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Data Analysis
The purpose of the data analysis is to have a more comprehensive view of our data. We performed a correlation and regression analysis in order to see if we could actually imply causation due to any correlation between variables. The regression analysis was done to see how influential variables were in our jump analysis. We also performed a sensitivity analysis to see how our variables in the project equations would tend to be affected by changing values. This would help us understand the best way to collect the most accurate data in the future. Our last analysis was one using ANNOVA and t-tests. This analysis would allow us to see if there were any statistically significant variables in respect to average jump height.
Regression and Correlation Analysis
By Daniel Ma
The desired outcome of conducting a regression and correlation analysis is to be able to draw conclusions about any possible connections between a people's physical traits and their jump height and power output. As mentioned in our uncertainty section, the power output was determined by the equation:
Overall, the equation predicts higher power output for participants with (1) higher jump height, (2) higher BMI, and (3) larger CTR (Calf Thigh Ratio).
(1) Higher jumps clearly indicates that people are able to generate more force and vertical displacement of their bodies. BMI plays a role in this equation by considering the overall physical shape of the participant's body. It is a measure of how mass is distributed throughout the body.
(2) A higher BMI can either indicate that a person is heavy and short or heavy enough to offset their taller height. Either of these attributes are not conducive to jumping high. The range of interest is the middle ground for BMI. A mid-range BMI, around 22-24, as seen in our results, seems to be the ideal range. The participants in this range might be light to medium weight and are tall. Lower weight and longer lower extremities seem conducive to producing larger jump heights. We acknowledge that BMI does not fully account for a persons muscle bulk or athletic ability. However, jump height seems to provide an adequate measure of athletic ability and muscular performance.
(3) The "Jump Master Constant" was determined by minimizing the difference between the output of the Johnson Bahamonde Peak Power equation and the output of the equation above for each subject. Our equation is a power function with the inverse of the "Thigh Calf Ratio" as the variable exponent. In hindsight, the exponent should have just been calculated as the "Calf Thigh Ratio," since it would have given the same result. The Calf Thigh Ratio will be referred to as the CTR for the remainder of this discussion. From a physiological perspective, jumping requires the a great deal of exertion by the leg muscles. In this study, only the quadricep and calf muscles were considered. Of the two, the quadriceps are capable of producing more power. Therefore, participants with larger calves might be able to produce more power, given that thigh circumference is relatively equal among all participants. The CTR would appear to predict higher power output because the constant value would increase if the CTR increases. The reasoning behind this would be that if participants have larger calves and assuming that larger calves are more muscular, they should have an edge on participants with smaller calves.
Figure 1. Trend line and 95% confidence interval for Johnson Bahamonde Peak Power equation output versus the product of BMI and jump height.
Figure 2. Trend line and 95% confidence interval for Jump Master Power equation output versus the product of BMI and jump height.
The trend lines for the equations in Figures 1 and 2 above were determined by making the power output a function of the product of BMI and jump height. A different method of combining the two variables, such as adding the BMI and jump height might result in a different Jump Master Constant. Given the basic SI units of power, performing dimensional analysis results in the units for the Jump Master Constant, shown in Figure 3 below. Adding BMI and jump height would produce a difficult equation to solve in order to calculate the proper units for the constant. Taking the product of the two gives us units that mesh well with the base units of power.
Figure 3. Dimensional analysis to determine the units of the Jump Master Constant. Future study could be done to calculate how to derive the constant value based on other leg measurements such as upper and lower leg length. Another point of interest would be to see if the leg muscles are fast twitch or slow twitch muscles.
By separating the product of the BMI and jump height and designating that value as the independent variable, we took the slope of the line to be 6^(1/TCR) or 6^CTR. From Figure 2, the power fit appears to favor the CTR values that are smaller. Many of the calculated power outputs lie in the confidence interval except for the subject with the largest CTR value. The slope of the line gives the power prediction for subject's whose CTR falls within a certain range. The ideal CTR calculated from this trend line is 0.71. This model allows us to predict the power output for a person if we know height, weight, jump height and their leg circumferences.
