79.99	14.96
Boost	droid-commander	1177	199.99	5.89
Brickheadz	thanksgiving-scarecrow	177	9.99	17.72
City	pizza-van	249	19.99	12.46
Classic	bricks-bricks-bricks	1500	59.99	25.00
Creator-3-in-1	deep-sea-creatures	230	14.99	15.34
Creator-Expert	taj-mahal	5923	369.99	16.01
Dc	1989-batmobile	3306	249.99	13.22
Disney	the-disney-castle	4080	349.99	11.66
Disney-Frozen-2	elsa-s-magical-ice-palace	701	79.99	8.76
Duplo	creative-fun	120	39.99	3.00
Fantastic-Beasts	newt-s-case-of-magical-creatures	694	49.99	13.88
Friends	underwater-loop	389	29.99	12.97
Harry-Potter	hogwarts-castle	6020	399.99	15.05
Hidden-Side	shrimp-shack-attack	579	49.99	11.58
Ideas	lego-nasa-apollo-saturn-v	1969	119.99	16.41
Juniors	city-central-airport	376	49.99	7.52
Jurassic-World	jurassic-park-t-rex-rampage	3120	249.99	12.48
Lego-Batman-Sets	1989-batmobile	3306	249.99	13.22
Lego-Spider-Man	spider-mech-vs-venom	604	49.99	12.08
Marvel	the-hulkbuster-smash-up	375	29.99	12.50
Mindstorms	lego-mindstorms-ev3	601	349.99	1.72
Minecraft	the-wool-farm	260	19.99	13.01
Minifigures	mf-set-ninjago-2019	59	12.99	4.54
Ninjago	ninjago-city	4867	299.99	16.22
Overwatch	dorado-showdown	419	29.99	13.97
Power-Functions	lego-power-functions-train-motor	7	13.99	0.50
Powered-Up	disney-train-and-station	2925	329.99	8.86
Powerpuff-Girls	mojo-jojo-strikes	228	29.99	7.60
Serious-Play	window-exploration-bag	4900	484.99	10.10
Speed-Champions	formula-e-panasonic-jaguar-racing	565	29.99	18.84
Star-Wars	yoda	1771	99.99	17.71
Stranger-Things	the-upside-down	2287	199.99	11.44
Technic	cherry-picker	155	9.99	15.52
The-Lego-Movie-2	pop-up-party-bus	1024	79.99	12.80
The-Lego-Ninjago-Movie	ninjago-city	4867	299.99	16.22
Toy-Story-4	woody-rc	69	9.99	6.91
Unikitty	unikitty-cloud-car	126	9.99	12.61
Xtra	traffic-lights	46	3.99	11.53

Now one important caveat should be mentioned here. Number of pieces is a great proxy for size and time to build, but it isn't perfect. Going through some sets I've noticed a tendency of adding more and more small decorative pieces. This trick can significantly and artificially bump up the piece number. Ideally we should also include weight to control for that aspect, but weight was not available and would probably not have been available at the desired granularity level. Take the new gingerbread house for instance. At $99 for 1477 pieces it seems like a great deal, but take a closer look at what's inside:

Look at all those small pieces! Especially the red and white candy canes around the doors and windows. Just tiny 1-by-1 round shapes staked together. Almost 100 of them, and I doubt they're worth $10!

Now of course, these were just models, and you can throw them all out the window when it comes Ole Kirk's house. Who? Ole Kirk as in Ole Kirk Christiansen, the man who created The Lego Company. His house is an extremely rare Lego set which was gifted only to employees (and a few special visitors). At 912 pieces we would roughly estimate it at $90, but good luck finding it at less than $300, prices of $500 are not uncommon for unopened sets!

How retro is Star Wars Episode 7: The Force Awakens?

Essentially all reviews I have read of the latest Star Wars installment, The Force Awakens, discuss how well J.J. Abrams was able to rekindle with the spirit of the original trilogy (Episodes 4, 5, 6).

It's sufficiently retro that even George Lucas spoke out against it for that reason!

Of course, as mentioned in Christopher Orr's critic excerpt, we do see many familiar faces we had grown attached to from the first trilogy and whom we missed in the second trilogy prequel. But can introducing some of our old friends back in the script be enough to recreate the environment we last saw over 20 years ago? Or was J.J. Abrams really able to go beyond the quick solution and put his heart in mind into reviving the magic?

There are many ways to approach the question, but I wanted one that would rely more on statistics than a degree in film studies. My angle was to perform a clustering analysis on the seven movies, solely relying on the dialogues. I found those at http://www.imsdb.com/scripts/. Formatting differed from movie to movie so slightly painful to get everything in a suitable format...

I then characterized each movie by the frequency of words, and looked at which movies were most similar to each other in terms of what is said. A few comments:

• frequency of occurence was normalized by number of words so comparisons were fair (episodes 5, 6 and 7 have much less dialogue than episodes 1 through 4)
• common english words typically referred to as stopwords in text analysis ("and", "I", "in", "was"....) were removed
• planet and character names were also removed (Han Solo doesn't even exist in episodes 1-3)

The results from the hierarchical clustering are plotted here:

What does this tell us? Well it seems that just from a frequency of word analysis, one could almost reconstruct the trilogies! Episodes 1, 2 and 3 were lumped together on the left, episodes 4, 5 and 6 together on the right. As for episode 7, the algorithm added it to the original trilogy cluster.

