In last year's conference semis, Lebron James, with just 1.5 seconds left and the game tied, hit a 21 foot shot from the corner to give the Cavaliers a victory over the Bulls, 86-84. Said James after the game:
I was supposed to take the ball out. I told coach, 'There's no way I'm taking the ball out unless I can shoot it over the backboard and go in.' So I told him to have somebody else take the ball out, give me the ball and everybody get out the way.
In clutch situations, teams tend to run their offense through their stars (or in Lebron's case, the star runs the offense). At its best, you get iconic moments like Michael Jordan's game winner in the '98 Finals and Lebron's aforementioned heroics. At its worst, you get low efficiency desperation, clock draining iso play, and heavily contested, off the mark mid-range jumpers (sarcastically referred to as "hero ball").

Earlier this season, I launched clutch shooting reports. This interactive report uses win probability to classify shots into four basic categories: garbage time, "normal" basketball, clutch, and "double" clutch. That fourth category, double clutch, represent the top 1% of shots in terms of potential impact on a team's chances of victory. The following chart summarizes these four phases as a function of score difference and time remaining.

For double clutch situations (that splotch of red on the right), shot attempts are more concentrated among a handful of a team's key starters. At the other end, garbage time, shot distribution is much more spread out. Teams put their bench players on the court, and, in general, starters and bench warmers alike are more prone to chuck it when the stakes are low.

Is there a way to quantify this shift in shot distribution? If so, we can also quantify which teams are most prone to "hero ball" - the tendency to run the offense through a single player (or players) when the stakes are the highest.

### I Dream of Gini

There are a multitude of ways to quantify how equally, or unequally, a resource is shared among a group of people. The most common of these is the Gini coefficient, and its most frequent use is to measure income inequality among nations. A Gini coefficient scales from 0 to 1, with a greater coefficient implying greater inequality. A nation in which each citizen had the exact same income would have a Gini coefficient of 0. A nation in which a single individual hoarded all the wealth would have a Gini coefficient of 1.

If, instead of income, we apply the Gini coefficient to player shot attempts, we get the following average Gini values for each of the four phases:

PhaseAverage Gini
Garbage Time0.263
Clutch0.343
Double Clutch0.524
All Shots0.292

For all shots, NBA teams spread the wealth about as well as Denmark, with a Gini coefficient of 0.292. Garbage time is the most egalitarian, with a Norwegian coefficient of 0.263. But for the biggest shots, things go from Nordic to tinpot. Double clutch shots have a Gini coefficient of 0.524, comparable to that of Guatemala's. A technical note: the Gini coefficient used here is the weighted version, where the weight used is minutes played (this prevents a player with just 15 minutes played over a season from skewing the overall Gini coefficient).

Here is a breakdown, by team, of the Gini coefficient for double clutch attempts.

Cleveland sits atop the hero ball index. 72% of "double clutch" shot attempts have been taken by a single player, Lebron James. The next two teams on the list, Miami and Oklahoma City, are similar in that two stars dominate in crunch time. For Miami, it is Chris Bosh and Dwyane Wade, with 34% and 49% of double clutch attempts, respectively. As one would expect, for the Thunder, it's Russell Westbrook and Kevin Durant, with 39% and 48% of attempts.

The Memphis Grizzlies are the "not it" team this year. No player on the Grizzlies accounts for more than 20% of double clutch shot attempts.

Here are double clutch shot attempts by player for the Cavaliers:

"Everybody get out the way" indeed.

The table below has Gini coefficients by team for the past seven seasons. The 2011-12 Oklahoma City Thunder have the highest "Hero Ball Index" over that time period. Kevin Durant and Russell Westbrook accounted for 95% of the team's double clutch shot attempts, leading to a Gini coefficient of 0.749. In fact, five of the top eight spots on the hero ball index belong to the Thunder.

