IB Math Studies Internal Assessment What is the Relationship between the Number of Legos in a Set and the Price of the Set? Exam Session: May 2017 Southwest Minneapolis High School Teacher: Mr Towne Date: 18 November 2016 William Towne
Introduction: While reading my son s Lego magazine, I saw that there was a contest to win a 3000 piece Lego set that wasn t sold in stores. I wondered how much it would cost if it was indeed sold. I know there must be some relationship between the number of Legos in a set and its cost, and my project will help me find it. I will also find out if there is a difference in the costs between different themes (Chima, Star Wars, Friends, etc.) My data is from Lego s website. On the website it lists the most recent sets by theme. I will write record for the latest 18 sets per theme: the theme the identification number of the set the name of the set the number of pieces in the set the cost of the set After collecting the data, I will first create a scatterplot of the number of pieces and the cost of the set. This will provide me with a description of the relationship between the two variables. Secondly, I will calculate the correlation between the number of pieces and the cost so that I can write about the reliability of the guess of the cost based on the number of pieces. Thirdly, I will find the line of best fit between pieces and cost so that I can use it to estimate the cost of the 3000 piece set. From the line of best fit I can also find out the cost of each brick and the cost of the packaging. Lastly, I will find separate lines of best fit for each theme to see if some themes cost more or less than other sets. Information There are many, many sets of Legos, but my data consists of 96 of the most recent sets. The data was taken from shop.lego.com. I chose six different themes of Lego sets: Star Wars, City, Friends, Ninjago, Classic, and Creator. I took the most recent 18 sets from each theme (they were ordered chronologically) since they were current. Some themes didn t have eighteen sets listed, so I chose what they did list. I chose the six themes based on what I perceived as being popular and they didn t include any sets that would have electronics since they would be priced more expensively and the 3000 piece set that I d like to know the cost of doesn t have any electronics. Here is a sample of the data, the rest of the data is in the appendix. Set ID Pieces Description Star Wars 75159 4016 499.99 Death Star Star Wars 75158 936 109.99 Rebel Combat Frigate Star Wars 75150 702 89.99 Vader's TIE Advanced vs. A-Wing Starfighter City 60125 1277 149.99 Volcano Heavy-Lift Helicopter City 60124 824 119.99 Volcano Exploration Base City 60128 184 24.99 Police Pursuit City 60118 248 19.99 Garbage Truck
Mathematical Processes From the scatter diagram below, I can see that the relationship between the number of pieces in a Lego set and the price of the set is a strong, positive, linear one. I can see that the relationship seems to be stronger if the number of pieces in the Lego set is less than 1500. After that, the prices seem to follow the same pattern but they are not nearly as strongly related. of Lego Sets Star Wars City Friends Ninjago Classic Creator 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 The correlation, r, between the number of legos in a set and the price of the set is r r n xy x y 2 2 2 2 n x x n y y 9614048239 839968225.04 2 2 r 96 154984360 83996 96 1405297 8225.04 657760484.2 782317054467257229 657760484.2 r 725372161.6 r.907 This confirms that the relationship is strong (since the correlation is near one) and positive (since r > 0). The line of best fit, y = mx + b, are
2 2 2 n xy x y 96 14048239 83996 8225.04 657760484.2 m 0.0841 n x x 96 154984360 83996 7823170544 y mx 8225.04 0.0841 83996 1162.78 12.1 b n 96 96 Price = 0.0841(Pieces) + 12.1 of Lego Sets y = 0.0841x + 12.1 Star Wars City Friends Ninjago Classic Creator 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Interpretation of Results The equation tells me even more about the relationship. The slope of the line, 0.0841 is the price per brick of the Lego sets. That is, each brick costs, on average, about 8.41 cents. The y-intercept of the line is 12.1 is the cost of a zero brick set. That probably represents the costs that go into making the sets that aren t the physical bricks themselves. For instance, this could mean the development (including the directions), packing, and shipping costs. With this equation I can also estimate the cost of the 3000 piece set: Price = 0.0841(3000) + 12.1 = $264 Since the prices of the sets all end in a.99 and generally go up by $5 increments, I would estimate a price of $259.99, or $264.99, or $269.99.
