Once I had my random samples I then recorded them into a table, and using Microsoft Excel I created a series of scatter graphs. I plotted graphs of each relationship, which included one for male and a separate one for female, as it would be unfair to compare opposite sexes. In total there were 8 scatter graphs plotted:
- Boys - Thumb v Wrist
- Boys - Wrist v Neck
- Boys - Neck v Waist
- Boys - Thumb v Waist
- Girls - Thumb v Wrist
- Girls - Wrist v Neck
- Girls - Neck v Waist
- Girls - Thumb v Waist
After completing the 8 scatter graphs, I made the following calculations for each relationship of each sex (again, using Microsoft Excel):
- the mean (average),
- the median (the middle measurement once all the measurements are lined up in order of size)
- the mode (the most frequent measurement)
- the co-efficient correlation (the strength of the relationship between the variables)
- the standard deviation (the measure of the spread of the data).
After plotting the scatter graphs and calculating the correlation, the decision as to whether to add a line of best fit was made. If the correlation was equal to 0.4 (shown in ‘Table 4’ on the following pages) or above then this meant there was a positive correlation and a line of best fit was therefore suitable. The only disadvantage of adding a line of best fit was that is was very subjective, as 10 different people will be using 10 different samples and will therefore end up with 10 different results, and also Microsoft Excel produced small scale graphs which often caused the line of best fit to appear inaccurate.
Once I had drawn my line of best fit on the graphs with a 0.4 or above correlation, I was able to then work out the equation of this straight line. All straight lines have the equation y = mx + c. m is the gradient of the line and c is the y intercept – in other words, the value of y where the line crosses the y axis.
To calculate the gradient of the line, I chose two points on the line fairly far apart where it is easy to read off the co-ordinates. Then using this following algebraic equation I was able to calculate the gradient, if the two points have the coordinates (x1, y1) and (x2, y2), then the gradient of the line is (y2 – y1) ÷ (x2 – x1).
So for example, if the line of best fit passed through the points (2,3) and (7,13), then the gradient of the line is (13 – 3) ÷ (7 – 2) = 10 ÷ 5 = 2. So in this case the equation of the line would be y = 2x + c.
To find the value of c, I simple extended my line of best fit until it crossed the y axis and read off the value of y at that point. The alternative method, was to substitute in a pair of values for x and y from a point on my line, then solve the resulting equation for c. In the example above, this would have given me c = -1, so the equation of the line would have been y = 2x – 1.
This method of finding the gradient of the line of best fit can be complicated and result in slightly inaccurate results due to the scale of the graphs I used. Fortunately using Microsoft Excel this process can be done by simple selected the equation to be shown on my particular graph. Obviously it would be pointless to include a line of best fit when the data represents a negative correlation therefore there are only a select few were I have included the gradient. At a correlation of 0.4 or above a line of best fit and gradient is suitable. I then recorded my findings in a table, (‘Table 5).
Before I could explore and compare my results I had to make a decision as to which value would be most reliable to represent the average value of each set of information. I had the chose of three of the calculations done earlier in the investigation:
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The mode, the most frequent measurement - This would have been a reasonable representative of the data if there were a small spread of values throughout the data, but as this is not the case in this investigation, it would be inappropriate and inaccurate.
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The median, the middle measurement once all the measurements are lined up in order of size. – This value is unsuitable for this investigation as, again, it does not account for a large spread of values within the data.
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The mean, this is obtained by dividing the sum of the values by the number of values - This is the most accurate form of representation as it accounts for a large spread. It is the most logical of the three as it produces a reliable average and therefore I chose to use this as the average value, which was to represent each set of data.
After all of the above was completed I was then able to explore any relationships between each of the body parts. I began by doubling the mean of the thumb for the boys and comparing it to that of the mean of the boys’ wrist. If Gulliver’s theory were correct then the mean of the thumb would’ve been identical to the mean of the wrist, which was highly unlikely. I continued my investigation by comparing the rest of my samples, doubling the mean of the wrist and comparing it to the mean length of the neck and so on for both the boys and girls. I recorded my findings into a suitable table (‘Table 2,’ on the following page). I then found the difference between the doubled mean of the smaller measurements and the mean of the larger measurement (e.g. the mean of the wrist subtracted by the mean of the thumb doubled). From this I worked out the percentage error, through the following calculation:
I recorded my findings into a suitable table (labelled ‘Table 2,’ in my analysis), which acted as a reliable source of evidence aiding me to come to a conclusion as to whether Gulliver’s theory was correct or inaccurate.
The standard deviation helped me to conclude the reliability of the mean, and acted as an indicator showing how bundled the data is around the mean. A small standard deviation would show that the mean was reliable. The standard deviation was used to produce a range that would highlight any anomalies. The mean of a particular characteristic added to double its standard deviation acted as the upper boundary, while the mean of that same characteristic subtracted by double its standard deviation acted as the lower boundary. The results that fall into this range can then be assumed reliable values. Any values that are outside the range are outliers and these can be identified on the scatter graphs, which are examples of unreliable data. If there are few anomalies, then this would lead us to believe the theory is true in the particular relationships in discussion. Little error would result in a strong correlation. Next, I recorded my results into a table (‘Table 3,’ in the analysis) and used this as a further source of evidence to help disprove or prove Gulliver’s theory.
