Regression and the PMCC
Regression and the product moment correlation coefficient (PMCC) are Higher-tier tools for analysing bivariate (paired) data. The PMCC is a number between -1 and 1 that measures the strength and direction of a linear relationship between two variables, while a regression line is the equation of the line of best fit, used to predict the value of one variable from a given value of the other.
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Method
- From the question, identify or calculate the summary totals needed: n (the number of pairs), the sum of x, the sum of y, the sum of x^2, the sum of y^2, and the sum of xy.
- Substitute these into Sxy = sum(xy) - (sum x)(sum y)/n, Sxx = sum(x^2) - (sum x)^2/n, and Syy = sum(y^2) - (sum y)^2/n.
- Substitute Sxy, Sxx and Syy into r = Sxy / sqrt(Sxx x Syy) to find the PMCC, rounding as instructed by the question.
- Interpret r in context: a value close to 1 or -1 shows strong linear correlation, a value close to 0 shows weak or no linear correlation, and the sign shows the direction (positive or negative).
- For the regression line, find the gradient b = Sxy/Sxx, then find the intercept using a = ybar - b x xbar, where xbar and ybar are the means of x and y, and write the line as y = a + bx.
- Use the regression equation to predict y for a given x, but check that x lies within (interpolation) rather than far outside (extrapolation) the range of the original data before trusting the prediction.
Worked example
For 6 pairs of data, x: 2, 4, 6, 8, 10, 12 and y: 3, 5, 6, 8, 9, 11. Calculate the product moment correlation coefficient, giving your answer to 3 significant figures.
- n = 6, sum x = 42, sum y = 42, sum x^2 = 364, sum y^2 = 336, sum xy = 348.
- Sxy = 348 - (42 x 42)/6 = 348 - 294 = 54.
- Sxx = 364 - (42)^2/6 = 364 - 294 = 70.
- Syy = 336 - (42)^2/6 = 336 - 294 = 42.
- r = 54 / sqrt(70 x 42) = 54 / sqrt(2940) = 54 / 54.2 = 0.996 (3 sf).
- Final answer: r = 0.996, showing a very strong positive correlation.
Practice questions
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Q1A PMCC value of r = 0.85 is calculated for two variables. Describe the strength and direction of the correlation.Show answer
Answer: Strong positive correlation
Q2A PMCC value of r = -0.2 is calculated. Describe the correlation.Show answer
Answer: Weak negative correlation
Q3The equation of a regression line is y = 3 + 2x. Find y when x = 5.Show answer
Answer: y = 13 (3 + 2 x 5)
Q4For a data set, Sxy = 30, Sxx = 50, Syy = 40. Calculate the PMCC to 3 significant figures.Show answer
Answer: r = 0.671 (30/sqrt(50 x 40) = 30/44.72)
Q5For a data set, Sxy = 45 and Sxx = 60. Find the gradient b of the regression line.Show answer
Answer: b = 0.75 (45/60)
Q6A regression line y = 12 + 1.5x is used to predict y when x = 200, even though the original x-values ranged only from 5 to 20. Explain why this prediction may be unreliable.Show answer
Answer: x = 200 is far outside the range of the original data (extrapolation), so the linear relationship found may not still hold there
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
For 5 pairs of data, Sxy = 18, Sxx = 24 and Syy = 20. (a) Calculate the product moment correlation coefficient, giving your answer to 3 significant figures. (b) State what this value indicates about the correlation between the two variables.
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A researcher collects data on the age, x years, and reaction time, y hundredths of a second, of 7 people. She finds n = 7, sum x = 210, sum y = 490, sum x^2 = 6510, sum y^2 = 35700, sum xy = 15030. (a) Show that Sxy = 330. (b) Find Sxx and Syy. (c) Hence calculate the product moment correlation coefficient, giving your answer to 3 significant figures.
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The regression line for predicting exam score, y, from hours of revision, x, is y = 42 + 3.5x, valid for 2 <= x <= 15. (a) Predict the exam score for a student who revises for 8 hours. (b) State, giving a reason, whether this prediction is reliable.
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Free printable worksheet
Want more practice on paper? Download the regression and the pmcc worksheet pack - 19 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 15 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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