It was one of the best highlights of the 2015 – 2016 NBA season. February 27 2016, Golden State vs Oklahoma, 4 seconds left in the overtime … game tied, 118 – 118. Golden State guard and league MVP Stephen Curry dribbles the ball up court along the left side. Three point three seconds left … Curry reaches half court. Two point nine seconds … Curry has taken about two steps past the half court line (about 10 feet behind the three-point line and about 33 feet from the basket) and decides he is close enough to the basket to launch a jump shot! Two point three seconds… the ball leaves Curry’s hands. Zero point six seconds … swish! 121 – 118 Golden State wins the game! The game winning shot was Curry’s 12th three pointer of the game, tying an NBA single game record (he finished the game with 46 points).
Despite losing to LeBron James and the Cleveland Cavaliers in the finals, Stephen Curry and the Golden State Warriors had a memorable 2015 – 2016 season. They broke over 25 NBA records and set ten franchise records including setting the best ever regular record of 73 – 9 eclipsing the 95 – 96 Chicago Bulls 72 – 10 record. There is no secret to the success of the 2015 – 2016 Golden State Warriors … three-point shooting. They set the following 3-point shooting records …
- Most three pointers made in a regular season (averaging an NBA record of 13.1 per game)
- Most three pointers made in a post season (306, breaking their own record set in the 2015 playoffs)
- Most three pointers made in a 7 game series (90 against the Oklahoma Thunder)
- Most three pointers made in a single NBA finals game 17 in game 4 against Cleveland
- Stephen Curry set the record for most three pointers made in a regular season (402), as well as most three pointers made in a playoff series (32), and most consecutive games (regular season and playoffs combined) with a three pointer made (191) this streak is still active!
- Forward Draymond Green tied the record for the most 3 pointers made in a game 7 of an NBA final (6)
Anybody who watched the Golden State Warriors during their remarkable 2015 – 2016 season can most certainly come to one common conclusion, Stephon Curry is a three-point shooting genius! The image of him firing up a three pointer then turning his back to the basket and celebrating before the shot even goes in will forever define Curry’s remarkable 2015 – 2016 MVP season, where he averaged 30.1 points per game and 6.7 assists per game while hitting more than 50% of his shots.
Sports fans and writers often embellish the accomplishments of their sporting hero’s using terms like “legendary”, “historic” or “genius” to describe them and their accomplishments when in reality a term such as “excellent” or “very good” may be more fitting. Just how much better is Curry at shooting from long range than the rest of his peers? Is he truly a genius when it comes to shooting from long distance or is he just a very good shooter? Let’s take a look at what the statistics tell us. Technically a “genius” is a classification given to anyone who scores in the top 2% of any approved standardized intelligence test, in other words geniuses do better than 98% of the people who also took the test. There is no approved standardized test for long distance shooting, but we do have statistics from 82 regular season games. According to stats.nba.com when shooting from a range of 20 to 24 feet, Curry averaged 2 made field goals a game the league average (mean) was 0.515 per game. When shooting from 25 to 29 feet Curry averaged 3.2 made field goals a game whereas the league average (mean) is only 0.297. Just by comparing those averages we get the sense that Curry is far superior to the average NBA joe when it comes to hitting shots from long range. He averages almost three times more field goals from 20 to 24 feet and more than 10 times the number of field goals from 25 to 29 feet!
In order to really know how much better Curry was than the rest of the league at hitting from down town we need to do more than just compare the league average with his stats, we also need to know the “standard deviation” of the league averages, which turn out to be 0.466 for shots taken from 20 – 24 feet and 0.362 for shots taken from 25 – 29 feet. If two people race and the winner crosses the finish line just barely ahead of the other, would you be convinced that the winner is truly the faster runner? No, the more reasonable assumption is that the two runners are about the same speed. The standard deviation is a statistic that can help us determine if one value is insignificantly larger/smaller than the other (think one runner just barely edging the other) or if one value is significantly larger/smaller (think the way Usain Bolt beats everybody else).
