Importance of Statistic Analysis

Sports analysis are probably the most evident way in which we use math in basketball. However, some ignore the importance of these mathematical uses. This science helps athletes and coaches know what game structures work better for their team, overall, and against opponents; as well as being vital for personal and collective improvement. Most of this maths are based on percentages, fractions and probability. For example, statistical analysis of NBA games show that the four most important keys for team success in basketball are:
1) Shoot a high field goal percentage (40%): The benefit of a high shooting percentage is obvious. NBA teams with the higher field goal percentage won 79% of the games.
The shooting factor is measured using Effective Field Goal Percentage (eFG%). The formula for both offense and defense is (FG + 0.5 * 3P) / FGA

2) Do not commit turnovers (25%): Turnovers are important because if team cannot shoot, never mind scoring. The NBA team with fewer turnovers wins about 58% of the time.
The turnover factor is measured using Turnover Percentage (TOV%). The formula for both offense and defense is TOV / (FGA + 0.44 * FTA + TOV).

3) Get offensive rebounds (20%): Offensive rebounding can make up for poor shooting percentage, particularly as shots off of offensive rebounds are more likely to result in scores than initial field goal attempts. Offensive re-bounders increase their field goal percentage from 41% to 48% and points per play from about 0.80 to 0.94. In the NBA, if shooting percentages are about equal, the team with more offensive rebounds wins 63% of the games.
The rebounding factor is measured using Offensive and Defensive Rebound Percentage (ORB% and DRB%, respectively). The formula for offense is ORB / (ORB + Opp DRB), while the formula for defense is DRB / (Opp ORB + DRB).

4) Get to the foul line frequently (15%): Interestingly, it is more important for teams to get to the foul line frequently than it is for them to hit a high percentage out of their foul shots. The NBA team that commits fewer fouls wins 67% of the games.
The free throw factor is a measure of both how often a team gets to the line and how often they make them. The formula for both offense and defense is FT / FGA.
Teams that consistently win basketball games do at least three of these things well. If a team doesn't shoot well, it must do the other three things better. Having this numbers is of vital importance to coaches and players during their training and practices.

Source: Basketball-Reference.com

Hang Time

Hang time is what we call the time that someone spends in the air. As a kid this is perhaps one of the most impressive parts of basketball that seem so hard to understand and make the game look magical. If everything that goes up must come right back down; how do these athletes defy the laws gravity and seem to fly? 

Sadly, basketball players don't really hang in the air; it is just an illusion. When a basketball player jumps in the air to make a shot they can appear to be suspended in mid-air during the high point of the jump, but this is merely consequence of projectile motion. Projectile motion states that when an object is thrown in the air it will spend a large percentage of time in the top part of the throw. 

If you analyze it, when you jump, there are vertical and horizontal forces acting at the same time on your body (that is why a jump is represented by an arc). The vertical component of the velocity at take-off will determine the time the you spend 'hanging', while the horizontal remains constant since it is not affected by gravity. This means that, as represented by the graph below, half of your hang time is spent on the highest 25% of the jump and this creates the illusion of you floating! 

Also, the more power you have in your takeoff, the higher the arc of your jump will be and the longer you will appear to be in the air. If you want to create the illusion of a longer hang time, try some of Michael Jordan’s moves: hang on to the ball longer, pull your legs up as the jump progresses, and train to develop your leg muscles so you will be able to jump higher and seem to hang longer.

Bank Shots

Generally there are three main differences to take into account when trying to bank a shot as compared to just going for a straight-up jump shot. Obviously the idea is to use the glass, but if you're farther than about 5 feet out, aiming isn't automatic. You need to factor in:
1) The arc of your shot: generally you'll shoot in a flatter arc to hit the square of the backboard.
2) The power: depends on the distance at which you are taking the shot, and
3) The area of the backboard you want to hit.
The first part, aiming at the backboard, is pretty easy; but getting the arc and power right is a little more difficult. If you want to bank a shot with a higher arc, you need the ball to come down and hit the right area of the backboard and at the exact angle, otherwise it will result in a miss. This is why when you take a bank shot, you want to shoot a flatter angle. Another vital part of bank shots is the energy or power in your shot, as the energy transfer from the backboard is what helps make a bank shot go in the net. The spin of a basketball when it hits the backboard is where all the difference is made. When a ball makes contact with a backboard, the spin reverses and helps transfer the energy from the basketball, to the backboard, to the basket. This is why a ball with backspin has a greater chance of deflecting off the backboard and passing through the hoop.

