Correlation & Regression – SS3 Further Mathematics Lesson Note (First Term)

Correlation is the measure of how strongly two variables, x and y, are related.

Behavioural objectives

By the end of the lesson, students should be able to:

1. Illustrate data on a scatter diagram.
2. Identify the type of relationship shown by a scatter diagram.
3. Rank a given distribution.
4. Calculate the rank correlation coefficient of two distributions.
5. Interpret correlation coefficients.

What resources are needed for the lesson?

The lesson needs real-life data that students can collect and tabulate.

• Scores: marks of students in two subjects.
• Ages: ages of students in the class.
• Heights: heights of students in the class.
• Graph paper, ruler and pencil: for scatter diagrams and lines of best fit.

How is the lesson presented?

The lesson is taught in four steps, and the teacher begins with questions from previous lessons, such as “What is data?”

1. Step I, identification of prior ideas (entire class): the teacher asks questions linked to the lesson, for example “State the types of data.”
2. Step II, regression using real-life data (entire class): students study the objectives using real-life data such as scores and ages.
3. Step III, Spearman’s rank correlation (entire class): the class discusses the formula and works examples.
4. Step IV, evaluation: the teacher asks questions based on the objectives and scores with a three-point rubric, where 3 is the highest and 1 is the lowest.

What is correlation?

Correlation is the relationship between two variables x and y, and the variables are said to be correlated when they are related.

• Bivariate data: data in which each item has two values, x and y, for example the mathematics and physics marks of one student.
• Correlation coefficient: a number that states how strong the relationship is.

What is a scatter diagram?

A scatter diagram is a graphic display of bivariate data, with each pair (x, y) plotted as one point.

• Use: it is the best way to find out whether any relationship exists between x and y.
• Reading it: the pattern of the points shows whether the correlation is positive, negative or zero.

What are the types of correlation?

There are three types of correlation: positive, negative and zero.

Positive correlation

• Meaning: in general, an increase in x is accompanied by an increase in y.
• Example: students who study longer tend to score higher marks.

Negative correlation

• Meaning: an increase in x is accompanied by a decrease in y.
• Regression line: the corresponding line has a negative slope.

Zero correlation

• Meaning: there is no apparent association between x and y.
• Pattern: the points are scattered with no direction.

What is a regression line?

A regression line is the line on or near which the points of a scatter diagram lie when the relationship between x and y is linear.

• Linear relationship: all the points lie on or near a straight line.
• Line of best fit: a line drawn by eye (or by calculation) to pass as close as possible to all the points, through (x̄, ȳ).

What is a regression curve?

A regression curve is the curve on or near which most points lie when they follow a parabolic shape.

• Correlation: it is non-linear in this case.

How do you draw a scatter diagram and a line of best fit?

Plot the pairs, find the mean point, then draw a line through the mean point that follows the trend of the points.

1. Draw the x-axis and y-axis with suitable scales.
2. Plot each pair (x, y) as a point.
3. Calculate x̄ = ∑x ÷ n and ȳ = ∑y ÷ n.
4. Plot the point (x̄, ȳ).
5. Draw a straight line through (x̄, ȳ) with about equal numbers of points on each side.
6. Use the line to read estimates of y for a given x.

Worked example: scatter diagram and line of best fit

The table shows the corresponding values of two variables A and B.

• (a) Plot a scatter diagram to represent the data.
• (b) Calculate x̄, the mean of x, and ȳ, the mean of y.
• (c) Draw the line of best fit to pass through (x̄, ȳ).
• (d) From your graph, determine (i) the relationship between x and y, and (ii) the value of y when x is 20.

Solution

1. (a) Plot the ten points (33, 4), (31, 6), (28, 4) and so on.
2. (b) ∑x = 228, so x̄ = 228 ÷ 10 = 22.8.
3. ∑y = 115, so ȳ = 115 ÷ 10 = 11.5.
4. (c) Plot (22.8, 11.5) and draw a line through it that follows the trend of the points.
5. (d)(i) As x decreases, y increases, so there is a negative correlation between x and y.
6. (d)(ii) The line has the form y = a + bx.
7. b = ∑(x − x̄)(y − ȳ) ÷ ∑(x − x̄)² = −329 ÷ 375.6 ≈ −0.876.
8. a = ȳ − bx̄ = 11.5 + 0.876(22.8) ≈ 31.47.
9. So y ≈ 31.47 − 0.876x.
10. When x = 20, y = 31.47 − 0.876(20) ≈ 13.95, which is about 14.

