Math SL Internal Assessment What is the relationship between free throw shooting percentage and 3 point shooting percentages?

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1 Math SL Internal Assessment What is the relationship between free throw shooting percentage and 3 point shooting percentages? fts6

2 Introduction : Basketball is a sport where the players have to be adept in many different skills such as: dribbling, shooting, passing, running, and jumping. These skills are very important if you want to be a successful in your basketball career. All of these skills take a lot of effort in order to develop. One skill in particular takes hours of practice, shooting. This the way teams earn their points during a game. Regular field goals within the 3 point line are worth 2 points, while field goals made beyond the 3 point line are all worth 3 points. The shots beyond the 3 point line are much harder to hit than the ones in the 3 point line. There is another type of shot that is worth only 1 point, these are the ones made by a free throw. You are given the opportunity to shoot a free throw shot if you are fouled. These can be the easiest points to earn, or the hardest. If you are good at free throw shots, which in turn improves your shooting in general, will you also be good at 3 point shots? The data collected from basketball players shooting percentages were used to analyze whether there is a relationship between free throw percentages and 3 point shooting percentages. Statement of Task : The main purpose of this investigation is to determine whether there is a relationship between free throw percentages and 3 point percentages of players in the NBA. The type of data that has been collected come from Basketball-Reference.com, a site which compiles statistics of basketball players. The stats are used to show if having a higher free throw percentage will result in a higher 3 point percentage. The data used consists of NBA basketball players, showing their free throw percentages and 3 point percentages for the NBA season. We go further to see whether this data stays consistent by averaging the season from Hypothesis : If you have a higher free throw throw percentage, then you will have a 3 point shooting percentage. Plan of Action : I am investigating the relationship between free throw percentages and 3 point percentages. I have collected data from many different NBA basketball players. I will use many mathematical processes to analyze the data including: a scatter plot of the data, a X 2 test on the data to show the dependence of free throw shooting percentage to 3 point shooting percentage of the NBA players. I will also calculate the least square regression line and correlation coefficient.

3 Mathematical Investigation Table 1: Free throw and 3 point shooting percentages & Free throw and 3 point shooting percentages Player Name Free Throw Percentage ( ) 3 Point Percentage ( ) Free Throw Percentages (Averaged between 5 seasons) ( ) 3 Point Percentages (Averaged between 5 seasons) ( ) Steve Nash 91.7% 33.3% 92% 40% Stephen Curry 88.5% 42.4% 88.3% 44.2% Ray Allen 90.5% 37.5% 90% 41.1% J.J. Redick 91.5% 39.5% 89.22% 39.6% Kevin Durant LeBron James Dwyane Wade Michael Beasley 87.3% % % % 35.7% 74.1% % 29.4% 75.6% 38.9% 73.9% 34.4% Tony Parker 79.2% 37.3% 78.4% 32.7% LaMarcus Aldridge DeAndre Jordan Dwight Howard Jason Maxiell 81.8% 27.8% 79.8% 22.2% 42.8% 0% 43.3% 0% 54.7% 28.6% 54.4% 9.1% % 54.8% 0%

4 Andrew Bogut 34.4% 0% 50.5% 0% Reggie Evans % 51.3% 0% This data table shows all of the free throw shooting percentages and 3 point shooting percentages of each player used in this investigation. I chose players which free throw shooting percentage were really good to poor, in order to better get a better range of data. The data is from the season. I also included an average of the season from in order to see if the trend is accurate. From the data about the 2 sets of data seems to be similar. ( and ) Graph #1 shows the relationship between free throw percentages to 3 point shooting percentages. It appears that there is a positive correlation. This might mean that the better your free throw percentage the better your 3 point shooting percentage is. (Graph was generated through Microsoft Excel) Table 2 Least Square Regression Values:

5 The linear fit that matches the pattern of a set of paired data as closely as possible. Out of all possible linear fits, the least - squares regression line is the one that has the smallest possible value for the sum of the squares of the residuals. It shows the relationship between the independent variable,x, to the dependent variable,y, by using the equation: xy Sxy y y = Sx 2 (x x) where Sxy = n xy and S x 2 = x 2 x 2 n X Y X Y X 2 92% 40% % 44.2% % 41.1% % 39.6% % % 35.7% % 29.4% % 34.4% % 32.7% % 22.2% % 0% % 9.1% % 0% % 0% % 0% = =366.6 = = x =72.23 y =24.44 xy = x 2 = S xy = xy n x y S xy =