The standard deviation is a statistic that gives us an idea of how much difference there is between the number of three pointers made per game from player to player. If the standard deviation is a large number (relative to the mean), it tells us that the difference between the number of shots made per game is generally very different from one player to another. On the other hand, if the standard deviation is a small number (compared to the mean) the majority of players made similar amounts of shots each game. If we look at the league average of shots made from 20 – 24 feet: average 0.515, standard deviation 0.362 it appears as though the number of shots made per game can be very different from player to player because the standard deviation’s value is relatively large when compared to the mean. This is also true for shots made from 25 – 29 feet (average 0.297, standard deviation 0.362).
It makes sense that there appears to be significant variation in the number of shots made from 20 + feet from player to player because some players usually guards and/or small forwards (Curry, Klay Thompson, Kyle Korver, Terrance Ross and company) are essentially paid to shot from long distance, while other players usually the big fella’s (power forwards and centers) like DeAndre Jordan, Andre Drummond and Dwight Howard are paid to be close to the basket so they can grab rebounds and score on put packs ( and probably get screamed at by their coaches whenever they attempt to shot from beyond 10 feet!). In summary there are “perimeter players” (players who shot from the outside and there are “interior players” (players who position themselves near the basket. Perimeter players will probably make many more shots from 20 feet plus than interior players. For example, JJ Redick the Los Angeles Clippers shooting guard averages about 3.5 field goals a game from 20 + feet while Dwight Howard the center for the Houston Rockets averages 0.
Now let’s talk about how the standard deviation can be used to help us determine if Stephen Curry is truly a superior shooter from the outside. Statisticians employ the standard deviation to get an idea of how much significant difference there is from a data point value (such as Golden States 13.1, three pointers per game) is from the overall average (such as 8.5 the league average for 2015- 2016). We can start of by assuming that the number of three pointers made per game is “Normally distributed” which means the data tends to be around a central value and no bias left or right. Data that are normally distributed have Histograms that look like a “bell curve”.
Figure 1: example Histogram data that is normally distributed. Note that “x” is the data values, f(x) is the frequency of data value “x” and is the mean or average of the data values. Also notice that is exactly in the center of the distribution.
Many things in the natural and social sciences follow a normal distribution, some examples are …
· Standardized testing results
· Measures of size of living tissue (length, weight, height, skin area)
· Some physiological measurements, such as blood pressure in adults
· Measurement errors in physical experiments
When data is normally distributed the following properties apply …
· Symmetry. Fifty percent of the data is less than the mean, and fifty percent of the data is greater than the mean, also the three measures of central tendencies are equal. In other words, the mean = median = mode.
· The 68-95-99 rule. Sixty-eight percent of the data lies within one standard deviation from the mean, ninety-five percent of the data lies within two standard deviation of the mean while ninety-nine percent of the data is within three standard deviations from the mean.
Note: denotes the sample mean, σ denotes the sample standard deviation
The 68 – 95 – 99 rule is true because the percentage of values that is greater than or lesser than any particular value is related to the standard deviation of the data. Remember the larger the standard deviation the more “spread out “the data is, the smaller the standard deviation the less spread out the data is. For example, lets say the mean score one test is 85 with a standard deviation of 5 while the mean score for another test is also 85 but has a standard deviation of 10. A score of 95 is less likely on the first test than the second test because the second test scores are more spread out. If you imagined the bell curves they would both peak at 85 but the first test would be a leaner bell indicating that most of the scores are close to 85 whereas the second test bell would be wider because not all scores are as close to the mean as they are in the first class.
It should be noted that when results from a standardized test with its mean and standard deviation is a “sample” of the population since not everyone wrote the test. Statisticians often times use sample results to estimate population results assuming the results will be normally distributed. So if the sample mean is 85 with a standard deviation of 5, it would be highly unlikely for anyone to score a 95 or better.
We can get an idea of how much significant difference there is from a data point value to the mean by calculating how many standard deviations larger or smaller that data value is than the mean. Based on the 69-95-99 rule if a data point is three or more standard deviations bigger or smaller than the mean it is reasonable to assume that that data value is exceptionally high or low. This is because if a data value is 3 standard deviations bigger or smaller than the mean only 0.5% of the data is larger or smaller than it. For example, if the scores on a standardized test was 65 with a standard deviation of 2, a score of 71 (which is 3 standard deviations, 3×2 = 6 larger than the mean) would be better than 99.5% of the other scores! It would therefore be safe to say that the person who scores a 71 did considerably better than the average. In fact, it would be fair to classify this person as a genius.
