Correlation and regression · Grade 11 · 4 units
In Correlation and regression (Grade 11 · 4 units) you practise 7 skills: Reading a scatter diagram · The mean point and the graph of means · Working out the correlation coefficient · Comparing correlation coefficients · What the correlation coefficient keeps · The regression line · 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.
The mean point and the graph of means
For example: The table shows five pairs of values.
| x | y |
|---|---|
| 2 | 17 |
| 7 | 18 |
| 8 | 21 |
| 12 | 25 |
| 16 | 9 |
Answer and solution
The answer: (9, 18)
The x values add up to 45 , so x̄ = 9 . The y values add up to 90 , so ȳ = 18 . The mean point is (9, 18)
Working out the correlation coefficient
For example: The table shows four pairs of values.
| x | y |
|---|---|
| 2 | 18 |
| 7 | 11 |
| 12 | 11 |
| 7 | 20 |
Answer and solution
The answer: −35
The means are x̄ = 7 and ȳ = 15 . The differences from the mean of x : −5, 0, 5, 0 . The differences from the mean of y : 3, −4, −4, 5 . The products of the differences in each pair: −15, 0, −20, 0 . They add up to −35 .
Comparing correlation coefficients
For example: Here are three tables of pairs of values. Table A:
| x | y |
|---|---|
| 3 | 26 |
| 5 | 20 |
| 10 | 5 |
| 11 | 2 |
| x | y |
|---|---|
| 3 | 8 |
| 9 | 23 |
| 10 | 26 |
| 11 | 22 |
| x | y |
|---|---|
| 1 | 31 |
| 9 | 25 |
| 10 | 26 |
| 11 | 21 |
- Table B
- Table C
- Table A
Answer and solution
The answer: Table A
In Table A every pair lies on one falling line: whenever x grows, y falls at the same rate. So there the correlation coefficient equals minus one.
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.
The regression line
For example: For a set of pairs: r = 0.9 , Sx = 2 , Sy = 11 , and the mean point is (x̄, ȳ) = (72, 68) . What is the gradient of the regression line of y on x ?
Answer and solution
The answer: 4.95
Substitute: b = 0.9 × 11 ÷ 2 = 4.95
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
Not on the site yet
Parts of this topic that are not built yet: Working out the correlation coefficient of large data sets with technology · Building the regression line from raw data.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).