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Regression Line of Y on X 📉

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

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

The Equation 📐

The algebraic equation is:

Yc = a + bX

Where:

  • Yc = Computed/Estimated value of Y
  • a = Y-intercept
  • b = Slope of the line (b_yx)

Calculating 'a' and 'b' (Least Squares Method) 🧮

To find the values of 'a' and 'b', we solve the following Normal Equations:

  1. ∑ Y = Na + b ∑ X
  2. ∑ XY = a ∑ X + b ∑ X²

Note: Since these are tedious to solve every time, we use a simpler formula based on deviations.

The Deviation Formula (Preferred Method) ✨

Instead of solving simultaneous equations, we use the regression coefficient b_yx and the means of X and Y.

The Formula:

(Y - Ȳ) = b_yx (X - X̄)

Where:

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

Calculating b_yx (Regression Coefficient)

There are multiple ways to calculate b_yx depending on the data:

1. Using Actual Means (X̄, Ȳ are integers):

b_yx = (∑ xy) / (∑ x²)

(Where x = X - X̄ and y = Y - Ȳ)

2. Using Assumed Mean (Shortcut Method):

b_yx = [N ∑ dx dy - (∑ dx)(∑ dy)] / [N ∑ dx² - (∑ dx)²]

3. Using Standard Deviation and Correlation:

b_yx = r * (σy / σx)
  • r = Correlation Coefficient
  • σy = Standard Deviation of Y
  • σx = Standard Deviation of X

Properties of b_yx 🧐

  1. It measures the change in Y for a unit change in X.
  2. If b_yx is positive, Y increases as X increases.
  3. Its value can be greater than 1 (unlike correlation r).
  4. It has the same sign as r and b_xy.

Example Calculation 📝

Data: X: 1, 2, 3, 4, 5 Y: 2, 5, 3, 8, 7

Step 1: Find Means X̄ = 3, Ȳ = 5

Step 2: Calculate Deviations (x, y)

XYx (X-3)y (Y-5)x²xy
12-2-346
25-1010
330-200
481313
572244
Sum001013

Step 3: Calculate b_yx

b_yx = (∑ xy) / (∑ x²) = 13 / 10 = 1.3

Step 4: Form the Equation Y - 5 = 1.3(X - 3) Y - 5 = 1.3X - 3.9 Y = 1.3X + 1.1

Now, if X = 6, then Y = 1.3(6) + 1.1 = 7.8 + 1.1 = 8.9.

Summary

To fit the line of Y on X:

  1. Calculate Means (X̄, Ȳ).
  2. Calculate b_yx.
  3. Substitute in Y - Ȳ = b_yx(X - X̄).

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