Finally, I plotted a bar chart on Microsoft Excel showing the means of each of the characteristics of both sexes, which can be explored and used as a further source of evidence to aid in proving or disproving the hypothesis.
Analysis:
The mean, median and mode were calculated for each group of data, for both sexes, as shown below in Table 1.
Table 1:
The mean, median and mode of the boys’ thumb and neck are identical when rounded to the nearest whole number, and the mean, median and mode of the girls’ thumb, neck and waist are exactly the same when rounded to the nearest whole number. The characteristics unmentioned only have slight variation between the mean, median and mode. The mode is the value that occurs most frequently and is too simplistic, the median cuts out the extreme values but doesn’t use all the information provided and takes account of the range. Therefore, I have chosen to use the mean as a single value representative of each group of data as I feel it is most reliable. The mean values are highlighted in red in the table above.
Table 2:
BOYS:
GIRLS:
As I pointed out in my planning, if Gulliver’s theory is correct then doubling the mean of the thumb should be equal to the mean of the wrist, and doubling the mean of the wrist equal to the mean of the neck, etc. Obviously the relationship between the ‘Thumb v Waist’ is different to the others and it works that the waist should be 8 times greater than the thumb (as shown on the following page) if Gulliver’s theory is true. My findings are recorded in the table above, Table 2.
Wrist = 2 x Thumb,
Neck = 2 x Wrist = 2 x 2 x Thumb,
Waist = 2 x Neck = 2 x Wrist = 2 x 2 x 2 x Thumb.
= 8 _ x _ Thumb
In every relationship there is a difference of at least 1, which is evidence that the theory is incorrect. Obviously the fact that we have used the mean as a single representative value may be the reason that none of the measurements fit exactly into one another. Therefore, we can compensate by considering results within 5% of the expected measurement to be correct. From this we can conclude that the most reliable application of Gulliver’s theory is the relationship between the wrist and neck for both boys and girls. The least reliable relationship is that between the thumb and the wrist, which has a percentage error of 16.35% in the boys and 21.10% in the girls. This would lead us to believe there may be a true relationship between the wrist and neck size, but no relationship between the thumb and wrist. If this is not the case the other reason for these results may be that it is easier to measure the wrist and neck, whereas the thumb is far smaller and there is little tolerance for error as the measurements are so precise.
In Table 2 there is a column, which states the standard deviation of the means. From this we can see how the data is distributed around the mean. The standard deviation of both of the boys and girls thumb measurements is very low, less than 1, and therefore from this we can conclude that the mean of the boys and girls thumb is extremely reliable. The standard deviation of the waist measurements for both sexes is quite high (in the boys’ case 6.63 and the girls 4.62), this would lead us to believe that the mean used for the waist measurement is unreliable and is an unfair value to use as a single representative value.
As explained in the planning, I was able to calculate a reliable range that would allow us to identify any outliers. The mean of a particular characteristic added to the double of its standard deviation represents the lower boundary while the mean subtracted by the double of its standard deviation represents the upper boundary, as shown in the table below.
Table 3:
BOYS:
GIRLS:
Using + or – 2 standard deviations will produce a beam effect either side of the line of best fit and this should cover 65% of the points on the scatter graph. If the data doesn’t lie between the 2 x standard deviation range then it is considered an outlier and has been circled on the graphs. These outliers may be due to inaccurate measuring or recording of the results. There may be some measurements that are inaccurate and if for example the measurement of a thumb is too small and a wrist is too big these errors will be amplified when plotted. This type of error may also occur if a boy or girl has a swollen thumb or wrist when the measurements are taken. Measuring both thumbs and both wrists would provide more accurate data.
The outliers will obviously affect the average and would therefore have an overall affect on our conclusion as to whether the theory is correct or incorrect. If there are few outliers, then this would lead us to believe the theory is true in the particular relationship in discussion. Little error would result in a strong correlation. The table below shows the correlation of each of the graphs. Each of the graphs with a correlation equal to or above 0.4 I have added a line of best fit to, as it is reliable in these instances.
Table 4:
The final source of evidence that disproves this hypothesis is the gradients and intercepts of the lines of best fit in the particular graphs (with a 0.4 or above correlation).
Table 5:
If the relationship between each of the applications was double, as Gulliver stated, then I would expect there to be a gradient of 2 and it would be fair to say this was not the case with any of these relationships. The equations of the lines of best fit are quite ridiculous in each graph and there is no evidence of a relationship between each of the characteristics.
Conclusion:
Although there seemed to be a slight relationship between the wrist and neck this is not enough to conclude that Gulliver’s theory is correct. Therefore on the basis of the data I used Gulliver’s theory is incorrect and I cannot accept this hypothesis to be true.
If I was to re-do this investigation then I would make the following improvements:
- I would gain my samples from a larger number of people. If I was calculating a mean over a greater amount of data then it would inevitable result in a more accurate, reliable mean.
- Instead of collecting data from youngsters I would use samples from adults. Youngsters have still not fully developed causing a variation throughout the data and between relationships. Adults, on the other hand, are fully-grown and therefore provide more accurate results.
- I would be sure to have just one person measuring the particular body parts so that all measurements are obtained through the same method and therefore resulting in a fairer investigation.
GCSE Maths Coursework – Statistics (Gulliver’s Theory)