The set would appear on my line of best fit like shown above. Since the points for the larger sets can be far away from the line, I know that my estimate could be off. For instance, the 4000 piece set, using my line of best fit, would cost $350. But the actual set, the Star Wars Death Star, sells for $499.99. So that one is off by $150 which is quite a large error. On the other hand, Lego Creator sets like Big Ben and Tower Bridge sell for $112 and $133 less than what the line would predict. Those are the two dots on the right side of the scatterplot above that are below the line of best fit. Finally, I notice that by looking at the themes separately, there are different types of patterns. So I did a scatterplot and the line of best fit for each theme, in Appendix A. Using these lines of best fit, I can find the cost per brick and the base cost for each theme. Theme per brick Base cost for the set Star Wars 11.11 $ 3.37 City 14.34 $ 4.54 Friends 8.54 $ 7.66 Ninjago 9.83 $ 3.19 Classic 6.13 $ 2.51 Creator 4.99 $ 59.28 The cost per Lego does change for the different types of sets which I found interesting. Perhaps some types of sets have more elaborate, specific pieces that needs to be made especially for that type whereas other themes have mostly the basic pieces. I also noticed that the Creator set base cost was much larger than the other themes. I thought that there was something special about the way that they were made or boxed but couldn t figure out why after doing some research. So I added in more sets from 2015 which cost much less than the current lineup and found that they were actually much more in line with the other themes that I looked at. This data is shown in Appendix B. This probably occurred because in the original data, the cheapest set was quite expensive at $100. Theme per brick Base cost for the set Star Wars 11.11 $ 3.37 City 14.34 $ 4.54 Friends 8.54 $ 7.66 Ninjago 9.83 $ 3.19 Classic 6.13 $ 2.51 Creator* 6.80 $ 6.23 * with various 2014 and 2015 models Validity I began this project trying to find the price of a 3000 piece Lego set that I saw in a contest. I wanted to find other sets of Legos so that I could find out the price structure that is, how much do different sizes of Lego sets cost. To do this, I needed a line of best fit. But in order to find the line of best fit, I first had to be sure that the data was in fact linear, which I found it was using a scatterplot.
But there are some possible warnings. First, the set that was in a contest didn t conform to one of the themes of Legos that is sold in stores. Because the lines of best fit for the different themes are slightly different, my estimated price may be incorrect. I believe that these are slightly different because of licensing fees or more elaborate bricks for certain sets. Without knowing what theme or what types of bricks were in the 3000 piece set, my guess may be inaccurate. In some of the larger models, the actual cost is quite far from the estimated cost, even when using interpolation. Perhaps if I knew what percentage of bricks in each set was standard rather than custom made, I could better fit my data, but this would introduce another variable. A second problem could be lack of data in my set. I only used six themes, but there are many more. I chose the themes I did because there are many sets from each, but I could have used more data in my scatterplot.
