Is there a correlation between the mass and the circumference of Gala apples?

Hi, First of for the sources just site things like the actual equation or the theory behind each acual method.

This paper is a reasearch papaer in maths where the data that I have recoreded will be analysed. The data along with a somewhat completed version of this assesment will be uploaded to you. I shall also upload a completed version done by a classmate so you see what it should look like. You will notice on his copy that there is a lot of writing, you need not worry about that. However what you need to write for each mathematical technique that you use will be a small explanation of what this method will show and what it is before using it. Then once you apply this technique to my data (remember always make a separate one for mass and circumference) you should write a few sentences re stating the result and what this proves to us.

Now as for the mathematical techniques that should be applied to the data they are the following.

• Cumulative Frequency graph
• Box and Whisker plot
• Standard deviation
• Pearson’s correlation coefficient
• The Chi-Squared test (if you can fully explain and add the graph with degrees of freedom this would be great!)
and lastly the T test (MUST BE DONE)

So just to re-enforce the fact for each of these operations you would be required to introduce the method and what it shows (maybe copy paste the equation as I did)
then once you have you results just make a few lines to explain what this result shows us about the data.
and at the end just write a small conclusion that relates back to the research question is there a correlation?

lastly it should be known that if any of the eqationsr require a hypothesis (maybe for chi squared) to be proved or disproved keep in mind that my personal opinion or my hypothesis was that there should be a positive correlation. So kindly keep it in mind that any null hypothesis should be written as if you want to prove a positive correlation.

There wil also be the official markshceme uploaded, its very long just scroll down to the statistics part this should prove to be useful.

One last thing as you will see on mine all the graphs should be done on excel and also for my sake try to format them all as in fornt of text. As I will have to then expand all of your small paragraphs into something similar to my classmates oing paragraphs.

Good Luck!

 

 

The Effectiveness of Projectile weapons over long range, in the hope to ward off an Intruder

 

 

 

 

 

Candidate Number:

 

 

 

 

 

Session:

 

 

 

Candidate:

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents

 

 

Introduction:

 

            Aim:

 

            Plan:

 

            Method:

 

            Mathematical Techniques:

 

            Hypothesis:

 

 

 

Data Collection and Interpretation:

 

           

 

Analysis:

 

 

 

Evaluation:

 

 

 

Conclusion:

 

 

 

Bibliography:

 

 

 

 

 

 

 

 

 

 

 

Aim:

 

Day to day situation vary from normal happenings to extraordinary occurrences.  Some of these events may be as frightening as an intruder in your home. In such situations, ones rational thinking is affected and it is during strange events such as this, that one needs help from friends, family or even essays such as the following. Therefore, the following is a contribution to society, as I will be investigating, “The Effectiveness of Projectile weapons over long range, in the hope to ward off an Intruder.”

 

Plan:

 

My task is simple. It is to evaluate different variables in the process of using the appliances. These variables are:

 

-accuracy

-damage.

 

In order to test accuracy, a life size target of around 1.60m has been recreated using cardboard box, paper and items used for supporting the structure. From calculating the area of rectangles, squares, circles and triangles that consisted of the targets body, it was concluded that it had an average surface area of 2700 cm squared.

 

Feet                 =                        117

Legs                =                      1092

Belt                 =                          50

Lower body    =                        590

Upped body    =                          44

Head               =                        707

Total               =                      2700

 

 

In order to test for further accuracy, there is a preferred target in the middle of the chest. It has a radius of 4.5cm and therefore an area of 64 cm squared. Therefore in order to hit within the boundaries of the target, there is only a 2.37% chance. This target symbolizes the optimum positioning for an effective attack. This will be used to measure accuracy.

 

Method:

 

I will measure the distance from the centre of the preferred area for accuracy and the size of the puncture to symbolize damage. Then from using the proportion of the target size, a percentage will be calculated, conveying the percentage miss of the knife. The worst possible hit is the one furthest from the centre and this is approximately 90cm. Therefore taking into account the worst possible hit, a percentage miss can be calculated. In order to test for damage, the wound of the puncture will be measured in cm, the greater the puncture, the greater the damage.

With both these factors in mind, an index, created by myself will rank the throws. It is called the accuracy to damage ratio. It is calculated by dividing the percentage of the knife off target by the puncture in cm. A value closest to 0 will score higher in the rank.

 

 

The plan was very simple, yet consisted of hours of vigorous field work and testing. The projectile I chose to use was a small, yet sharp cheese knife. The reason I chose this knife was because of its size a weight. Its short design allowed for the spinning effect to work at its best. Also the light design allowed for a fast throw, with minimum drop. Also with a smaller blade, the impact chance increases. I set up the target as seen above on a metal structure for support against the incoming projectile. As seen below.