 

Free Throws

Back to the fundamentals of basketball, I will refer to the game-changing free throw. Free throw shooting is a skill that every basketball player has to work on. To improve performance and free throw percentage, it is vital to understand the maths behind it.
3 key variables can influence you shot:
1) Release height: starting position of the ball as it leaves you hand should be in line to your eyes.
2) Ball's initial vertical velocity: a rate that includes the angle and speed of the ball as it leaves your hand , and
3) Gravity: makes the basketball accelerate as it falls.
If you manage to make this three elements work together and mix them with the right shot mechanics, you will have a consistent free throw shot.
To achieve this you need to practice applying the same force to your shots every time so that release is automatic. Try to have an open, shoulder-with, stance that maximizes body balance and release in line with the eye. Power will come from your lower body muscles of your bent legs, so there is no need to jump. Also the release is a push with you elbow locking and a flick of the wrist to 'cradle' the ball and deliver it on the line dictated by the arm´s motion; it is not meant to add any power. Finally, use your fingertips to add backspin to the free throw. This will cause the ball to follow an arc toward the hoop spinning counterclockwise. When the ball hits the back of the rim or the backboard, the velocity of the ball changes as the spin of the ball is reversed. A ball with backspin is more likely to rebound into the net due to the change in velocity, angle and spin direction. The result of all this combined is a high arching shot with great predictability and repeat-ability so you free throw shots can become mechanical.

The Ewing Theory

The other day I found something very interesting in a physics blog that I wanted to share here, so here it is.
Patrick Ewing is Jamaican-American retired Hall of Fame basketball player. During his years in the NBA and college basketball Ewing enjoyed a more than successful career being eleven times named an NBA All-Star and also part of the original Dream Team at the 1992 Barcelona Olympic Games. He was also given the honor of being named one of the 50 Greatest Players in NBA History. He was a warrior and at a yearly basis, fought for the NBA championship, but it was denied every time.
So.. why are we talking about this star baller? Well, an interesting phenomenon known as 'The Ewing Theory'.
Basically, The Ewing Theory states that a team that has a 'big man' or star player which is vital to their playing scheme, is more likely to win when that player is benched. This theory (really more like a paradox) is based on a famous the famous Traffic Theory that explains how by closing a road you can improve average traffic. The article below is written by the great physics blogger,  Brian Skinner, who is also a postdoc in theoretical condensed matter physics, and explains all this in a very straightforward and engaging way.
Gravity And Levity - Braess' Paradox and The Ewing Theory

Defense

Defense is the easiest skill to learn for a basketball player, in weeks you can become an expert. However, every defense is got to know how geometry is implied so they can know how they can (1) steal more balls, (2) avoid being tricked and (3) stop their mark from scoring. 

1) Stealing: defensive body position enables you to move quickly in any direction. The feet are slightly greater than shoulder-width apart. The body is low to the ground with the knees bent. The back is straight, not bent. Put your weight on the balls of your feet. The fingers are spread apart ready to catch the ball.

2) Confidence: in defensive position, an instant of improper movement like sliding gives the offense an opening. You need to practice staying with the offensive player. Practice maintaining a distance of from two meters to as little as one foot from the mark.

3) Not losing your mark: always force the offense to one side or the other whether or not they have the ball. You need to be aware of which way, right or left, the offense likes or wants to move, and then prevent them from doing so. Forcing involves setting up at about a 45º angle to the mark, one foot to the side. This positioning makes it easier for the mark to go around you one way (the way you want them to) and more difficult to go the other way.

Once again, it is VERY IMPORTANT to be careful with the defensive angles at which you stand your man, when you are half-sided, you have an advantage, as he cannot get pass you as easy as if you faced him directly with no body angle. The same is applied to the angle of you legs on defensive position, the more folded your legs the faster you are. Finally realize that when you displace with the balls of your feet, you also are faster, as there is faster contact with the floor.

Rebounding

60% of rebounding is pure geometry, the rest is wanting to get the ball and timing your jump. In a field shoot, whenever the ball is shot from one side, it will end on the other side most of the time; when shot from the free throw zone it will rebound on the same side. Also, when more force is applied, logically, the rebound will get further. On the other hand, the smaller the force applied, the ball will fall nearer to the place it hit the board or rim. So familiarize yourself with rebounding.
In addition to all this, you should "box-out" (push out your mark so your the closest to the board) to rebound. Anticipation is a great factor in your effectiveness, you not only need to predict where a missed shot will go to be a good rebounder, but also move toward it long before the shot hits the rim.