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Bivariate Data

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Introduction

Bivariate Data

        I am going to carry out an investigation into a set of bivariate data. The data I will investigate are a previous year groups KS3 and GCSE point score averages. I will see, whether or not there is a correlation between the KS3 and GCSE result scores.

        By finding a correlation or not, I will be able to determine if the scores obtained at KS3 will allow teachers to predict the student’s score at GCSE. If there is a strong correlation, this will be very useful for teachers and students to give them an idea on what they can be expected to score. Grade Predictions would be easier and probably more accurate.

        For example, if there is a correlation, a student could predict their GCSE score by using the KS3 results they obtained, and with this would provide a target score to reach or beat. This will also be useful for the teacher where they will be able to overview any additional help or teaching that a student may or may not need.

The Population (presented in table 1) shows last year’s groups, KS3 and GCSE point score averages. There are a total of 90 pieces of data. This is a fairly small population but it was the only set easily available. From the 90 I will randomly sample 50 pieces to investigate.

The way I randomly sampled the population, was to number each set one to ninety (1-90), while also numbering ninety pieces of paper.

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Middle

x and y to show that I have knowledge on how to calculate the mean.

Table 2, shows the sample, in relation to x and y, where x = KS3 and y = GCSE point scores. It also shows, x2, y2, xy, and the means of x and y. These have been calculated for the Pearson’s Product Moment Correlation Coefficient. From this, I can use the formula and calculate if there is a linear correlation or not. If there is a correlation (r) will take a value close to zero. The nearer ‘r’ is to +1 or –1 the stronger the correlation. From observation alone I know that the correlation will be positive.

I have calculated the correlation coefficient and the value I have ended up with is 0.935. As said before, the nearer ‘r’ is to +1 or –1 the stronger the correlation. For my sample, ‘r’ = 0935 to 3 s.f. This shows that there is a very strong positive linear correlation. This means that a line of best fit would be suitable for the data, as the line will be fairly accurate. The line of best fit has been drawn on graph 1.

The value calculated, ‘r’, can be used as an estimate for ‘ρ’. Where ‘ρ’ = the Correlation Coefficient of the Parent Population. ‘r’ can also be used  to carry out a hypothesis test on this value of ‘ρ’. The test consists of a Null hypothesis, where there is no correlation with the parent population. This is denoted as H0: ρ = 0. There are also three alternatives.

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Conclusion

        I believe the only errors with the data, were that that the scores given were all averages. The sample of 50 from the parent population was too big considering that the parent population consisted of 90 pieces of data. The parent population itself were KS3 and GCSE point scores from just one particular year group. It would have been more appropriate to gather data from several year groups to give the school a more general idea on the educational level through a period of time.

        To improve the investigation I would have sampled from a much bigger population to get more varied pieces of data. Once finding the linear correlation, if it there was a strong linear correlation I would have calculated the Least Square’s Regression line to find an equation for the line of best fit. This equation could then be used by itself to calculate GCSE scores by just being given the KS3 scores. I believe that using average point scores generalises a student’s ability. They may be weak in certain subjects and stronger in others. To improve this investigation, I would take the point scores of each subject and investigate their relation to the GCSE score points of the same subject. This would help show the student’s abilities in each subject and not his overall ability that may seem weaker because of a singular weakness in once subject which would bring their average point score down.

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