The knife was thrown at distances of 2, 4 and 6 meters. The position of the target never moved, only I did as I switched from distances. After 10 successful hits, I calculated the distance from the center and the puncture width and wrote them in a table. I did this two more time from both 4 meters and 6 meters. To distinguish between which punctures have already been measured, a small tick was placed beside them. The technique used to throw the blade is very common. It is holding the tip of the blade with the thumb and index finger, and throwing it at the target like a baseball. After collecting all the data, and puncturing the target 30 times, the target looked like the above.

 

Mathmatical Techniques:

 

Mathematical processes I learnt in class that will be involved within the project consist of standard deviation. This will calculate the spread of my data to convey its significance. Pearson’s correlation regression is also used to calculate how connected my data is.

 

 

 

 

I will be doing correlation tests with many different variables such as:

 

-Accuracy at Different Distances

-How far the Knife Missed the Target at Different Distances

-The Puncture Size at Different Distances

-The Accuracy to Damage Ratio with Different Distances

 

Furthermore, I will use mathematical techniques of distinguishing data such as mean, median and mode. To classify the damage, I will use the discrete data for the punctures in cm and create box-plots and cumulative frequency graphs. With all the above investigations in mind, a reader of such an essay will be more than aware as to the optimum distance they can accurately throw a knife, the probability that they hit the target and how badly they injure the target.

 

Hypothesis:

 

I predict that the closer the thrower is from the target the most effective the attack will be. This is as distance is a factor affecting accuracy, the smaller the distant, the greater the accuracy. Furthermore, distance is also a factor affecting power and thus damage. The closer to the target the thrower is the higher change of impact and thus potential damage.

Therefore I predict at a range of 2 meters, the damage, accuracy and correlation will be at its highest.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Data Collection and Interpretation:

 

Distance of Two Meters:

 

Miss (cm) Puncture (cm) % Miss Accuracy to Damage Ratio

4.00

1.50

4.44

2.96

14.50

2.50

16.11

6.44

8.50

2.50

9.44

3.78

3.50

2.50

3.89

1.56

12.50

2.50

13.89

5.56

13.50

2.50

15.00

6.00

16.00

2.50

17.78

7.11

17.50

3.00

19.44

6.48

2.00

3.00

2.22

0.74

10.50

3.50

11.67

3.33

 

Mean of Miss (cm) Accuracy (%) (10/17)

10.25

59

 

Ratio:              % Miss / Puncture

Accuracy:       Hits / Attempts

Mean:             Sum of data / Number of data

Standard Deviation:

 

 

Puncture Interval

Frequency

0.50

0

1.0

0

1.50

1

2.00

0

2.50

6

3.00

2

3.50

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Distance of Four Meters:

Miss (cm) Puncture (cm) % Miss Ratio

25.00

0.50

27.78

55.56

14.00

0.50

15.56

31.11

40.50

1.00

45.00

45.00

30.00

1.00

33.33

33.33

17.00

1.00

18.89

18.89

19.50

1.50

21.67

14.44

21.00

2.50

23.33

9.33

28.00

2.50

31.11

12.44

40.00

2.50

44.44

17.78

29.00

3.00

32.22

10.74

 

 

Ratio:              % Miss / Puncture

Accuracy:       Hits / Attempts

Mean:             Sum of data / Number of data

Standard Deviation:

 

Mean of Miss (cm) Accuracy (%) (10/71)

26.40

14

 

 

 

Puncture Interval

Frequency

0.50

2

1.0

3

1.50

1

2.00

0

2.50

3

3.00

1

3.50

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Distance of Six Meters:

 

Miss (cm) Puncture (cm) % Miss Ratio

40.50

0.50

45.00

90.00

12.00

1.00

13.33

13.33

17.00

1.00

18.89

18.89

20.50

1.50

22.78

15.19

47.00

1.50

52.22

34.81

20.00

1.50

22.22

14.81

35.00

2.00

38.89

19.44

60.50

2.50

67.22

26.89

33.00

3.00

36.67

12.22

56.00

3.50

62.22

17.78

 

Ratio:              % Miss / Puncture

Accuracy:       Hits / Attempts

Mean:             Sum of data / Number of data

Standard Deviation:

 

Mean of Miss (cm) Accuracy (%) (10/201)

34.15

5

 

 

 

Puncture Interval

Frequency

0.50

1

1.0

2

1.50

3

2.00

1

2.50

1

3.00

1

3.50

1

 