Appendix A Scatterplot of Number of Legos versus for different themes of Legos. Star Wars Linear (Star Wars) City Linear (City) y = 0.1111x + 3.369 R² = 0.964 y = 0.1434x + 4.5374 R² = 0.8354 Friends Linear (Friends) Ninjago Linear (Ninjago) y = 0.0854x + 7.6583 R² = 0.9427 y = 0.0983x - 3.1888 R² = 0.9793 Classic Linear (Classic) Creator Linear (Creator) y = 0.0613x + 2.512 R² = 0.9519 y = 0.0499x + 59.28 R² = 0.5863
Appendix B - Scatterplot of Number of Legos versus for the Creator theme before and after adding, cheaper and smaller, 2015 sets. 2016 Sets (n = 10) 2015 and 2016 Sets (n = 34) Creator Linear (Creator) Creator Linear (Creator) y = 0.0499x + 59.28 R² = 0.5863 y = 0.068x + 6.2259 R² = 0.9176 Appendix C Sums for calculating Pearson s Correlation Coefficient and the Regression Line ΣPieces Σ ΣPieces 2 Σ 2 Σ (Pieces)() 874.9583333 85.6775 14048239.04 154984360 1405297.49
Appendix D Raw data Set ID Pieces Description Predicted Error Star Wars 75159 4016 499.99 Death Star 349.8456 150.14 Star Wars 75158 936 109.99 Rebel Combat Frigate 90.8176 19.17 Star Wars 75150 702 89.99 Vader's TIE Advanced vs. A-Wing Starfighter 71.1382 18.85 Star Wars 75156 863 89.99 Krennic's Imperial Shuttle 84.6783 5.31 Star Wars 75155 659 79.99 Rebel U-Wing Fighter 67.5219 12.47 Star Wars 75154 543 69.99 TIE Striker 57.7663 12.22 Star Wars 75148 530 59.99 Encounter on Jakku 56.673 3.32 Star Wars 75153 449 39.99 AT-ST Walker 49.8609-9.87 Star Wars 75152 385 29.99 Imperial Assault Hovertank 44.4785-14.49 Star Wars 75119 104 24.99 Sergeant Jyn Erso 20.8464 4.14 Star Wars 75120 169 24.99 K-2SO 26.3129-1.32 Star Wars 75121 106 24.99 Imperial Death Trooper 21.0146 3.98 Star Wars 75059 3296 299.99 Sandcrawler 289.2936 10.70 Star Wars 10236 1990 249.99 Ewok Village 179.459 70.53 Star Wars 75098 2144 249.99 Assault on Hoth 192.4104 57.58 Star Wars 75060 1996 199.99 Slave I 179.9636 20.03 Star Wars 75095 1685 199.99 TIE Fighter 153.8085 46.18 Star Wars 75105 1329 149.99 Millennium Falcon 123.8689 26.12 City 60125 1277 149.99 Volcano Heavy-Lift Helicopter 119.4957 30.49 City 60124 824 119.99 Volcano Exploration Base 81.3984 38.59 City 60128 184 24.99 Police Pursuit 27.5744-2.58 City 60118 248 19.99 Garbage Truck 32.9568-12.97 City 60116 183 19.99 Ambulance Plane 27.4903-7.50 City 60115 176 19.99 4x4 Off Roader 26.9016-6.91 City 60120 83 9.99 Volcano Starter Set 19.0803-9.09 City 60106 90 9.99 Fire Starter Set 19.669-9.68 City 60113 104 9.99 Rally Car 20.8464-10.86 City 60127 92 9.99 Prison Island Starter Set 19.8372-9.85 City 60114 95 9.99 Race Boat 20.0895-10.10 City 60126 47 6.99 Tire Escape 16.0527-9.06 City 60105 64 6.99 Fire ATV 17.4824-10.49 City 60098 984 199.99 Heavy-Haul Train 94.8544 105.14 City 60052 888 199.99 Cargo Train 86.7808 113.21 City 60097 1683 189.99 City Square 153.6403 36.35 City 60051 610 149.99 High-Speed Passenger Train 63.401 86.59 City 60095 717 119.99 Deep Sea Exploration Vessel 72.3997 47.59 Friends 41130 1124 99.99 Amusement Park Roller Coaster 106.6284-6.64 Friends 41134 774 79.99 Heartlake Performance School 77.1934 2.80 Friends 41133 424 39.99 Amusement Park Bumper Cars 47.7584-7.77 Friends 41129 243 29.99 Amusement Park Hot Dog Van 32.5363-2.55
