Evaluating Total Game Time Progression & Overall Success Rate for the Computer Game, Sprinter

Introduction 

The following literature would mathematically procure recorded time measurements from a computer game called sprinter and then categorically map the results to evaluate total game time progression and the overall success rate for individual players. The preceding mentioned video game known as ‘Sprinter’ is played when an individual rapidly inputs data into a simulation environment and concurrently controls the sprinting speed of a two dimensional character along a virtual Olympic track. This simplistic feat is carried out using a computer keyboard via the tapping of the directional arrow keys while at the end of this simulated hundred meter course, the time is recorded for each attempt which is the score of an individual playing that particular event or level. 

For my research question and to fully identify the progression of distinct players I invited a group of ten willing participants over the course of five days who would sit for a couple of hours and then try to play sprinter without pause. The data sets generated were of the time they took to complete the virtual hundred meter race, furthermore, the participants were grouped into two categories and the progress marked was segregated with players tapping directional keys using one hand (the index and middle finger) and those that used both hands (index fingers from both right and left hand). These data sets are subsequently then utilized in establishing inference and creating a progressive trend on how playing techniques which differ from individual to individual is one of the major reasons for the percentage success of any given player. 

The data recorded is further evaluated using statistical metrics such as mean, median, standard deviation, and a chi squared test which ideally provides a very profound basis on whether there might be any given trend for the techniques used, or due to the time spent by a given player playing sprinter for an extended run. Conclusively, I determined whether the procured measurements are geometrically and or arithmetically sequenced. 

Sprinter Gameplay 

The Sprinter gameplay is set in a simulated Olympic track upon which the user has control of a single two dimensional character that is made to sprint across a hundred meter lane using directional arrow keys on the keyboard. The speed of the sprinter is directly proportional to the input speed by the user and if, upon successful completion of the race; where the user manages to beat AI computer generated runners, he or she is allowed to progress to the next level in the game. The difficulty of the game increases with each successful race the player completes and the game concludes when at the final level, the user is tasked in beating simulated AI runners that complete the hurdle in less than 8 seconds. 

The premise of the game is relatively straightforward and captures the simplicity of the task at hand however, there is marked difference in player performance when an individual is said to have played the latter for an extended period of time. To establish precedence on my claims I recorded best time taken for any given level by a player based on the number of days that he or she may have concurrently played Sprinter for, the tabulated results are as follows: 

Figure 1 

Best Recorded Time (Seconds) for Players Using One Hand (Index & Middle Finger)
	Day 1	Day 2	Day 3	Day 4	Day 5	Total
Player 1	8.52	8.30	8.16	8.01	7.78	40.77
Player 2	8.33	8.21	8.03	7.80	7. 63	40.00
Player 3	8.95	8.54	8.32	8.11	8.01	41.93
Player 4	8.22	8.01	7.98	7.72	7.55	39.48
Player 5	8. 30	8.01	7.78	7. 63	7.54	39.26
Total	42.32	41.07	40.27	39.27	38.51

Figure 2 

Best Recorded Time (Seconds) for Players Using Two Hands (Index Finger for Both)
	Day 1	Day 2	Day 3	Day 4	Day 5	Total
Player 1	8.33	8.24	8.15	8.06	7.99	40.77
Player 2	8.29	8.21	8.11	8.08	8.01	40.70
Player 3	8.32	8.29	8.20	8.13	8.05	40.99
Player 4	8.21	8.14	8.05	7.95	7.91	40.26
Player 5	8.19	8.13	8.07	7.99	7.96	40.34
Total	41.34	41.01	40.58	40.21	39.92

Obtaining the Survey Results (How the Data was collected) 

The survey was conducted when I gathered ten willing participants who would play the Sprinter game for a good couple of hours, all of whom were trying to beat their best scores for the given day. The game is spanned out in a way which makes it harder for players to compete when they successfully clear a stage of the game and proceed to the next level. The time showcased in figure 1 and figure 2 are every player’s best time of that particular day. I have recorded the best time irrespective of whichever level they might’ve obtained it from because the trend that I will be looking at the moment pertains to the effect on time, based on the hand technique that is used for playing the game rather than the proficiency at which it is to be played. 

