# Correlation and regression

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Introduction

Ibrar KhanAS Use of mathematics coursework

Correlation and regression

## Background information

In statistics, correlation often indicates the strength and direction of a linear relationship between two random variables. If there is no correlation between the two sets of data, the points will be widely scattered and will require a circular shape to enclose them. If the two sets of data have weak positive correlation, the points can be modelled in a broad eclipse sloping upwards from bottom left to top right. Furthermore if the two sets of data have a strong positive correlation, the points lie within a narrow eclipse sloping upwards.

If the two sets of data have weak negative correlation, where one quantity generally increases as the other one decreases, the points can be modelled in a broad eclipse sloping downards from top left to bottom right. If the two sets of data have a strong positive correlation, the points lie within a narrow eclipse sloping downwards.

Finally ‘perfect’ positive correlation (a correlation co-efficient of +1) implies that as one variable moves, either up or down, the other variable will move in lockstep, in the same direction. Alternatively, ‘perfect’ negative correlation means that if one variable moves in either direction the variable that is perfectly negatively correlated will move by an equal amount in the opposite direction, also when all the data points on a scatter diagram accumulate on a straight line we say that they are perfectly positively or negatively correlated.

A correlation coefficient is a number between -1 and +1, which measures the degree to which two variables are linearly related. If there is perfect linear relationship with positive slope between the two variables, we have a correlation coefficient of +1; if there is positive correlation, whenever one variable has a high (low) value, so does the other.

Middle

After finding the median marks for the non-calculator written paper (Paper 1) of the male and female populations, I can quite honestly articulate that so far my prediction of ‘the male population will have higher marks on the non-calculator written paper (Paper 1) and the calculator written paper (Paper 2) than the female population’ is true and accurate because the median marks for the non-calculator written paper (Paper 1) of the male population was ‘35’ marks, whereas the female population was only ‘29’ marks.

Next, I am going to find the median marks on the calculator written paper (Paper 2) for both the male and female populations by doing a similar process as above.

Starting with the calculator written paper (Paper 2) for the male population.

0.5 (29 + 1) = 15 th value

Then you start with the highest mark, which for the male population on the calculator written paper (Paper 2) was ‘46’ marks and count along, next being ‘45’ and then ‘45’ and so on until you reach the 15 th value. The 15 th value in my case is ‘30’ marks, so therefore, the median mark for the male population on the calculator written paper (Paper 2) is 30 marks.

Next, the calculator written paper (Paper 2) for the female population.

0.5 (35 + 1) = 18 th value

Then you start with the highest mark, which for the female population on the calculator written paper (Paper 2) was ‘51’ marks and count along, next being ‘47’ and then ‘42’ and so on until you reach the 18 th value. The 18 th value in my case is ‘26’ marks, so therefore, the median mark for the female population on the calculator written paper (Paper 2) is 26 marks.

After finding the median marks for the non-calculator written paper (Paper 1) and the calculator written paper (Paper 2)

Conclusion

Hypothesis

For my new hypothesis I am going to compare the correlation coefficients of the male and female population in terms of the mental test and the calculator written paper (Paper 2) results. On this occasion I predict that ‘the correlation coefficient for the female population will be larger than the correlation coefficient for the male population. Furthermore I again predict the values of the correlation coefficients for both the male and female population to be between 0.5 < r ≤ 1, which would indicate strong positive correlations’. I will be able to find the values of the correlation coefficients by using a CASIO graphical calculator (CFX-9850GC PLUS).

After carrying out the method I explained in the previous hypothesis I found the correlation coefficient of the mental test and the calculator written paper (Paper 2) for the male population to be ‘r = 0.775’ and the correlation coefficient for the female population to be ‘r = 0.698’. Now that I have found the correlation coefficients of the mental test and the calculator written paper

(Paper 2) for the male and female populations I can quite truthfully articulate that my hypothesis ‘the correlation coefficient for the female population will be larger than the correlation coefficient for the male population. Furthermore I again predict the values of the correlation coefficients for both the male and female population to be between 0.5 < r ≤ 1, which would indicate strong positive correlations’ was incorrect and erroneous because the correlation coefficient for the male population, which was ‘r = 0.775’ was larger than the correlation coefficient for the female population, which was ‘r = 0.698’. A fraction of my hypothesis that was proved correct was both correlation coefficients that I acquired were in the region of 0.5 < r ≤ 1, which would indicate strong positive correlations.

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