Types of relationship that I am looking for, for the PMCC I would be looking for
Within the scatter graphs I am looking for a positive correlation this indicates, that there is a positive relationship between the predicted and achieved. A
I would not expect a negative relationship, because this would be saying the better the prediction the worse they would do in the exam B
I also would not expect no correlation, because both prediction are based on an element of ability. But this is the student ability at the time, of Yellis; this could change positively or negatively over the time before the final gcse exam. C
But if there was not this distribution if it looked like D1 then the relation ship would be rank ???? this means that the variables would not be normal. So if I got this result I would be very surprised
The type of data I am looking at.
The three sets of data I am looking at are all independent data. This means that the data is not dependent on one another, an example of dependent data would be course work mark compared to gcse grad, where the gcse grad is made up of the course work and a exam results, the course work grad would be the independent variable and the overall gcse grade would be dependent because it is based on the interpreted the course work grade.
All of the predictions and the results are independent of each other, but they are dependent on the students abilities. Normally the independent values are placed on the x-axis, as these are all independent variables I have chosen to place the gcse results on the x-axis because its is in both comparisons, also the GCSE’s achieved is in integer value so it is more appropriate to have it as the X-axis
Random
If all of the data is a normal distribution, eg D where the mean and the mode are the same or similar in a practical situation ?????? WHAT IS NORMAL? the scatter graph will have a elliptical, shape to it. With the 2 extremes, with the least number of students and the middle with the most number.
The gcse are awarded with a normal distribution nationally, so if stoke Dameral was a “normal” school I would expected the distribution of results to be normal, but as stoke Dameral is a state inner-city school with many privet school in the vicinity, the level of the students is on average below the national average, therefore I would expected a negatively scewed distribution. Eg E or shifted F
The nature of the data for all set of data is random but for the staff predicted grad and the gcse result it was only random within grad classifications, so it is not totally.. An example of random data would be yellis
The gcse grads achieved and predicted by staff are given in alphabetic values, so they need to be converted to numerical ones for analyses, because there are regions for getting grades eg A* 100% to 91% but someone getting 91% would get the same grade as 100% so to make the grades fare I will have to make all of the values in the middle of the grade boundaries.
I would have preferred to use the raw result before grades were assigned but this was not available to me so I have to use the mid value for the achieved gcse result, also the staff predictions can only really be given in grade values.
All of the scatter graphs fitted to the elliptical positive correlation shape, this tells me that I can apply the PMCC
It is clear from the scatter graphs that the math and science are more of a normal distribution, where English Yellis appears less of a ellipse and more of a circle, this means it is a more of a random distribution so does not fit a perfected normal distribution I believe there will still be correlation measurable with PMCC, but it should be less then the rest.
I have made a table of more critical interpretations of the scatter graphs
I will now calculate the PMCC values for all of the predictions.
What is PMCC?
PMCC is a way to measure the correlation between bivariate data (just meaning changing values associated with one measurement)
PMCC = r
r = Sxy/SxSy
Sx = √((Σx2/n)-x2 )
Sx = square root of the average x squared value subtracted the means squared.
Sx = there average variation from the mean.
Sx is the same as Sy but for the other variable involved. This is always a positive value because of the squaring,
Sxy = (Σxy/n) –xy
Sxy = the square root of average x multiplied by y value subtract the two means multiplied together.
Sxy = the average difference in the data multiplied together, to the means multiplied. This is what determines if there is positive or negative correlation. If this value is negative then the correlation is too.
r = is the correlation coefficient for the sample, it with zero being no correlation and 1 or –1 being perfected correlation. The signs say if it is positive or negative, the values in-between are different degrees of correlation. This is only the correlation of the sample, but if the sample is a random sample, like it is, then it can be used for a estimate of the parent populations correlation coefficient (ρ).
ρ can be tested by performing a hypothesis test, using the value n, the number in the sample and by working to a significance level, the significance level I have chosen to use is 5% because the data was integer values, I would have used a smaller significance level if I had raw exam scores.
The hypothesis test gives a critical value, where the justification of the claim that there is correlation in the parent population.
H0: ρ = 0 There is no correlation between predicted and achieved GCSE grades
H1: ρ > 0 There is positive correlation between predicted and achieved GCSE grades (one tail)
I have worked out the value of r using Microsoft Excel:
From looking up the PMCC table and extracted, the values when n = 50 on a one tail test, (it is one tail because there is no negative correlation)
The smallest r value is 0.24
Since 0.241 > 0.2353, the critical value, the alternate hypothesis is accepted, that there is significant evidence for correlation.
This shows that there is correlation between the predictions and the achieved results. This was expected as the ability at the Yellis test would be greater if there was a greater ability of the student they are more likely to do well in there GCSEs and the staff ability to predicted the grade achieved is also likely to be reflective of the student overall grade.
Interpretation
I have took a sample of 50 student out of the parent population of ~250. from the PMCC values being higher then the null hypothesis one, I can say that there is correlation between the predicted and the achieved.
The significant discovery is that staff predicted grades have greater correlation then those by Yellis. This is shown by the significantly higher PMCC values for staff predictions then Yellis. This concurs with my original aim to prove that the staff predicted grades are much more accurate then those of Yellis. I can also draw up more conclusions from the statistical evidence.
From looking at the means I can concluded that, on average “Stoke Dameral” achieves lower then the national average in mathematics and slightly lower in English Language but just above the national average in Science. All of the Yellis test under predicted the ability of students or that of the teaching staff. Mathematics staff predicted over the students achievements on average, but English and maths staff prediction were lower.
Looking at the PMCC values individually, I can conclude that the Yellis mathematic prediction showed a small amount of correlation if the significance level would have been smaller the 5% then there would not be enough evidence to say that there is correlation. I am therefore concluding that Yellis predictions for English based on this investigation should not be used to access a students ability.
PMCC for Yellis mathematics and science are above the level of English PMCC but still lower then those by staff, I would conclude from this that although there is more correlation there is still not enough to make accurate predictions for the parent population, but I would say that it would be enough to make predicted grades that are banded, an example would be:
The PMCC of the staff prediction for English and Science showed good correlation, these values are well about the 0.5% significance level so correlation is deferent. With this level of correlation I can say that there is a deferent like to a student ability to when Yellis was taken to the results archived at the end of year 11.
The staff predictions of mathematics showed the greatest level of correlation, this suggest that the level of work is maintained over the course of the GCSE because the GCSE staff predictions are based on the students natural ability, and their work rate. It could also mean that the work rate has less of a effect on the overall outcome.
On the whole I believe that this data was worth collecting and the analysis was very useful, it shows although Yellis’s predictions do correlate with the GCSE achieved, they are not a accurate as staff predictions.
One of the major sources of error in this investigation was not having raw data, although I tried to counteracted this by taking a large sample, I feel that there would have been greater correlation if I would have used the raw scores. I restricted my self to one year group so one set of results I would have like to expanded the investigation so that I incorporated other years, this would insure that my conclusions are true for all years like I expected.
Another problem I found I could only test students with all of the data, this meant that I didn’t use student that didn’t turn up for the exam, these students where most probably likely to acquire poor grades, so on the whole the average grade was increased. chose
Although my parent population is stoke Dameral, the conclusions have implications for all schools that use yellis as a form of predictions, it surely have implications for inner city public schools, with relatively the same standards as Stoke Dameral. I would like to improve this investigation so that the parent population would be the England. To do this I would randomly select 10% of the schools taking part in Yellis, and perform the same statistical analysis as I have already preformed on Stoke Dameral