Set ID Pieces Description Predicted Error Friends 41132 176 29.99 Heartlake Party Shop 26.9016 3.09 Friends 41124 286 29.99 Heartlake Puppy Daycare 36.1526-6.16 Friends 41131 218 29.99 LEGO Friends Advent Calendar 30.4338-0.44 Friends 41128 195 19.99 Amusement Park Space Ride 28.4995-8.51 Friends 41127 174 14.99 Amusement Park Arcade 26.7334-11.74 Friends 41123 77 9.99 Foal's Washing Station 18.5757-8.59 Friends 41114 54 4.99 Party Styling 16.6414-11.65 Friends 41101 1552 129.99 Heartlake Grand Hotel 142.6232-12.63 Friends 41109 692 89.99 Heatlake Airport 70.2972 19.69 Friends 41122 726 79.99 Adventure Camp Tree House 73.1566 6.83 Friends 41095 706 69.99 Emma's House 71.4746-1.48 Friends 41126 575 59.99 Heartlake Riding Club 60.4575-0.47 Friends 41106 682 59.99 Pop Star Tour Bus 69.4562-9.47 Friends 41135 597 49.99 Livi's Pop Star House 62.3077-12.32 Ninjago 70751 2028 199.99 Temple of Airjitzu 182.6548 17.34 Ninjago 70596 1253 119.99 Samurai X Cave Chaos 117.4773 2.51 Ninjago 70738 1253 119.99 Final Flight of Destiny's Bounty 117.4773 2.51 Ninjago 70595 1093 99.99 Ultra Stealth Raider 104.0213-4.03 Ninjago 70732 1069 99.99 City of Stiix 102.0029-2.01 Ninjago 70605 754 79.99 Misfortune's Keep 75.5114 4.48 Ninjago 70594 767 69.99 The Lighthouse Siege 76.6047-6.61 Ninjago 70736 658 69.99 Attack of the Morro Dragon 67.4378 2.55 Ninjago 70590 666 59.99 Airjitzu Battle Grounds 68.1106-8.12 Ninjago 70737 754 59.99 Titan Mech Battle 75.5114-15.52 Ninjago 70593 567 49.99 The Green NRG Dragon 59.7847-9.79 Ninjago 70604 450 49.99 Tiger Widow Island 49.945 0.05 Ninjago 70735 547 49.99 Ronin R.E.X. 58.1027-8.11 Ninjago 70592 439 39.99 Salvage M.E.C. 49.0199-9.03 Ninjago 70589 406 39.99 Rock Roader 46.2446-6.25 Ninjago 70731 386 39.99 Jay Walker One 44.5626-4.57 Ninjago 70734 575 39.99 Master Wu Dragon 60.4575-20.47 Ninjago 70402 350 29.99 Jay's Elemental Dragon 41.535-11.55 Classic 10705 1000 59.99 LEGO Creative Building Basket 96.2-36.21 Classic 10698 790 59.99 LEGO Large Creative Brick Box 78.539-18.55 Classic 10596 484 34.99 LEGO Medium Creative Brick Box 52.8044-17.81 Classic 10702 583 29.99 LEGO Creative Building Set 61.1303-31.14 Classic 10693 303 19.99 LEGO Creative Supplement 37.5823-17.59 Classic 10694 303 19.99 LEGO Creative Supplement Bright 37.5823-17.59 Classic 10692 221 16.99 LEGO Creative Bricks 30.6861-13.70 Creator 31046 222 19.99 Fast Car 30.7702-10.78 Creator 31045 213 14.99 Ocean Explorer 30.0133-15.02 Creator 31044 202 14.99 Park Animals 29.0882-14.10
Set ID Pieces Description Predicted Error Creator 31042 100 9.99 Super Soarer 20.51-10.52 Creator 31043 124 9.99 Chopper Transporter 22.5284-12.54 Creator 31041 64 4.99 Construction Vehicles 17.4824-12.49 Creator 31040 65 4.99 Desert Racers 17.5665-12.58 Creator 10234 2989 319.99 Sydney Opera House 263.4749 56.52 Creator 10253 4163 249.99 Big Ben 362.2083-112.22 Creator 10214 4295 239.99 Tower Bridge 373.3095-133.32 Creator 10247 2464 199.99 Ferris Wheel 219.3224-19.33 Creator 10246 2262 159.99 Detective's Office 202.3342-42.34 Creator 10243 2469 159.99 Parisian Restaurant 219.7429-59.75 Creator 10244 1746 149.99 Fairground Mixes 158.9386-8.95 Creator 10232 2194 149.99 Palace Cinema 196.6154-46.63 Creator 10218 2032 149.99 Pet Shop 182.9912-33.00 Creator 10252 1167 99.99 Volkswagen Beetle 110.2447-10.25