Gini Coefficient for Shot Attempts
TeamTotal Grbg Nrml Cltch DblCl
OKC (11-12) 0.417 0.410 0.412 0.491 0.749
CLE (15-16) 0.333 0.262 0.347 0.369 0.737
NYK (11-12) 0.304 0.246 0.317 0.421 0.719
MIA (15-16) 0.281 0.300 0.272 0.385 0.694
OKC (12-13) 0.379 0.308 0.388 0.516 0.682
OKC (15-16) 0.341 0.269 0.350 0.416 0.664
OKC (13-14) 0.397 0.325 0.407 0.524 0.660
OKC (14-15) 0.388 0.313 0.399 0.480 0.652
TOR (15-16) 0.312 0.307 0.310 0.389 0.650
POR (14-15) 0.375 0.360 0.383 0.399 0.640
DAL (09-10) 0.370 0.346 0.364 0.480 0.639
CHI (09-10) 0.293 0.211 0.306 0.357 0.638
LAC (12-13) 0.263 0.256 0.279 0.448 0.637
CHA (11-12) 0.266 0.273 0.275 0.377 0.635
CHA (15-16) 0.280 0.302 0.267 0.411 0.633
UTA (14-15) 0.295 0.269 0.299 0.292 0.631
CHI (10-11) 0.359 0.302 0.360 0.450 0.630
OKC (10-11) 0.380 0.335 0.377 0.439 0.627
IND (13-14) 0.312 0.302 0.315 0.362 0.627
HOU (14-15) 0.351 0.319 0.360 0.360 0.627
DAL (14-15) 0.345 0.316 0.352 0.416 0.625
NOP (14-15) 0.347 0.280 0.355 0.383 0.624
DAL (10-11) 0.324 0.335 0.312 0.485 0.624
CLE (13-14) 0.313 0.303 0.311 0.426 0.624
CHA (13-14) 0.367 0.348 0.368 0.388 0.616
CHA (14-15) 0.285 0.253 0.294 0.340 0.609
LAL (12-13) 0.330 0.322 0.331 0.413 0.607
WAS (11-12) 0.337 0.314 0.349 0.393 0.606
LAL (14-15) 0.202 0.171 0.216 0.278 0.605
PHX (15-16) 0.254 0.292 0.252 0.386 0.604
SAS (12-13) 0.265 0.118 0.289 0.341 0.602
NOH (12-13) 0.296 0.273 0.293 0.407 0.602
CLE (11-12) 0.358 0.271 0.382 0.329 0.602
CHA (12-13) 0.328 0.275 0.330 0.460 0.600
LAL (11-12) 0.393 0.399 0.389 0.406 0.598
LAC (14-15) 0.294 0.238 0.309 0.380 0.597
SAS (15-16) 0.258 0.193 0.284 0.300 0.594
CLE (09-10) 0.332 0.322 0.336 0.424 0.594
LAL (13-14) 0.276 0.252 0.295 0.321 0.594
GSW (09-10) 0.378 0.336 0.394 0.423 0.592
MIN (15-16) 0.330 0.382 0.318 0.405 0.588
GSW (15-16) 0.351 0.282 0.377 0.362 0.587
CHI (11-12) 0.229 0.188 0.256 0.269 0.586
MIL (12-13) 0.352 0.312 0.352 0.429 0.585
NYK (13-14) 0.382 0.383 0.377 0.458 0.585
HOU (12-13) 0.317 0.231 0.328 0.387 0.584
MIL (15-16) 0.267 0.205 0.275 0.255 0.584
NYK (12-13) 0.396 0.397 0.399 0.477 0.583
MIN (10-11) 0.307 0.248 0.320 0.330 0.581
DAL (11-12) 0.316 0.325 0.303 0.466 0.581
DAL (13-14) 0.305 0.274 0.299 0.432 0.578
OKC (09-10) 0.345 0.313 0.350 0.407 0.578
MIL (11-12) 0.337 0.284 0.345 0.416 0.576
MIA (14-15) 0.353 0.318 0.363 0.426 0.576
MIA (12-13) 0.345 0.308 0.347 0.413 0.575
SAC (14-15) 0.302 0.291 0.313 0.256 0.574
CLE (14-15) 0.370 0.309 0.381 0.458 0.572
MEM (13-14) 0.301 0.261 0.322 0.334 0.571
ATL (12-13) 0.336 0.285 0.350 0.370 0.570
CLE (12-13) 0.290 0.248 0.293 0.372 0.570
LAC (13-14) 0.318 0.275 0.326 0.369 0.569
SAS (11-12) 0.280 0.204 0.299 0.376 0.569
ATL (10-11) 0.317 0.258 0.328 0.378 0.569
MIA (13-14) 0.342 0.297 0.356 0.378 0.568
UTA (13-14) 0.232 0.239 0.236 0.310 0.568
UTA (11-12) 0.309 0.267 0.324 0.285 0.565
HOU (13-14) 0.288 0.223 0.301 0.346 0.565
DEN (14-15) 0.286 0.223 0.306 0.338 0.565
BKN (13-14) 0.290 0.237 0.313 0.304 0.565
