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CA Foundation · Quantitative Aptitude · Correlation and Regression

A scatter diagram of five points (x, y) is: (1, 5), (2, 3), (3, 1), (4, 3), (5, 5). It forms a symmetrical V or U shape. Using the data, what are the value of Σ(x - x̄)(y - ȳ) and the correct interpretation by the scatter diagram method?

The sum of products of deviations is 0, so r = 0, even though the points clearly form a U-shaped curve. The scatter diagram reveals a strong non-linear relationship that r cannot detect. Zero correlation therefore does not mean the variables are independent.

  1. AΣ(x - x̄)(y - ȳ) = 0, so r = 0 although a clear non-linear relationship existsCorrect
  2. BΣ(x - x̄)(y - ȳ) = 8, so r is positive and strong
  3. CΣ(x - x̄)(y - ȳ) = -8, so r is negative and strong
  4. DΣ(x - x̄)(y - ȳ) = 0, so the variables are independent of each other

Explanation

x̄ = 3 and ȳ = 3.4. Deviations in x: -2, -1, 0, 1, 2. Deviations in y: 1.6, -0.4, -2.4, -0.4, 1.6. Products: -3.2, 0.4, 0, -0.4, 3.2, which sum to 0. So r = 0, yet the points follow a clear curve. Zero r means no linear relation only, not independence, so the last option is wrong.

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