Curriculum
Course: Statistics — Third Secondary
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Curriculum

Statistics — Third Secondary

Unit 1: Correlation and Regression

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Unit 2: Advanced Statistical Measures

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Unit 3: Probability

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Unit 4: Random Variables

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Unit 5: The Normal Distribution

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Text lesson

Correlation

Statistics — Third Secondary

Correlation

Worked examples

Worked examples

Example 1→Find r for x: 1, 2, 3, 4, 5 and y: 2, 4, 6, 8, 10.
  • n = 5, Σx = 15, Σy = 30, Σxy = 2 + 8 + 18 + 32 + 50 = 110.
  • Σx² = 55, Σy² = 4 + 16 + 36 + 64 + 100 = 220.
  • Numerator 5 × 110 − 15 × 30 = 550 − 450 = 100.
  • Denominator √[(5 × 55 − 15²)(5 × 220 − 30²)] = √[(275 − 225)(1100 − 900)] = √(50 × 200) = √10000 = 100.
  • r = 100 ÷ 100 = 1. This is perfect positive correlation: the points lie on the line y = 2x.
  • Example 2 Find r for x: 1, 2, 3, 4, 5 and y: 5, 4, 4, 2, 1.
  • n = 5, Σx = 15, Σy = 16, Σxy = 5 + 8 + 12 + 8 + 5 = 38.
  • Σx² = 55, Σy² = 25 + 16 + 16 + 4 + 1 = 62.
  • Numerator 5 × 38 − 15 × 16 = 190 − 240 = −50.
  • Denominator √[(275 − 225)(5 × 62 − 16²)] = √[50 × 54] = √2700 ≈ 51.96.
  • r ≈ −50 ÷ 51.96 ≈ −0.96. This is a strong negative correlation.
  • Example 3 Hours of study (x) and exam score (y) for six students: x: 1, 2, 3, 4, 5, 6 and y: 40, 46, 47, 55, 58, 62.
Comparison aspectMain ideaKeywords
•
  • n = 6
  • Σx = 21
  • Σy = 308
  • Σxy = 1155
  • Σx² = 91
  • Σy² = 16158
115516158
Numerator6 × 1155 − 21 × 308 = 6930 − 6468 = 462115569306468
Denominator√[(6 × 91 − 21²)(6 × 16158 − 308²)] = √[(546 − 441)(96948 − 94864)] = √(105 × 2084) = √218820 ≈ 467.7816158308²9694894864
•r ≈ 462 ÷ 467.78 ≈ 0.99. This is a very strong positive correlation: more study hours go with higher scoresverystrongpositivecorrelation
Example 4
  • Find r for x: 3
  • 9 and y: 12
  • then interpret the result
Findtheninterpretresult
  • n = 4, Σx = 24, Σy = 30, Σxy = 36 + 50 + 42 + 18 = 146.
  • Σx² = 9 + 25 + 49 + 81 = 164, Σy² = 144 + 100 + 36 + 4 = 284.
  • Numerator 4 × 146 − 24 × 30 = 584 − 720 = −136.
  • Denominator √[(4 × 164 − 24²)(4 × 284 − 30²)] = √[(656 − 576)(1136 − 900)] = √(80 × 236) = √18880 ≈ 137.40.
  • r ≈ −136 ÷ 137.40 ≈ −0.99. This is a very strong negative correlation: as x increases, y decreases almost in a straight line.
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