NBA Real Plus-Minus

ESPN just came out with a new advanced metric called Real Plus-Minus (RPM). It is intended to be a perfected form of the normal plus-minus stat, which measures the gains or losses of points while a given player is on the field. The issue with the original plus-minus stat is that it doesn't factor in the impact of the other players on the court at the same time as the given player, so mediocre players who play alongside superstars will often excel in this stat as much as their superior teammates.

Nothing has been revealed about the actual calculation of this stat yet, but it allegedly factors in numerous variables on both offense and defense to calculate the number of points a player contributes to or takes away from his team per 100 possessions. Although it's difficult to do anything to calculate using this metric because the algorithm is not given, we can use the already-calculated RPM numbers to compare and contrast different teams.

Comparison of the Oklahoma City Thunder and the Miami Heat based on the RPM of starters:

OKC:

Russell Westbrook: 4.09

Thabo Sefolosha: 0.37

Kevin Durant: 6.77

Serge Ibaka: 3.10

Kendrick Perkins: -3.19

Total: 11.14

MIA:

Mario Chalmers: 1.48

Dwyane Wade: 2.13

Shane Battier: 1.60

LeBron James: 8.11

Chris Bosh: 3.75

Total: 17.07

Interestingly, Nick Collison, a backup big-man for the Thunder, is the sixth-best player in the NBA according to this statistic with an RPM of 5.81. Collison is the same height as Kendrick Perkins, whose RPM is a lowly -3.19; if they inserted Collison into the starting lineup for Kendrick Perkins, their RPM total for starters would be increased to 20.14! 

Although this stat is new and certainly imperfect to some degree, it does bring the value of some underrated players to light. Maybe this switch would improve OKC and make them even better than the defending-champ Heat.

Source:
http://espn.go.com/nba/statistics/rpm/_/sort/RPM

11.2 Vectors In Space

This section is about the properties of vectors in 3-D space. The component (regular) form of a vector is
v = <v1,v2,v3>. As we know from earlier chapters, a vector is the difference between a terminal and initial point. Since this is in three dimensions, the three numbers inside the angle brackets are the differences between the x-values, y-values, and z-values, respectively, of a terminal and initial point.


Length of a vector: ||v|| = squareroot(v1^2 + v2^2 + v3^2)

• This equation makes sense, as the length of a vector would be the distance between its initial point and its terminal point. Each vn value is the difference between the x-, y-, or z-value of the terminal and initial points, so this length equation is the same as the basic distance formula we all know and love!

11.4 Lines and Planes In Space

In this lesson, we learned how to find the parametric and symmetric equations of a line in a 3-D space and how to find the angle and distance between two planes. In this blog, I'm going to focus on planes. The basic equation is:

a(x-x1) + b(y-y1) + b(z-z1) = 0

In this equation:

• a,b,c are the x, y, and z components of a perpendicular (normal) vector. n = <a,b,c>
• x1, y1, and z1 are the coordinates of a point on the plane.
• x,y, and z are variables. 

Many problems will give hints and ask you to find the equation of a plane. In the following example, a point on the plane and a perpendicular vector are given:

Cool 3D Shapes

The Importance of Steals

I read an interesting article about the importance of steals in the NBA last night. The writer, Benjamin Morris, measured the performance of teams with and without certain players. He compared the team's points per game with and without the player to the player's average stats and a replacement player's average stats; the chart below shows his results. For example, if a team is playing without a player that gets 12 rebounds per game who is being replaced by a player with just 2 rebounds per game, the team will probably average 17 fewer points.

As you can see in this chart, a steal is 9 times more valuable than a point! It's high value is understandable, as a steal will almost always result in two easy points in the resulting fast-break scoring opportunity and also shifts the momentum, but it's amazing that such an overlooked stat is so valuable. It's easier to glorify high-scoring players because the numbers are so much grander; Kevin Durant is leading the league with 32.2 points per game, while Chris Paul leads the league in steals with 2.53 steals per game. But let's look at the stats of these two players with the logic of this chart:

Durant:

32.2 PPG

1.3 STLPG

0.8 BLKPG

5.6 APG

7.6 RPG

3.6 TOPG

32.2(1) + 1.3(9.1) + 0.8(6.1) + 5.6(2.2) + 7.6(1.7) - 3.6(5.4) = 54.71

Chris Paul:

18.5 PPG

2.5 STLPG

0.1 BLKPG

11.0 APG

4.3 RPG

2.4 TOPG

18.5(1) + 2.5(9.1) + 0.1(6.1) + 11(2.2) + 4.3(1.7) - 2.4(5.4) = 60.41

With the logic of a player's contributions compared to a replacement player, it seems like our own Chris Paul of the Los Angeles Clippers is an even better player than Kevin Durant, the leading MVP candidate!

Works Cited:

http://espn.go.com/nba/player/_/id/2779/chris-paul

http://espn.go.com/nba/player/_/id/3202/kevin-durant

http://fivethirtyeight.com/features/the-hidden-value-of-the-nba-steal/

10.8 Polar Equations of Conics

In this section, we are finding the polar equations of conics and graphing them on polar graphs. In this example, we are given the type of conic (parabola), the eccentricity (1), and the directrix (y=-2). With a horizontal directrix below the origin, we can tell that the equation's denominator will be (1-esinØ). The directrix is also 2 units away from the focus [which is always at (0,0) for these problems], so we know that p is 2. The bottom row of this picture displays how to insert those numbers and find the final equation (boxed on the right).

10.6 Polar Coordinates

This section introduces us to graphing using polar coordinates rather than the usual rectangular (x,y) coordinates. Polar coordinates are (r,Ø) [that Ø is the closest thing to theta I could get]. The four equations in the picture below are used to transfer polar equations into rectangular equations and vice versa. 

This next picture is an example of changing a polar equation into a rectangular one. Knowing the angle, you can plug it into the top-left equation from the above picture. The final answer is boxed below:

This printable paper is helpful to use for graphing polar equations:

http://www.embeddedmath.com/downloads/files/polargraph/polargraph-letter.pdf