6 S xy = S x = S x = x 2 x 2 n S x = Sxy y y = Sx 2 (x x ) y = (x 72.23) (266.93) 2 y = x Calculation of Pearson s Correlation Coefficient A statistic measuring the linear interdependence between two variables or two sets of data. Table 3: Value of Pearson s Correlation Coefficient X Y (x-x ) 2 (y-ȳ) 2 92% 40% % 44.2% % 41.1% % 39.6% % % 35.7% % 29.4% % 34.4% % 32.7% % 22.2% % 0% % 9.1%

7 54.8% 0% % 0% % 0% = =366.6 = = x =72.23 y =24.44 μ X: μ Y: XY: r = Sxy (x x) 2 (y y) where S, Sy = 2 Sxy is the Covariance (Sx)(Sy) x = Sxy is the Covariance S xy = S x = xy n x y S y = S y = r = (16.33)(16.89) r =. 95 r 2 =. 9 The correlation coefficient value is.95 this shows that the is a strong positive correlation for free throw shooting percentages to 3 point shooting percentages.

8 This shows the best fit line for the set of data. It shows that there is a positive correlation. This is reinforced by the correlation coefficient, which is over a correlation coefficient of.7 which also shows that there is a positive correlation. Chi-Squared Test The Chi-Squared test is used to measure if the two variable are independent of one another, or if there is a relationship. X 2 = (fo fe) 2 fe Calculation of the Chi-Squared Test df =(Rows-1)(Columns-1) Degrees of Freedom: Measure the number of values in the final calculation that are free to vary. Table 4: Observed Values Total Total 9 6

9 Table 5: Calculations for Expected Values Total (5)(9) (10)(9) (5)(6) (10)(6) 5 10 Total 9 6 Table 6: Expected Values B1 B2 Total A A Total 9 6 X 2 = (fo fe) 2 fe X 2 (5 3) = ( 2 (0 2) ) + ( 2 (4 6) ) + ( 2 ) + ( ) X 2 = d f = ( rows 1)(Columns 1 ) d f + ( 2 1)(2 1 ) d f = 1 Null Hypothesis: Free throw shooting percentage and 3 point shooting percentage are independent of each other. Alternative Hypothesis: Free throw shooting percentage and 3 point shooting percentage are not independent on each other. The Chi-Squared value has a significance value of.05 or 5% with 1 degree of freedom, the critical value is The Chi-Squared Number is greater than the critical value, 4.999>3.841, this means that the null hypothesis was rejected. 3 point shooting (6 4) 2

10 percentage is not independent of the free throw shooting percentage Continuing Investigation This graph shows the correlation coefficient to be.78 This shows that there is a very strong positive correlation to the graph for the game seasons The Chi-Squared value for the game seasons is 6.56 Null Hypothesis: Free throw shooting percentage and 3 point shooting percentage are independent of each other. Alternative Hypothesis: Free throw shooting percentage and 3 point shooting percentage are dependent on each other. This is a scatter plot from the season This shows the free throw shooting percentages to 3 point shooting percentages. This graph helps reinforce that there is a positive relationship between free throw percentages and 3 point percentages. The correlation coefficient is.78. This shows that there is a strong positive correlation. This is also reinforced by the Chi-Squared test, the Chi-Squared value was 6.56, which was greater than the critical value of Discussion/Validity Limitations: There are some limitations from this investigation of whether there is a correlation between free throw percentages to 3 point percentage. One limitation of this experiment is that it only tested the basketball players that were in

11 the NBA. It neglected to include basketball players that were in other countries, or in other basketball leagues. Another limitation is that there are different basketball positions in basketball. This means that in games these players may not shoot 3 point shooters in general. This would be because each position has a certain job, and those jobs don t all consist of shooting 3 point shots. So they might have the ability to shoot the 3 point shots or even the free throw shots, but not have the opportunity to. Lastly, this doesn t take into consideration other factors such as: sickness, injury, homecourt advantage etc. These factors may play a big role in their shooting percentages in both 3 point and free throw shooting. Conclusion Despite the limitations, the data shows that there is a strong positive relationship between free throw shooting percentages to 3 point shooting percentages. The correlations coefficient for both sets of data were higher than.50. Also for both sets of data it showed a Chi-Squared number of, 6.565, for the season, and, 4.99, for the averaged seasons from These Chi-Squared values reject the null hypothesis, free throw shooting percentage and 3 point shooting percentage are independent of each other, and accepts the alternative hypothesis, free throw shooting percentage and 3 point shooting percentage are not independent on each other. This data strongly supports that there is a correlation between free throw shooting percentages and 3 point shooting percentages.

12 Cited Sources "Basketball Statistics and History Basketball-Reference.com." N.p., n.d. Web. 21 Oct

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