English
Menu

What is Correlation coefficient?

A measure expressing the strength and direction of the linear relationship between two quantities as a value from −1 to 1. It is calculated by dividing the covariance of x and y by the product of the standard deviation of x and the standard deviation of y. The closer it is to 1, the stronger the positive correlation; the closer to −1, the stronger the negative correlation; close to 0, the linear relationship is weak. Curved relationships hardly show up in the correlation coefficient, and correlation does not mean that one quantity causes the other. The slope of the regression line is the correlation coefficient multiplied by the ratio of the standard deviations of the two quantities.

Japanese name: 相関係数

Correlation coefficient appears in 5 practice questions across 3 topics, including Probability, Statistics and Queueing, Yield and quality control, Business Strategy and Accounting Basics.

Field: Probability, Statistics and Queueing

Try it: questions about Correlation coefficient

Which of the following correctly describes the correlation coefficient of two quantities?

  1. AEven if the correlation coefficient is close to 0, there may be a curved relationship between the two quantities
  2. BWhen the correlation coefficient is −0.9, there is almost no relationship between the two quantities
  3. CThe larger the correlation coefficient, the larger the slope of the regression line predicting one quantity from the other, without exception
Show answer and explanation

Answer: A. Even if the correlation coefficient is close to 0, there may be a curved relationship between the two quantities

The correlation coefficient expresses the strength and direction of the linear relationship between two quantities as a value from −1 to 1. Close to 1, the points lie along a line rising to the right; close to −1, along a line falling to the right; close to 0, the linear relationship is weak. For example, for y = x² with x from −1 to 1, there is a strong valley-shaped relationship, yet the correlation coefficient can be close to 0. The slope of the regression line depends not only on the correlation coefficient but also on the ratio of the standard deviations of the two quantities.

Which of the following is closest to the correlation coefficient of the five data pairs (x, y) = (1, 2), (2, 4), (3, 5), (4, 4), (5, 5)?

  1. A0.50
  2. B0.60
  3. C0.77
Show answer and explanation

Answer: C. 0.77

The mean of x is 3 and the mean of y is 4. The sum of the products of the differences from the means is (−2) × (−2) + (−1) × 0 + 0 × 1 + 1 × 0 + 2 × 1 = 6, the sum of the squared differences of x from its mean is 10, and that of y is 6. The correlation coefficient is 6 ÷ √(10 × 6) = 6 ÷ √60 ≈ 0.77, a fairly strong positive correlation.

Correlation coefficient in the explanations

Questions that mention Correlation coefficient (5)

  1. Which of the following correctly describes the correlation coefficient of two quantities?
  2. Which of the following is closest to the correlation coefficient of the five data pairs (x, y) = (1, 2), (2, 4), (3, 5), (4, 4), (5, 5)?
  3. Two quantities x and y have a correlation coefficient of 0.8, the standard deviation of x is 5, and the standard deviation of y is 10. Which of the following is the slope of the least-squares regression line predicting y from x?
  4. Which chart plots pairs of two characteristic values to examine their correlation?
  5. At a fab, the monthly number of wafer starts x (thousand wafers) and electricity cost y (¥ million) were (x, y) = (2, 5), (4, 9), (6, 11) and (8, 15). If the regression line y = ax + b is found by the least squares method and the slope a is taken as the variable cost per thousand wafers and the intercept b as the fixed cost, which combination is correct?

Books on Probability, Statistics and Queueing Ad

As an Amazon Associate I earn from qualifying purchases.

Related terms

Standard deviation Variable cost Fixed cost Break-even point Bayes' theorem Binomial distribution Conditional probability Confidence interval Expected value Exponential distribution

Back to the glossary