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Multiple correlation: expenditure, income, family size

Short answer

R_y.x1x2 = 0.718. Calculated from simple correlations: r_yx1=0.6614, r_yx2=0.3546, r_x1x2=0.7939 using formula R² = (r_yx1² + r_yx2² - 2r_yx1 r_yx2 r_x1x2)/(1 - r_x1x2²).

A house survey on monthly expenditure on food yield following data:

Monthly expenditure (100 Rs.)10152025303540
Monthly income (1000 Rs.)2457665
Size of the family457108114
Obtain the multiple correlation coefficient.

Let Y = monthly expenditure on food (dependent), X₁ = monthly income, X₂ = family size. n = 7.

Step 1: Sums

     y    x1    x2   x1y   x2y  x1x2    y²   x1²   x2²
    10     2     4    20    40     8   100     4    16
    15     4     5    60    75    20   225    16    25
    20     5     7   100   140    35   400    25    49
    25     7    10   175   250    70   625    49   100
    30     6     8   180   240    48   900    36    64
    35     6    11   210   385    66  1225    36   121
    40     5     4   200   160    20  1600    25    16
------------------------------------------------------
   175    35    49   945  1290   267  5075   191   391

Step 2: Simple correlation coefficients

r = Sab/√(Saa·Sbb), with Sab = Σab − ΣaΣb/n.

  • r_yx₁: Sx₁y = 70.000, Sx₁x₁ = 16.000, Syy = 700.000 → 0.6614
  • r_yx₂: Sx₂y = 65.000, Sx₂x₂ = 48.000 → 0.3546
  • r_x₁x₂: Sx₁x₂ = 22.000 → 0.7939

Step 3: Multiple correlation

R_y.x₁x₂ = √[(r_yx₁² + r_yx₂² − 2·r_yx₁·r_yx₂·r_x₁x₂) / (1 − r_x₁x₂²)]

  • r_yx₁² + r_yx₂² = 0.4375 + 0.1257 = 0.5632
  • 2·r_yx₁·r_yx₂·r_x₁x₂ = 0.3724
  • Numerator = 0.1908; denominator = 1 − 0.6302 = 0.3698

R² = 0.1908 / 0.3698 = 0.5161

Answer: R_y.x₁x₂ = 0.7184 ≈ 0.718.

Interpretation: income and family size together explain about 51.6% of the variation in monthly food expenditure; the relationship is strong and positive.

✓ Answer checked by Claude Sonnet 5.5

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