Ratio:              % Miss / Puncture

Accuracy:       Hits / Attempts

Mean:             Sum of data / Number of data

Standard Deviation:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Calculations for Punctures:

 

 

Puncture Interval

Cumulative

Frequency

 0.50

3

1.0

8

1.50

12

2.00

13

2.50

23

3.00

27

3.50

30

 

MinX

0.5

Q1

1

Median

1.25

Q3

2.5

MaxX

3.5

Mode

2.5

Range

3

 

Correlation:

 

 

Distance (m)

 

2

4

6

1.50

1

0.50

2.50

0.50

1.00

2.50

0.50

1.00

2.50

1.00

1.50

Puncture

2.50

1.00

1.50

2.50

1.50

1.50

2.50

2.50

2.00

3.00

2.50

2.50

3.00

2.50

3.00

3.50

3.00

3.50

 

Distance (meters)

Accuracy

Distance (meters) Mean of Ratio Distance (meters) Mean Miss

 2

59

2

4.40

2

10.25

4

14

4

24.86

4

26.40

6

5

6

26.34

6

34.15

 

Accuracy at Different Distances:

 

R^2 = 0.871

 

How far the Knife Missed the Target at Different Distances:

 

R^2 = 0.9605

 

The Puncture Size at Different Distances:

 

R^2 = 0.7273 at 2 meters

R^2 = 0.8473 at 4 meters

R^2 = 0.9353 at 6 meters

 

The Accuracy to Damage Ratio with Different Distances:

 

R^2 = 0.8001

 

Figure 1.0

 

Figure 1.1

 

 

 

 

 

Figure 1.2

 

 

Figure 1.3

 

Total frequency                       =30

Q3 or 75th Percentile               =22.5              = around 2.5 cm

Median or 50th Percentile        =15                = around 2.0cm

Q1 or 25th Percentile               =7.5                = around 1.0cm

 

 

 

 

Box and Whisper Plot for Punctures

0

0.5

1

1.5

2

2.5

3

3.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.4

 

MinX   =0.5

Q1       =1

Med     =2.5

Q3       =2.5

MaxX  =3.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Analysis, Validity and Evaluation:

 

 

In figure 1.0, one can see a line graph, with two variables. They are accuracy and % misses. These are plotted on the Y axis against the other factor affecting both the variables which is range on projectile thrown. One can see as the accuracy decreases with in strong negative correlation of 0.87 that % misses actually increase as seen with the strong positive correlation of 0.96. Therefore accuracy and % miss are inversely proportionate. This means as one decrease the other increased and vice versa. Accuracy seems to be very high with minimal misses, however as the range lengthens accuracy starts to decrease and with the following statement that the two variables are inversely proportionate one can predict that the % miss will increase, and it does. Unfortunately, 10 data points was not sufficient, however as difficult as it was obtaining results for longer ranges, it would be an aid to accuracy is more data points were used. This would result in a smoother less jagged and easier to interpret graph, with more chances of being lifelike and not just because of coincidence. I used such a graph as it was an easy way in order to interpret both variables against each other against range, which better allowed me to analyze my data.

 

Figure 1.1 conveys weapon punctures measured in cm plotted on the Y axis against successful attempts or in other word successful puncture. Furthermore there are three lines plotted on these axis and they are punctures from ranges 2, 4 and 6 meters. The blue line which signifies the 2 meter range seems to always be on top of the other two ranges except for the last few data points. This range is above the rest as the closest one is to the target, the easier it is to accurately puncture the target at such range as one gets familiar to the technique. This familiarity is seen with the constant puncture of 2.5cm for 60% of the successful attempts for that range. The other two ranges on average seem to have lower puncture levels probably due to the increased difficulty to hit the target accurately and with enough power to sustain an effective attack. This however is disproven at end points when both the 4 and 6 meter range seem to have greater puncture wounds. This may be down to the fact that by the end of the 4 and 6 meter testing, I had spent a lot of time actually getting 7 or 8 successful attempts so exaggeration due to impatience may have led to the extra force to create the larger puncture for the last two data points of ranges 4 and 6 meters. Furthermore the correlation between punctures seems to be increasing as the range increases in length. For 2 meters the correlation is seen to be 0.72, whereas correlation is 0.85 for 4 meters and 0.94 for 6 meters. A value of 1 conveys full correlation. When measuring the puncture, I used a 30 cm ruler and rounded to the nearest one unit. However this is inaccurate, and may be the reasons why the ranges at greater lengths have larger punctures at their last data points. If I were to use a smaller ruler and measure exactly without rounding then maybe my results would be more accurate. Therefore in order to ameliorate my inaccuracy, in future or while recommending to another student, I would measure all small punctures with small utensils for easier maneuverability and accuracy. I used this sort of graph with three separate curves in order to compare them with each other in an easy visable form so that I and other could understand my analysis and acknowledge the fact that it is easier to hit the target at a closer range.