This model has limitations because the independent variable is dependent on both jump height and BMI, both of which are also dependent on factors we did not measure, such as athletic ability, muscle distribution, and overall health. These factors are confounders in our study.
Correlation Analysis
Before collecting data, we predicted that people with larger legs would be able to jump higher than people who did not. Through our correlation analysis, we were able to determine the mathematical strength of the relationships between variables.
Figure 4. The highlighted values represent correlations that had both coefficient of correlation greater than 0.75 and statistical significance of correlation that was less than 0.0001.
From analysis of the results in Figure 4 above, it would seem that mass, calf circumference and jump height are the most influential variables in the Jump Master equation. Jump height correlated the most with power output. Interestingly, BMI was correlated with calf circumference and strongly correlated with thigh circumference. Neither thigh nor calf circumference correlated strongly with jump height, but both correlated with BMI and calf circumference correlated with the Jump Master power output.
Figure 5. From the table in Figure 4 above, the correlation coefficients of thigh and calf circumference to BMI are 0.910 and 0.773, respectively. This relationship strongly suggests that thigh size increases with BMI. The correlation between thigh circumference and BMI was statistically significant for a p-value of 0.000001. This relationship makes sense if we assume the that mid-to-high range BMI (22-27) indicates that someone's legs are rather muscular. This is a safe assumption to make since the BMI of many professional athletes categorize them as obese.
Figure 6. From the table in Figure 4 above, the correlation between calf circumference and power output was 0.7534, which was statistically significant. This makes sense because our Jump Master equation favors high calf circumferences, since the CTR increases as calf circumference increases.
Conclusions:
Through our regression analysis, we calculated an ideal Calf Thigh Ratio that we believe will predict power output generated by a vertical jump given measurements of height, weight and jump height. Calf Thigh Ratios that differ significantly from the ideal CTR value 0.71 will lead to deviation from the linearized model.
Through our correlation analysis, we found that power is most correlated with how high someone can jump. This makes sense because a higher jump requires greater takeoff speed and force generation due to the limited time for concentric and eccentric muscle contractions of a stationary jump.
Overall, we found that people with higher BMI and greater vertical jumps were able to produce the most power. Additional study with a force plate would have been very useful in determining max force applied. Additionally, if we measured the time of leg muscle contraction, we could see if quicker contractions could enable a person to jump higher.
Sensitivity Analysis
By Luke Betteridge
The purpose of the sensitivity analysis was to find out which variables in our project equations would be more sensitive to changing values.
The analysis was preformed by finding nominal, low, and high values for each of the variables in the different equations so that the calculations could be preformed at each of those different low to high values. Only one variable was changed at a time while the others were held at their nominal values.
The low to high values we chose for the Jump Master Power equation are shown in Figure 7. These were found by looking at the low and high values in our testing and also looking up low, high, and average values for the different parameters online. There is also a sample calculation shown in Figure 7 from our power predictor equation.
Figure 7. Low, medium and high values for our variables
The first sensitivity analysis I preformed was on our Jump Master Power equation which utilizes the equations on the right.
Graphing the results of the sensitivity analysis we can see that the calf and thigh measurements seem to have the highest impact on the max power in our equation. All of the variables have a decent impact of effecting the power that is produced, but the thigh size looks like the most important variable from this analysis.
Figure 8. Straight line graph of the the max, medium and low values for each of our variables
By dividing all of the Max powers by the medium power that was found in Figure 7, using the medium values for all our variables, I produced a bar graph of the percent change in Max power from our equation. We can see that as the thigh calf ratio gets more disproportionate the power goes down. We can also see the trend of each variables as they increase. As calf size, mass, and vertical jump go up the power goes up; while as the thigh size, and height go up the power goes down.