Additionally, the survey group was segregated into two distinct groups; the first group was given the task to compete in Sprinter using one hand only. They were instructed to utilize the index and middle finger for the whole two hours that they played the game. The second group was instructed to play the game using both the right and left hand by tapping the directional arrow keys using the index finger of both hands. A successful attempt was recorded only when the individual player, attempting the Sprinter level cleared the latter stage successfully, without falling while the time for completing that level was then recorded manually. 

After every successful attempt the individual player was given a choice on whether to proceed to the next level or to compete again on the same and try to beat his or her own high score. This process was repeated until the highest score was recorded by an individual player for a day, which primarily comprised of a two hour session. The recorded numbers that were below the high score were then discarded as they are not part of the metricized results in this study while for five concurrent days this particular method was employed and the progress marked as well. 

Both set of groups, the ones using a single hand and also the two hand group, were exposed to the same variables i.e. time spent on the game (2 hours) and the choice of proceeding to the next level or staying and beating the high score set etc. In all of the five days no single individual player successfully completed the entire game until the end and there wasn’t any player as well who couldn’t complete minimum one level of Sprinter. The tabulated results in figure 1 and figure 2 are summarized metrics that will be further exposed to statistical tests for more in depth analysis. 

Running Mean Test for Individual Players 

The mean value for any given set of data showcases the average of the entire data set. This metric can be effectively utilized to make many simplistic inferences on a set of recorded observations and generally speaking, these inferences tell us basic trends along with other comparison points for a group of values. To start off the arithmetic mean for any given dataset can be expressed in the following notation: 

Based on the above expression I will proceed to compare players who use one hand with players who use both hands and then see if the mean time in each group is lesser or more than their respective counterparts. 

Figure 3 

One Hand Group		Two Hand Group
Player	Five Day Mean	Player	Five Day Mean
1	8.15	1	8.15
2	8.00	2	8.14
3	8.39	3	8.20
4	7.90	4	8.05
5	7.85	5	8.07
Total	40.29	Total	40.61

The above representation of the mean clearly show that players using one had get better results in mean values than players who might be using both hands for Sprinter, furthermore arithmetic means also identifies that the difference between individuals of a particular group are much more similar/closer to each other in the two hand group, than in the one hand group. This further showcases the simple fact that two hand technique does not handicap or give unfair advantage to any particular playing making it a fairer contest for each participant. 

Step by Step Formulation for Calculating Arithmetic Mean 

Step 1: Tabulate the Data 

To completely understand how I came about the mean time that is displayed in the figure 3 above I will isolate player three sample set from the one hand group. In order to start calculating the mean I isolated a sample set of the recorded data that was noted whenever an individual player completed a Sprinter level. This recorded data was marked in units of time (seconds) and tabulated as per the below table: 

Table (A) 

Player #	Day 1 (s)	Day 2 (s)	Day 3 (s)	Day 4 (s)	Day 5 (s)	Total
Player 3	8.95	8.54	8.32	8.11	8.01	41.93

Step 2: Write down the Formula 

Where = Arithmetic Mean, N = Total Number of Observations, i = First observation, n = last observation and = sum of all the observations. 

Step 3: Summing all observations 

Based on the formula I will start solving from the right hand side where represents the sum of all the recorded values in the above table. During this calculation I will add: 

= 8.95 + 8.54 + 8.32 + 8.11 + 8.01 = 41.93 

Step 4: Dividing by total number of observations 

The is then divided by N which gives my study the arithmetic mean for the entire data set, the working for step four is as follows: 

Mean = / N, where N = 5 

Mean = 41.93 / N 

Mean = 41.95 / 5 

Mean = 8.39 

Using the above method I calculated arithmetic mean for all the values in figure 3 and cross checked them with values in figure 1 and figure 2. 

Standard Deviation and Median Tests for the Procured Data 

In the above represented Figure 3 I isolated mean values to understand which participants were performing better as a group. There are two other statistical metrics, the standard deviation and the median, which additionally delve into the ascertaining on how much deviation is there in a particular group and whether individual players have large or small performance gaps. 

For standard deviation, I would ideally use the following expression of the recoded set of data from figure 1 and figure 2: 

Based on the above inference my data will categorically show how much deviation is there between values in a given group. This type of a metric enables me to see if the values in the data set are uniform among each other or are the skewed and have large extremes. Standard deviation also provides this study with the ability to mark normal values along with data outliners. The below Figure 4 is the summarized results on my recoded observations in figure 1 and figure 2. 

