Correlation and regression · Year 12
In Correlation and regression (Year 12) you practise 3 skills: Reading a scatter diagram · What the correlation coefficient keeps · Predicting with the regression line. A scatter diagram shows pairs of values as points. When the points rise from left to right the correlation is positive, when they fall it is negative, and when they have no direction there is no correlation. The closer the points lie to one straight line, the stronger the linear correlation, and when they all lie on one line the relation is perfect.
The linear correlation coefficient measures the correlation in one number, always between minus one and one. Its sign is the direction of the correlation and its size is the strength. It does not change when a constant is added to the values or the units change, and it is the same number when the two variables swap roles. Multiplying one variable by a negative number changes its sign. To work it out, find the means and the standard deviations, multiply the differences from the means in each pair and add. The correlation coefficient is that sum divided by the number of pairs times the two standard deviations. For standardised data it is simply the mean of the products. The regression line passes through the mean point, and its gradient is the correlation coefficient times the standard deviation of the vertical values, divided by the standard deviation of the horizontal values. Substitute a value into it to get a prediction. The prediction is not exact, and it is sound only within the range of the data. A strong correlation does not prove that one variable causes the other.
Tip: The regression line always passes through the mean point.
What you learn
Reading a scatter diagram
For example: The scatter diagram shows pairs of values.
What correlation is there between the two variables?- Negative correlation
- No correlation
- Positive correlation
Answer and solution
The answer: Positive correlation
As one value grows, the other tends to grow too: the points rise from left to right. That is a positive correlation.
What the correlation coefficient keeps
For example: The linear correlation coefficient of a set of pairs is r = −0.18 . Add 20 to every y value. What happens to the correlation coefficient?
- Its size grows
- Its size falls
- It changes sign
- It does not change
Answer and solution
The answer: It does not change
Adding a constant moves every point up and leaves the correlation as it was: the correlation coefficient does not change.
Predicting with the regression line
For example: The scatter diagram shows 8 points and their regression line, y = 0.81x + 2.71 .
What is the predicted value of y for x = 4.5 ? Round to 1 decimal place.Answer and solution
The answer: 6.4
Substitute: y = 0.81 × 4.5 + 2.71 ≈ 6.4
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).