What does this tell us? Well, it would appear that purely from a dialogues perspective, J.J. Abrams and other writers did a very impressive job in maintaining the original look and feel of the first trilogy by using the same lexical field. As an example, in Episodes 1, 2 and 3, the word "jedi" appears in dialogues approximately 11.9 times for every 1000 (non-stop) words. In contrast, that value is 2.6 for Episodes 4, 5 and 6. For The Force Awakens, the value is 2.4. This is one of the many signals which led the clustering algorithm to find Episode 7 "closer" to the first trilogy than the second.

From a modest stat perspective of dialogues, kudos to J.J. Abrams for a perfectly well-executed "retro" style!

Stay posted, this post will be updated quite regularly in the next few years...

Lego Pricing: For which partnerships do consumers pay the most?

As for most parents out there, a Lego set was part of the Christmas wishlist, and I found myself in front of an impressive display of options. As I was looking through the boxes, I noticed something while comparing these two boxes:

Extra information: the two sets are identically priced at $39.99....
So it's not completely easy to spot on the boxes, but despite the same price, the Ninjago box contains twice as many pieces (575) as the Frozen one (292). Ninjago is a line of set produced by Lego and therefore owned by Lego, while Frozen is the result of a partnership with Disney. Quickly scanning the different boxes, I seemed to some trend there: similarly priced sets appeared to have fewer pieces for themes that were the result of external as opposed to internally derived.

Having some free time on my hands during the Christmas break, I extracted as much data as I could for the LEGO site, pulling for each set the theme, the number of pieces and the price. I was able to identify close to 700 sets which provides a reasonable size for exploring some trends. Here are all the data points with number of pieces on the x-axis and price on the y-axis, and some jitter was added but not particularly necessary (prices tend to take discrete levels but not number of pieces).

A few observations:

• the data is densely concentrated around the origin, the outliers on the scale make it hard to determine what exactly is going on there
• there appears to be quite some variability in number of pieces for a given price point, which confirms my initial impression from the Lego store. Looking at the $200 vertical line, we see that there are boxes at that price with fewer than 1000 pieces, and others with over 2500!
• overall, the relationship seems pretty linear along the lines of pieces = 10 * price, every $1 gets you about 10 pieces. I was more expecting a convex shape where each incremental piece costs a little less than the previous one, similarly to Starbucks where the larger the drink, the better the size-to-price ratio). I guess this can somewhat make sense: with food/drinks, two one-size units are equivalent to a two-size unit (if a gallon costs too much I'll just buy two half-gallons), but two 300 pieces Lego sets are not equivalent to a 600 Lego set, and so I guess Lego can afford maintaining the linear relationship.

And if you're wondering about the two data points in the upper right corner:

• at 3808 pieces and $399, we have the tough-to-find Star Wars Death Star
• at 4634 pieces and $349, we have the Ghostbusters Firestation (to be released into 2016)

Let's focus a little more around the origin where most of the data resides (92% of sets are priced less than $100):

Along the x-axis there appears to be a category of sets (green dots) consisting of just a few pieces but priced incredibly high. These are actually of the Mindstorm category. They are actually very sophisticated Lego pieces allowing you to build robots containing touch / light sensors that are sold separately at high price points. In the rest of this post, we will exclude the Mindstorm category, as well as the Power Functions category for the same reason. The Dimensions category was also excluded given that the pieces, while not as sophisticated as for Mindstorm and Power Functions, were quite elaborate based on their interaction with the Playstation console (average pieces-to-price ratio is about 3).

There appears to be another category with it's own specific piece/price relationship (red dots). While overall it seemed that every $1 was equivalent to about $10 pieces, this category seems to have a steep $1 for 1.5 pieces. This is actually the Duplos category for younger children, and the pieces are much larger than regular Legos. That being said, I'm wondering if Lego isn't taking advantage of all the parents eager to give their toddlers a head start in the Lego environment... Duplos are also thrown out for the rest of the post.

Back to our original question, how do the different themes compare to each other, and is there a price difference between internal and external brands?
The following boxplot provides some insight in the pieces-to-price ratio within each category. I've sorted them by decreasing median (higher median is synonym with a 'good deal', many pieces for every dollar). I've also color-coded them based on whether the theme was internal (red) or external (blue) to Lego.

Glancing at the graph, the two main take-aways are that:

• there is strong variability within each category (in Star Wars for instance, the Troop Carrier set has 565 pieces for $40, while Battle on Takodana has fewer pieces (409) for a 50% higher price)
• there does nonetheless seem to be a trend that internal themes have a better pieces-to-price ratio

We can try to explore the difference between the two types of themes via a linear regression, with a different slope and different intercept for each type:

The conclusion of the regression analysis is that the slopes for the two lines is not statistically significant (9.67 pieces/$ for external brands, 10.15 pieces/$ for internal brands), but there was a significant difference in intercept (50 fewer pieces for an external brand at the same price).

So in summary, don't feel Lego is completely overpricing you Disney Princesses or Star Wars figurines although there is a small pricing difference. If you do want the biggest bang for your buck, take a look at the Creator theme, and in particular here's the overall pieces-to-price winner (which I ended up getting my kid!):

Happy building!