Figure 4 

One Hand Group		Two Hand Group
Day	Standard Deviation (Players)	Day	Standard Deviation (Players)
1	0.278433116	1	0.0574108
2	0.198454025	2	0.060464866
3	0.18062115	3	0.05425864
4	0.156684396	4	0.064311741
5	0.193002591	5	0.047159304

The Figure 4 results clearly indicate that even though the arithmetic mean provided me with the indication that group one performed better (as a group), the standard deviation show that it is the two hand group which has a significantly lesser deviation making it a much more accurate representation of the better performing group as well. 

In addition to taking the standard deviation, my study will also partake a smaller base metric in the form of obtaining the median from both the original data sets that are represented in Figure 1 and Figure 2. Generally median values are the middle values of a particular data set and can be represented by: 

The formula gives a value each for both the group, the one hand group and the two hand group and those can be tabulated as follows: 

Group	Median
One Hand Group	8.01
Two Hand Group	8.13

The median values can then additionally be compared with the aggregate mean of both the samples groups as well, the closeness of the mean value to the median value will further determine the accuracy of the results in question. In my preceding study and based on the values I obtained from both the ‘one hand’ and ‘two hand’ groups, the mean of all values for both the data sets are 8.05 and 8.12 respectively. This categorically implies that the two hand group even though having a larger mean time is relatively closer to the median; 0.01 units’ deviation than the one hand group which has a 0.04 deviation from the median value. 

The comparison of these two latter metrics are proof enough that for a beginner player with little to no experience in playing Sprinter the technique to use two hands instead of one is a much more productive and efficient way to attempt the game. 

Step by Step Calculation for Standard Deviation and Median  

Step 1: Tabulating Data set for Standard Deviation 

In addition to doing a step by step formulation for the arithmetic mean, I will further showcase the step by step process on how Standard deviation is calculated in this research paper. To begin my inquiry I will first take a sample data set of five values from Figure 1 to use as a prop for my explanation. In this example I have taken individual time recoded by players 1 to 5 on the Day five time (in seconds), below is the data representation: 

Day	Player 1	Player 2	Player 3	Player 4	Player 5	Total
Day 5	7.78	7.63	8.01	7.55	7.54	38.51

Step 2: Stating the Standard Deviation Formula 

Where N = Last Observation, I = First observation, = observation recoded = mean and N = Total Number of Observations. Using this formula I can calculate the last figure under the column one hand group in Figure 4. 

Step 3: Calculating Mean 

Mean = Sum of all values divided by Total number of observations 

Mean = 7.78 + 7.63 + 8.01 + 7.55 + 7.54 / 5 

Mean = 38.51 / 5 

Mean = 7.702 

Step 4: Finding (Observation – Mean) ^ 2 for all values 

(7.78 – 7.70)^2 = (0.08)^2 = 0.0064 

(7.63 – 7.70)^2 = (-0.07)^2 = 0.0049 

(8.01 – 7.70)^2 = (0.31)^2 = 0.0961 

(7.55 – 7.70)^2 = (-0.15)^2 = 0.0225 

(7.54 – 7.70)^2 = (-0.16)^2 = 0.0256 

Step 5: Sum all values for (Observation – Mean) ^ 2 Divide by (N -1) 

Standard Deviation = (Sum / N -1)^0.5 

Standard Deviation = (0.0064 + 0.0049 + 0.0961 + 0.0225 +0.0256 / 5-1)^0.5 

Standard Deviation = (0.038875)^0.5 

Standard Deviation = 0.19716 

To find the step by step median value I will first distribute all values in ascending order as pictured in the table below. For my observation I will take five values from Day 3 of the second hand group in ascending order. 