LAC (10-11) 0.341 0.305 0.352 0.366 0.565
IND (09-10) 0.236 0.204 0.259 0.216 0.563
SAC (13-14) 0.364 0.332 0.374 0.376 0.563
NYK (15-16) 0.252 0.244 0.263 0.316 0.562
HOU (15-16) 0.288 0.243 0.289 0.385 0.560
IND (12-13) 0.279 0.237 0.287 0.352 0.560
BKN (15-16) 0.282 0.162 0.311 0.232 0.558
WAS (14-15) 0.282 0.226 0.302 0.341 0.558
LAC (11-12) 0.331 0.327 0.339 0.401 0.557
MIL (13-14) 0.306 0.277 0.311 0.353 0.552
MIA (10-11) 0.428 0.414 0.430 0.494 0.552
LAL (10-11) 0.315 0.273 0.323 0.376 0.552
BOS (11-12) 0.320 0.233 0.334 0.385 0.552
PHX (14-15) 0.290 0.254 0.297 0.351 0.551
SAC (09-10) 0.275 0.280 0.271 0.349 0.551
ATL (09-10) 0.256 0.268 0.247 0.396 0.550
GSW (13-14) 0.325 0.260 0.330 0.414 0.549
PHI (13-14) 0.329 0.266 0.345 0.365 0.546
WAS (10-11) 0.331 0.314 0.348 0.326 0.546
LAL (09-10) 0.255 0.293 0.275 0.365 0.544
GSW (10-11) 0.353 0.310 0.361 0.421 0.543
NOP (15-16) 0.340 0.324 0.345 0.395 0.541
GSW (14-15) 0.309 0.234 0.334 0.319 0.541
ORL (10-11) 0.287 0.226 0.306 0.277 0.541
TOR (12-13) 0.312 0.314 0.321 0.357 0.537
SAS (10-11) 0.238 0.210 0.261 0.298 0.537
MIN (09-10) 0.266 0.223 0.280 0.307 0.536
DEN (09-10) 0.318 0.274 0.325 0.420 0.533
POR (10-11) 0.309 0.253 0.318 0.333 0.532
NJN (11-12) 0.346 0.304 0.360 0.423 0.531
PHX (09-10) 0.256 0.152 0.288 0.265 0.531
UTA (12-13) 0.274 0.236 0.297 0.313 0.530
DEN (10-11) 0.250 0.323 0.248 0.344 0.528
MIN (13-14) 0.313 0.237 0.335 0.313 0.527
MEM (14-15) 0.277 0.166 0.297 0.287 0.525
ATL (14-15) 0.251 0.186 0.269 0.262 0.524
LAC (09-10) 0.337 0.283 0.359 0.315 0.522
POR (09-10) 0.332 0.266 0.342 0.365 0.522
MIA (11-12) 0.365 0.329 0.374 0.372 0.521
BOS (12-13) 0.270 0.229 0.268 0.420 0.520
SAC (12-13) 0.271 0.241 0.289 0.320 0.520
TOR (13-14) 0.337 0.362 0.330 0.386 0.517
POR (15-16) 0.310 0.287 0.312 0.405 0.516
ATL (13-14) 0.264 0.204 0.278 0.265 0.516
IND (15-16) 0.299 0.295 0.294 0.342 0.516
PHI (11-12) 0.193 0.234 0.186 0.310 0.514
UTA (10-11) 0.338 0.312 0.342 0.375 0.514
BKN (12-13) 0.315 0.286 0.330 0.350 0.509
CHI (15-16) 0.277 0.253 0.284 0.319 0.509
MIL (09-10) 0.289 0.268 0.303 0.337 0.508
IND (10-11) 0.312 0.254 0.325 0.372 0.507
SAC (11-12) 0.308 0.259 0.324 0.328 0.506
ORL (09-10) 0.208 0.113 0.239 0.264 0.506
HOU (09-10) 0.312 0.341 0.315 0.290 0.505
DET (13-14) 0.243 0.200 0.259 0.254 0.505
DET (15-16) 0.232 0.241 0.222 0.299 0.504
DET (14-15) 0.293 0.314 0.305 0.251 0.504
MEM (12-13) 0.265 0.219 0.278 0.275 0.503
ORL (12-13) 0.265 0.225 0.282 0.317 0.501
MIN (14-15) 0.336 0.352 0.344 0.327 0.501
TOR (10-11) 0.336 0.318 0.346 0.369 0.500
BOS (09-10) 0.262 0.217 0.272 0.288 0.499
TOR (09-10) 0.266 0.167 0.286 0.335 0.499
DAL (15-16) 0.205 0.229 0.217 0.287 0.498
DET (09-10) 0.270 0.241 0.277 0.362 0.496
POR (11-12) 0.304 0.301 0.308 0.386 0.495
WAS (13-14) 0.310 0.264 0.316 0.322 0.495
BOS (14-15) 0.256 0.215 0.270 0.249 0.494
BKN (14-15) 0.283 0.228 0.290 0.394 0.494
ATL (11-12) 0.342 0.206 0.378 0.361 0.493
