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Regression Line of X on Y 📈

The Regression Line of X on Y is the best fit line that estimates the average value of X for a given value of Y.

  • Dependent Variable: X (We want to predict this)
  • Independent Variable: Y (We know this)

The Equation 📐

The algebraic equation is:

Xc = a + bY

Where:

  • Xc = Computed/Estimated value of X
  • a = X-intercept
  • b = Slope of the line (b_xy)

The Deviation Formula (Preferred Method) ✨

Just like the other line, we use the regression coefficient b_xy and the means.

The Formula:

(X - X̄) = b_xy (Y - Ȳ)

Where:

  • X̄ = Mean of X series
  • Ȳ = Mean of Y series
  • b_xy = Regression Coefficient of X on Y

Calculating b_xy (Regression Coefficient)

1. Using Actual Means:

b_xy = (∑ xy) / (∑ y²)

(Note: Denominator is sum of y² here, unlike x² in b_yx)

2. Using Assumed Mean (Shortcut):

b_xy = [N ∑ dx dy - (∑ dx)(∑ dy)] / [N ∑ dy² - (∑ dy)²]

3. Using Standard Deviation and Correlation:

b_xy = r * (σx / σy)

Properties of b_xy 🧐

  1. It measures the change in X for a unit change in Y.
  2. If b_xy > 1, then usually b_yx < 1 (because their product r² must be ≤ 1).
    • Exception: Both can be > 1 only if r > 1, which is impossible. So if one exceeds 1, the other must be less than 1. (Wait, strictly speaking b_yx * b_xy = r² ≤ 1. So yes, if one is > 1, other MUST be < 1).

Comparison of Formulas 🔄

ItemLine Y on X (b_yx)Line X on Y (b_xy)
Formula (SD)r * (σy / σx)r * (σx / σy)
Formula (Mean)Σxy / Σx²Σxy / Σy²
EquationY - Ȳ = b_yx(X - X̄)X - X̄ = b_xy(Y - Ȳ)
Used whenFind Y given XFind X given Y

Example Calculation 📝

Data: (Same as previous) X: 1, 2, 3, 4, 5 (X̄=3) Y: 2, 5, 3, 8, 7 (Ȳ=5)

From Deviation Table: ∑ xy = 13 We need ∑ y² (from Y-5 deviations: -3, 0, -2, 3, 2). Squares: 9, 0, 4, 9, 4. ∑ y² = 26

Step 1: Calculate b_xy

b_xy = (∑ xy) / (∑ y²) = 13 / 26 = 0.5

Step 2: Form the Equation X - 3 = 0.5(Y - 5) X - 3 = 0.5Y - 2.5 X = 0.5Y + 0.5

Now, if Y = 10, then X = 0.5(10) + 0.5 = 5.5.

Summary

To fit the line of X on Y:

  1. Use the formula X - X̄ = b_xy(Y - Ȳ).
  2. Calculate b_xy using deviations of Y in the denominator.
  3. Use this line ONLY to predict X.

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