Step 1: Tabulate Result in Ascending Order 

Time in Seconds
7.91
7.95
7.96
7.99
7.99
8.01
8.05
8.05
8.06
8.07
8.08
8.11
8.13
8.13
8.14
8.15
8.19
8.2
8.21
8.21
8.24
8.29
8.29
8.32
8.33

Step 2: Write Median Formula 

Median = A / 2 (If Total Observations are Even) 

Median = A + 1 / 2 (If Total Observations are Odd) 

Step 3: Plugging in the values 

Since the total number of observations are twenty five recoded time measurements from the two hand group I will use the following Notation: 

Median = A + 1 / 2 

Median = 25 + 1 / 2 

Median = 13 th Value 

Median = 8.13 

The Null Hypothesis Question 

Every statistic needs a null hypothesis which acts as the fulcrum of the entire inquiry. In my research for the sprinter game, I concluded that different hand techniques will yield different results therefore my research is primarily focused in finding whether there is any relationship between using one hand for playing the game and or using both hands. For this fundamental inquiry I established the following null hypothesis question: 

Distribution and accuracy of the results acquired from the data set in the two hand group is more superior to the ones in the one hand group. 

To prove the latter thesis I took mean, median, standard deviation and multiple p-values from the below chi squared tests. The results are formulated in the conclusion section of this particular assignment. 

Chi Squared Test 

Using a Chi Squared Test to determine inferential statistics for an entire distribution, can provide us the relationship and comparison points of two different data sets. It also gives my study the accuracy of the results in question and enables me to provide results that are fundamentally and substantially accurate. Before I begin calculating the Chi Squared value, I need to form a null hypothesis based on a question that can be inferred using statistics obtained from the earlier metricized results. For this study, my null hypothesis question would be that results from the two hand test are more accurate and reliable than getting a sample set of individual players attempting with one hand. To prove this notion I would initially calculate the Chi Squared value which can be denoted using the following expression: 

This expression can further be broke down in a tabulated result which helps us calculate the Chi squared value in a simplistic manner. Below is the standard table that I use to manually come up with the latter result while additionally for both groups the total population count was 25 recorded instances in each category. 

One Hand Group 

Category	Observed	Expected	Residual (Obs – Exp)	(Obs-Exp)^2	Component=  (Obs-Exp)^2/Exp
7.51 – 7.80	8	5	3	9	1.8
7.81 – 8.10	6	5	1	1	0.2
8.11 – 8.40	8	5	3	9	1.8
8.41 – 8.70	2	5	-3	9	1.8
8.71 – 9.00	1	5	-4	16	3.2

The expected value was calculated by dividing the population count but the total number of groups that are listed in the category column. Using the above expression and the tabulated results the p value is 0.066298 which at a 0.05 significance level is not an acceptable result. 

Two Hand Group 

Category	Observed	Expected	Residual (Obs – Exp)	(Obs-Exp)^2	Component=  (Obs-Exp)^2/Exp
7.51 – 7.80	0	5	-5	25	5
7.81 – 8.10	11	5	6	36	7.2
8.11 – 8.40	14	5	9	81	16.2
8.41 – 8.70	0	5	-5	25	5
8.71 – 9.00	0	5	-5	25	5

Based on the two hand group the p value is 0.000001 which implies that the result is significant and can be used as a proper metric. 

The Significance of Chi Squared Test  

Chi tests are conducted for various reasons but one of the primary aspects of doing a chi test is to identify and test the distribution recorded as something that is likely done by chance or by design using patterns that follow a user trend. The Chi square basically, is a ‘goodness of fit’ statistic that enables and measures how well the distributed data fits while also ascertaining on whether the variables are independent of each other or not. 

Conclusion 

In my research, I have took four major statistical tests that helped me verify the null hypothesis of the data set in question. Based on the results obtain I can safely conclude that due to the P-value being extremely small the chance of error in the data set is reduced (overall) however the group of players who attempt the sprinter game with two hands have more accurate results and are a more reliable metric than the one hand group. This can be due to many reasons however, with the representation of mean, median and standard deviation as well there is a lot less chance in the two hand group as compared to in the one hand group. 

References 

Kolmogorov, A.N., 2018. Foundations of the Theory of Probability: Second English Edition. Courier Dover Publications. 

Rao, J.N. and Scott, A.J., 1981. The analysis of categorical data from complex sample surveys: chi-squared tests for goodness of fit and independence in two-way tables. Journal of the American statistical association, 76(374), pp.221-230. 

Pitman, E.J., 2018. Some Basic Theory for Statistical Inference: Monographs on Applied Probability and Statistics. Chapman and Hall/CRC. 

Shumway, R.H. and Stoffer, D.S., 2011. Time series regression and exploratory data analysis. In Time series analysis and its applications (pp. 47-82). Springer New York.