Figure 9. Bar graph of the percent change in max power for each of the the high, medium and low values for each of our variables
The second sensitivity analysis I preformed was the Johnson Bahamonde equation which we found while doing research on jump height and power. We used this equation as a guide to create our own equation (JMP equation) that incorporates thigh and calf measurements. The Johnson equation is shown below.
In my Johnson equation sensitivity analysis the low to high values that I used for the variables (Figure 10) are the same as the ones I used for the jump master equation, but the height is in centimeters and not meters.
Figure 10. Low, medium and high values for the Johnson Equation
In the straight line graph (Figure 11) of the changing power for each variable we see that as vertical jump and mass increase the power increases and as the height increases the power decreases. This is the same trend that we saw in our Jump Master Equation.
Figure 11. Straight line graph of the the high, medium and low values for each of the Johnson Equation variables.
In the bar graph (Figure 12) that I produced for the percent difference of the power as each variable changed shows that vertical jump is the most sensitive input to this equation. The mass is close in sensitivity and the height looks like the least sensitive variable in this equation.
Figure 12. Bar graph of the percent change in max power for each of the the max, medium and low values for each of the Johnson variables.
Conclusions
In this analysis, I found that in our Jump Master equation the thigh size is the most sensitive variable, followed by the subject’s height being the second most sensitive. An important take away is that as the calf to thigh size gets more disproportionate that power goes down. The most sensitive variable in the Johnson equation was vertical jump. It is also important to note that we have similar trends in our Jump Master equation and the Johnson equation for vertical jump, mass and, height.
Click here to see all the sensitivity data
Statistical Analysis
By Josh Charney
Figure 13. This is the condensed version of the t-test analysis with each variable and its respective probability of statistical difference.
The purpose of this statistical analysis is to compare many different sets of our data in respect to participants jump heights to each other. The data sets compared were BMI, height, weight, thigh-calf ratio, knee angle, arm angle, power, engineers and non engineers, current athletic participation, and past athletic participation. If a set was considered statistically significant then we would be able to infer a special relationship between the jump height and the other variable. I preformed both a t-test and ANOVA analysis on our data set. The t-test was a separate from the ANOVA for our own curiosity to see if there was any significant difference in a variable rather than across the whole data set.
The t-test analysis concluded that the only statistically significant comparison to jump height was the participants power output. There were some comparisons that came close to being significant in our analysis but did not meet the threshold probability of 0.05. Variables we assumed to be important factors in jumping were actually not statistically significant. Jump technique (knee angle, arm angle) had almost no significance. The arm angle was closest between these two, but due to how large the knee angle probability of difference was, we can assume that jump technique has little to no significance with respect to jump height.
Figure 14. This is the graph of each data group and their average jump heights, as well as their standard deviation (shown in the error bars). The ANNOVA test compared all of these means and standard deviations.
The ANOVA test was compared the entirely of our data set. Similarly to the t-test, I had the data groups we were interested in as different columns. The ANOVA test concluded (see Fig 14) that there was not a significant difference in the average jump heights of any specific data group. In addition to our P-value being well above 0.05, the F-value was less than our F-critical-value as well. This allowed us to conclude that even though the t-tests showed that there was possible statistical significance, there actually wasn't any specific measured variable that stood out as statistically significant in respect to average jump height.
Figure 15. This is the results of the ANOVA test done on the data set.
The biggest limitation in this analysis is simply the lack of data. This may have changed if we had many more participants and especially if those participants were part of sports teams where jumping is a key to success. Because we only were able to survey the average population of jumpers, we did not have any large outliers like maybe a volleyball, basketball, and high jumper may have been. But since we don't have any jump specific athletes, we can more accurately describe the average person and their jumping abilities.
With the data we collected, there were no statistically significant variables. In our t-test calculations, the average jump height and power above and below 5000 Watts seemed to be significant, but after completing the ANOVA test we concluded that there were no statistically significant groups.
Click here for the full Excel sheet with the statistical analysis
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