 

 

In figure 1.2, one can see a line graph where my accuracy to damage ratio is plotted on the y axis against range on the x axis. The steepness of the graph stipulates the increase in the ratio and therefore how inaccurate and poor in damage the attack was, as a lower value conveys higher accuracy and power. This is seen as between data point 2 and 4, the steepness of the graph is very large. However the rate of increase decrease dramatically almost stopping at becoming constant between data points 4 and 6. This is due to the fact that at distant ranges, there may still be a good accuracy to damage ratio, but not at the pace and speed of a ratio’s calculation at a closer range. I expected there to be a steep difference between 4 meters and 6 meters but there was hardly any. There is a strong correlation of 0.8 which conveys the level of connectivity between the data points for the accuracy to damage ratio. The graph is very ridged and there may be inaccuracies with my unofficial ratio. This may have caused problems and may be the reason why the ratio still increases even slightly at the greater range where in all rationality is should increase by a large amount. If I was to have gathered more data, then maybe the curve would be smoother and more accurate with the steepness of the curves come to be discussed. It is unusual to why from 4 to 6 meters the change is almost nothing, and if next time I were to repeat the experiment, I would probably see a clearer change between 4 and 6 meters if I took the time to collect more data points for increase accuracy with my interpretations. I used this graph to convey my concept of this unofficial ratio. It is easy to visualize and because of the line of best fit and correlation, I can easily analyze and discuss what the graph is actually saying.

 

Moreover in figure 1.3, one cans see a cumulative frequency graph which dictates the cumulative frequency of all punctures to the 2700cm2 target. The total frequency seems to be 30, with the median being equal to 2cm, the Q1 being equal to 1cm and the Q3 being equivalent to 2.5cm. as one can see, the steeper the parts of a cumulative frequency, the greater their cumulative frequency for those data points. Between 2cm and 2.5cm there is the steepest rise, and this means the greatest frequency of punctures is seen to be around 2.5cm in length. The most level part of the graph is from 1.5cm to 2cm, therefore the least frequent puncture size seems to be 2cm in length. The data point for the 2cm is almost as if it was an outlier, as with that as an exception, the curve runs almost in a straight diagonally curved line. The reason I used such a graph as the steepness of the curve effectively conveys the frequency of the data points in an easy visual interpretation which I then can effectively develop and describe. There is a problem however, and this is the outlier of 2.5cm. This decreases to correlation and unfortunately looks like a wrong data point. It may have been caused by windy weather, and this could have affected the positioning of the knife. Moreover it was cold, and after a few hours outside, it was very difficult to continue to throw the knife as efficiently as I was for example with the first 20 throws. This may be why the values decrease so much. For the outlier, a large amount of data values come up with a value of 2.5 and this may be the average puncture if I throw the knife as hard as I can and for many attempts I did. Therefore if I was to correct my problems, I would firstly test my experiment in neutral conditions, where both the test and tester does not get affected by outside forces and secondly I would try and throw the knife at the same speed every time to decrease the chance of inaccurate data from overpowers throws.

 

Figure 1.4 shows a box and whisper plot, and takes into account all punctures on the target. Minx is seen to be 0.5c, and this is the starting point of the data. The maximum value or MaxX is equal to 3.5cm. The 25th percentile or Q1 is equal to 1cm, whereas the median, middle value of the data, and Q3, 75th percentile is equal to 2.5cm. This conveys there is a slight spread with the data and that correlation is not incredibly high like previous values. However data points are still in many ways close to each other and thus there is still enough correlation for a fair analysis of my project. I decided to use this form of mathematic techniques as one it is very simple to understand and two it is effective in its role. I wanted to see, by using all my data, how the correlation was without calculating R squared. However there were problems with the technique. Firstly values were put on the scale according to eye and not machine. This means there could be error in the process making the values inaccurate as I simply created the visual aid with boxes and lines. This may be due to the fact that excel or other forms of mathematical arranging sites and software do not facilitate box plots. If I were to increase the accuracy of such a mathematical technique, I would use software that actually made the plots accurately instead of trusting ones eyes and quality of the computer.

 

Conclusion:

 

 

 

 

 

-refer to aim

-review

-has the project been a success or a failure

-how does this help the world in some form

-give last advice continuing about intruder in house and say what is the best range, what is the chance they will miss, and what is the chance they will injure the intruder ect.

 

 

 

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