TOR (14-15) 0.211 0.259 0.198 0.340 0.493
ORL (14-15) 0.322 0.283 0.338 0.252 0.487
UTA (09-10) 0.257 0.235 0.269 0.246 0.485
WAS (15-16) 0.258 0.252 0.271 0.353 0.485
NOH (09-10) 0.268 0.320 0.280 0.304 0.483
ORL (13-14) 0.237 0.240 0.245 0.272 0.482
POR (13-14) 0.287 0.288 0.282 0.323 0.482
DAL (12-13) 0.263 0.239 0.262 0.377 0.482
PHI (14-15) 0.249 0.251 0.261 0.247 0.481
NOH (10-11) 0.261 0.226 0.267 0.342 0.479
PHI (09-10) 0.251 0.244 0.254 0.314 0.479
NYK (14-15) 0.266 0.258 0.288 0.316 0.475
SAS (14-15) 0.225 0.190 0.255 0.272 0.474
CHI (14-15) 0.247 0.231 0.263 0.277 0.473
MEM (10-11) 0.256 0.222 0.263 0.302 0.473
SAS (13-14) 0.236 0.187 0.273 0.255 0.472
MIN (11-12) 0.280 0.240 0.296 0.320 0.471
CHA (10-11) 0.310 0.240 0.318 0.353 0.470
POR (12-13) 0.322 0.258 0.332 0.333 0.470
WAS (09-10) 0.309 0.370 0.312 0.355 0.466
NJN (10-11) 0.317 0.240 0.328 0.365 0.466
CLE (10-11) 0.315 0.259 0.333 0.355 0.465
NOH (11-12) 0.262 0.266 0.270 0.294 0.464
LAC (15-16) 0.286 0.290 0.286 0.347 0.463
PHI (15-16) 0.139 0.165 0.145 0.240 0.462
HOU (10-11) 0.303 0.267 0.321 0.259 0.461
TOR (11-12) 0.295 0.299 0.316 0.311 0.461
MIL (10-11) 0.236 0.190 0.242 0.320 0.459
DEN (15-16) 0.255 0.334 0.258 0.337 0.458
ORL (15-16) 0.241 0.191 0.257 0.260 0.458
CHA (09-10) 0.330 0.242 0.342 0.373 0.458
SAS (09-10) 0.246 0.219 0.264 0.278 0.453
PHI (12-13) 0.289 0.276 0.289 0.352 0.452
ATL (15-16) 0.248 0.206 0.256 0.296 0.450
ORL (11-12) 0.224 0.176 0.243 0.247 0.449
BOS (10-11) 0.316 0.330 0.318 0.359 0.445
NJN (09-10) 0.303 0.271 0.316 0.315 0.445
MIL (14-15) 0.249 0.259 0.254 0.283 0.444
NYK (10-11) 0.346 0.362 0.339 0.438 0.442
SAC (15-16) 0.241 0.220 0.243 0.311 0.439
DET (11-12) 0.264 0.203 0.296 0.289 0.439
PHX (10-11) 0.234 0.221 0.248 0.264 0.438
MEM (09-10) 0.291 0.260 0.302 0.290 0.437
PHX (11-12) 0.199 0.186 0.244 0.193 0.437
BOS (13-14) 0.313 0.295 0.319 0.317 0.433
SAC (10-11) 0.228 0.205 0.237 0.232 0.432
MIA (09-10) 0.348 0.303 0.371 0.338 0.429
DEN (11-12) 0.273 0.241 0.282 0.361 0.425
NOP (13-14) 0.299 0.314 0.300 0.357 0.422
DET (12-13) 0.269 0.256 0.294 0.305 0.419
GSW (12-13) 0.307 0.274 0.310 0.394 0.418
IND (14-15) 0.206 0.163 0.224 0.188 0.416
UTA (15-16) 0.309 0.264 0.315 0.349 0.415
PHX (13-14) 0.218 0.220 0.226 0.251 0.414
DEN (12-13) 0.180 0.182 0.180 0.312 0.414
DEN (13-14) 0.219 0.187 0.239 0.191 0.412
DET (10-11) 0.232 0.195 0.249 0.272 0.411
CHI (12-13) 0.249 0.238 0.263 0.311 0.410
GSW (11-12) 0.327 0.310 0.345 0.359 0.404
BOS (15-16) 0.247 0.222 0.259 0.273 0.404
IND (11-12) 0.247 0.186 0.264 0.279 0.404
MIN (12-13) 0.205 0.288 0.225 0.341 0.400
PHI (10-11) 0.187 0.213 0.193 0.297 0.394
LAL (15-16) 0.267 0.223 0.284 0.368 0.391
HOU (11-12) 0.193 0.214 0.205 0.229 0.389
MEM (11-12) 0.281 0.272 0.290 0.348 0.370
MEM (15-16) 0.227 0.159 0.250 0.265 0.367
WAS (12-13) 0.222 0.245 0.234 0.254 0.365
PHX (12-13) 0.219 0.183 0.235 0.203 0.345
CHI (13-14) 0.176 0.171 0.194 0.285 0.318
NYK (09-10) 0.339 0.334 0.348 0.318 0.314