The aim is to explore and examine how to handle errors and other problems associated with making experimental measurements. This will be carried out through a series of activities involving simple scenarios to conduct numerous trials in and apply statistical methods for analysis.
Two paper aeroplanes and additional material for design if required
Bag of 50 rods
Pendulum system with adjustable string
5 metal beams of different sizes and bench clamp
Design and build two paper aeroplanes and conduct flights repeatedly towards a target 5m away. 29 trials (to have a median value) were carried out and the distances travelled by the two planes were recorded.
The results are shown in Table 1. We can make a statistical analysis for the results of the planes by assessing the distribution of each set of data.
Aside from making a visual estimate of whether the data is normally distributed or not, one of the many ways of verifying this estimate is by using a box-and-whisker plot. The results are sorted according to the frequency of each value, and the peak is noted relative to the data distribution.
The data ranges from 7 to 76 with a median of 36.5. For a normal distribution this would be exactly in the middle of the range but as the exact middle would be 41.5, the distribution of plane 1’s data is skewed. The box-and-whisker plot would look like:
The data ranges from 0 to 52 with a median of 9. For a normal distribution the median would be exactly 26 so the distribution of plane 2’s data is heavily skewed. The box-and-whisker plot would look like:
A U-test must be performed on the data as opposed to the t-test because the U-test is a testimonial to the fact the data is not normally distributed. This factor along with there being two continuous groups of data is enough to meet the condition of the u-test. The Mann-Whitney U-test on the data shows that:-
- a) The distribution of plane1 > distribution of plane 2; therefore there is a hypothesis
- b) The data can be ordered
Part a) relates a hypothesis to the design of the planes and what in the design makes the difference in data distribution. The box-and-whisker plot of the two planes show that planes 1 is more consistent in travelling far, whilst though plane 2 does travel far on occasion, the frequency is less. This suggests that plane 1 is more successful because it has been designed to have more of its weight nearer the front to avoid drag, maintain more balance and have more thrust.
Measure the length of the 50 rods and place in a bag. Compute the initial mean and standard deviation of the 50. Take samples of 1, 2, 3, 5, 10, 15 and 20, and each time measure the lengths of the rod. Calculate the mean and standard deviation for each sample, and compare against the initial values.
The 50 rods were recorded to have lengths of:-The data has a mean of 54.26 and a standard deviation of
The samples and corresponding mean and standard deviation of them are constructed in table 3 below.
Table 3 exemplifies how the more the elements in each sample, the closer the mean is to the initial mean of the whole population, in this case being a length of 54.26 from all 50 rods. Also, as the median of both the initial population and the samples are not the same as the mean, it can be said that neither sets of data are normal distributions, albeit the whole population has a median close to the mean. This infers that although there is a computational standard deviation for each set of data, in practice this will not be consistent on both sides of the mean – it is a generalised attribute of the data. However, as previously stated, the more the sample population, the closer the standard deviation is to that of the whole population because of the increase in correlation between the sample data and initial data.
Using the pendulum system with variable string length, note the length of the string and the time taken to complete 10 cycles after being initially swung. Repeat the process 5 times, varying the length each time. The results are used to determine the value of g in the equation T = 2π√(l/g), where ‘T’ is the time recorded and ‘l’ is the length of the string; g is therefore a constant.
To calculate g either each value of T and l can be implemented into the equation manually with an overall value of g calculated by taking the average of all the results. However, an alternative is to plot T² v L to uncover the value of g.
The reason T² v L is chosen as opposed to for example T v L is because if the latter were taken, the square root would be disregarded which plays a significant factor. The former is better than even T v √L as although this is also where T is proportional to √L, the result would be 1/√g which complicates things unnecessarily. Therefore, if T² v L is plotted, this would result in a straight line graph, whereby the gradient is simply 4π²/g, and thus g can easily be recovered.
The error and uncertainty in measuring the lengths in the metal beam, which is clamped to a bench and applied a load of varying weights is to be implemented into calculations of deflection, the error in the deflection and finally examining Young’s Modulus of the metal beam.
Cross section = 12mm * 3mm = 36mm2
Distance from block = 608mm