### Intercept

The intercept of the least-squares line.

The formula:

Intercept = [∑(x²)∑(y) - ∑(x)∑(xy)]/[n∑(x²) - ∑(x)²]

Sample 1:

Given the set of X and Y values where X and Y can represent any correlated values below:

X |
Y |

1 |
2 |

2 |
4 |

3 |
6 |

4 |
8 |

5 |
10 |

6 |
12 |

7 |
14 |

8 |
16 |

9 |
18 |

10 |
20 |

Sample table of correlated values

Steps:

1. Solve the parts of the formula.

Intercept = [∑(x²)∑(y) - ∑(x)∑(xy)]/[n∑(x²) - ∑(x)²]

n =count of items, equal to 10

∑(x²) = get the square of all x items and sum up the values. To square a number also means to multiply the number by itself.

1x1 + 2x2 + 3x3 + 4x4 + 5x5 + 6x6 + 7x7 + 8x8 + 9x9 + 10x10 = 385

∑(y) = sum of y items = 110

∑(x) = sum of x items = 55

∑(xy) = multiply all x and y items and get the sum = 770

1x2 + 2x4 + 3x6 + 4x8 + 5x10 + 6x12 + 7x14 + 8x16 + 9x18 + 10x20 = 770

∑(x)² = get the sum of all items in x and get the square = 55 * 55 or 3025

2. Substitute the known values in the formula and compute for the intercept:

Intercept = [385(110) – 55(770)/[10(385) - 3025]

Intercept = [42350 – 42350]/[3850 – 3025]

Intercept = [0]/825]

Intercept = 0