• Note: a line drawn by eye gives values close to these but not exactly equal.

Students’ activities

1. The table gives the population (in 000’s) of a city over 80 years.
   

   Year (Y)	10	20	30	40	50	60	80
   Population (P)	9.0	10.2	12.0	13.9	15.9	20.1	22.7

   The population P is linearly related to the year Y, that is P = h + kY, where h and k are constants.

   • (a) Plot the scatter diagram.
   • (b) Draw an eye-fitted line of best fit (least-square line).
   • (c) Use your graph to estimate (i) the values of h and k, and (ii) the population in the year 90.
2. The table shows values of quantities x and y from a laboratory experiment.
   

   X	0.0	0.5	1.0	1.5	2.0	2.5	3.0
   Y	0.5	0.7	1.0	1.5	1.5	1.5	2.4

   • (a) Plot a scatter diagram for the data.
   • (b) Draw an eye-fitted line of best fit.
   • (c) Using your graph, estimate the value of y when x is 3.5.

How is regression used with real-life data?

Regression uses a line fitted to real data to describe the relationship between two quantities and to estimate one from the other.

Teacher’s activities

The table shows the price of a commodity in 2015 (₦x) and the price of the same commodity in 2016 (₦y).

1. Plot a scatter diagram for the data and draw a line of regression on it.
2. Find the regression equation and obtain the coefficient of regression of y on x.

Students’ activities

Collect and tabulate any data, such as scores, ages or heights, and use it to do the following.

1. Plot a scatter diagram.
2. Draw an eye-fitted line of best fit.
3. Determine the relationship between the data collected.

What is Spearman’s rank correlation coefficient?

Spearman’s rank correlation coefficient, r_k, is a number between −1 and +1 that measures how closely two sets of ranks agree.

• Formula: r_k = 1 − [6∑D²] ÷ [N(N² − 1)]
• D: the difference between the pair of ranks, D = R_x − R_y.
• N: the number of pairs.
• ∑D²: the sum of the squared differences.

How do you rank a distribution?

To rank a distribution, give rank 1 to the highest mark, rank 2 to the next, and so on.

1. Arrange the marks in descending order.
2. Write ranks 1, 2, 3 and so on beside them.
3. Write each rank beside its mark in the original table.
4. If two marks are equal, give each the mean of the ranks they share (for example, two marks sharing ranks 3 and 4 each get 3.5).

How do you calculate r_k?

1. Rank the x values to get R_x.
2. Rank the y values to get R_y.
3. Find D = R_x − R_y for each pair.
4. Square each D and add to get ∑D².
5. Substitute ∑D² and N into the formula.

Example: calculating the coefficient

The marks of four candidates in mathematics and chemistry are given. Calculate the Spearman’s rank correlation coefficient.

Solution

First, arrange each set of marks in descending order and rank them.

Next, write the marks in their original order with the rank of each mark beside it.

• r_k = 1 − [6(4)] ÷ [4(4² − 1)]
• r_k = 1 − 24 ÷ 60 = 0.6
• Interpretation: there is a moderate positive correlation between mathematics and chemistry.

How do you interpret a correlation coefficient?

The sign of the coefficient gives the direction, and its size gives the strength of the relationship.

Students’ activities

The marks of ten students in mathematics and physics in an examination are as follows.

Calculate the Spearman’s rank correlation coefficient. (Answer: 0.8061)

Evaluation

1. State the formula for the rank correlation coefficient.
2. What is a scatter diagram?
3. Describe positive, negative and zero correlation.
4. Rank the marks 4, 2, 1 and 3 in descending order.
5. What does a rank correlation coefficient of −0.9 tell you about two sets of marks?

Assignment

Calculate the Spearman’s rank correlation coefficient in each of the following.

1. The ranks of the marks of 7 candidates in statistics and economics are given below.
   

   Subject	Ranks
   Statistics	4	7	5	1	2	6	3
   Economics	2	5	6	3	1	7	4

   (Answer: 0.714)

2. D is the difference between the ranks of 5 paired variables and ∑D² = 4. (Answer: 0.8)
3. The table shows the marks of ten students in mathematics and further mathematics.
   

   Mathematics	51	25	33	55	65	38	35	53	61	44
   Further Mathematics	20	65	25	36	51	50	77	31	60	05

   (Answer: −0.079, a very weak negative correlation)

References

• Further mathematics for SS3 by P.N Lassa and S.A Ilori
• Hidden Facts in Further Mathematics, page 273
• New Further Mathematics Project 2, page 313