It should be noted that the number of standard deviations a value is from the mean is known as the “z – score” or “standard score” and is computed with the following formula;
Where x is the data value, is the mean and is the standard deviation. For example, if the mean score on a test is 70 and the standard deviation of 2 then a score of 75 has a z score of z = (75-70) / 2 = 5/2 = 2.5, in other words 75 is two standard deviations greater than the mean. The 68 – 95- 99 rule can be stated using z scores the following way; about 68 % of the data have a z score between – 1 and 1; about 95% of the data have a z score between -2 and 2; and about 99% of the data have a z score between -3 and 3. There are “z – score” tables that can be used to determine the percentage of data less than any z -score between -3.4 and 3.4, for example a if a data value has a z – score of 1.25 approximately 89.44% of the data will be less than or equal to it (according to the z – score tables).
Now let’s get back to basketball and Steph Curry. In our case for shots made between 20 – 24 feet Stephen Curry’s 2 made shots per game is larger than the league average (0.515) by 2 – 0.515 = 1.485 made shots per game which is equivalent to 1.485 ÷0.466 = 3.2 standard deviations (recall the standard deviation is 0.466). The fact that Curry’s shooting from 20 – 24 feet was 3.2 standard deviations better that the average would lead us to believe he is exceptionally effective at shooting from that distance. If we look at shots taken from 25 to 29 feet Curry is even more impressive, he made 3.2 shots per game from that distance while the league average was only 0.297 with a standard deviation of 0.362 which means that he is 8 standard deviations greater than the average! There can be not doubt he is way ahead of the rest in terms of shooting from this distance! A z – score of 8 is not on the z score table because it is certain that 100% of the data is less than any data value with a z – score of 3.4 or greater. Based on this analysis is appears as though labeling Curry as a 3-point shooting genius is appropriate.
Before we conclude that Curry is indeed a genius shooter, we should look carefully at the data we used to come to the conclusion. If you think about it there are a few problems that stand out about the data we used (stats from all players in the NBA were used to calculate the mean’s and standard deviations). The data is NOT normally distributed, take a look at the two Histograms of the data for shots from 20 – 24 feet and 25 – 29 feet. Note: FGM stands for “Field Goals Made per game” which means how many shots they hit per game.
It is clear to see that in both cases the data is not at all normally distributed as the mean is larger than the mode (the most frequently occurring value(s)) and the shapes of the histogram is not a bell curve. Instead we have what we call a “Right – skew” distribution because the mode (the highest bump) is not in the middle but on the left side instead and the tail is on the right side. If we think about it the right skew distribution makes sense, the reason why the median is on the lower end is because they would include the players who either never shoot from 20 plus feet (the power forwards, and centers) and players who simply don’t get much playing time (the non starters and bench warmers), these two types of players make up the great majority of every team’s roster. The tail exists because of the 3 or four players (who get considerable playing time) on each team that do shoot from long range.
In order to apply the 69 – 95 – 99 rule we can “normalize” our data by taking the log of all our data values. The resulting histograms are shown below.
As you can see still not what you would call normal, but more normal than the original data. Another tweak to the data that occurred when we logged all the data values is that we eliminated all players who scored 0 field goals per game because the log (0) is undefined. This is a good thing because the players who scored 0 shots from So while the 68-95-99 rule and the z – score table is still not entirely accurate it would be still more accurate with the log of the data than the original data. So let’s see what it tells us. For shots taken from 20 to 24 feet Curry’s log (2) = 0.301 shots made per game has a z score of (0.301 – (-0.309))/0.349 = 1.75 which according to the z score table means his statistic is better than 96% of the other players! Much better than the average NBA player but just short of genius. From 25 – 29 feet Curry’s log(3.2) = 0.505 shots made per game has a z score of (0.505 – (-0.485))/0.342 = 2.89 which makes his stat better than 99.81% of his peers. This is conclusive evidence that he is a genius even when compared to the world’s best basketball players when it comes to shooting from 25 to 29 feet!
The verdict: The statistics seem to show that Curry is an outstanding shooting from 20 – 24 feet, but is truly special when it comes to shooting from 25 – 29 feet.