The use of linear regression (least squares method) is the most accurate method in segregating total costs into fixed and variable components.

The total fixed cost and variable cost per unit are determined mathematically through a series of computations.

Like the other methods of cost segregation, the least squares method follows the same cost function:

**y = a + bx**

where: y = total cost; a = total fixed costs; b = variable cost per level of activity (or units); x = level of activity (or number of units).

## The Normal Equations in Differential Calculus

∑y = na + b∑x

∑xy = ∑xa + b∑x²

Note that through the process of elimination, these equations can be used to determine the values of a and b. Nonetheless, formulas for * total fixed costs (a)* and *variable cost per unit (b)* can be derived from the above equations.

## Variable Cost per Unit (b)

Using the normal equations above, a formula for *b* can be derived. The variable cost per unit or slope is computed using the following formula:

b = | n∑xy – (∑x)(∑y) |

n∑x² – (∑x)² |

## Total Fixed Costs (a)

Once *b* has been determined, the total fixed cost or *a* can be computed using the formula:

a = ȳ - bx̄

where: | ȳ = | ∑y | and | x̄ = | ∑x |

n | n |

Or, it is the same as:

a = | ∑y – b∑x |

n |

## Example

The following data was gathered for five production runs of ABC Company. Determine the cost function using the least squares method.

Batch |
Units (x) |
Total Cost (y) |

1 |
680 |
$29,800 |

2 |
820 |
$34,000 |

3 |
570 |
$27,500 |

4 |
660 |
$29,000 |

5 |
750 |
$31,900 |

**Solution:**

Batch |
Units (x) |
Total Cost (y) |
xy |
x² |

1 |
680 |
29,800 |
20,264,000 |
462,400 |

2 |
820 |
34,000 |
27,880,000 |
672,400 |

3 |
570 |
27,500 |
15,675,000 |
324,900 |

4 |
660 |
29,000 |
19,140,000 |
435,600 |

5 |
750 |
31,900 |
23,925,000 |
562,500 |

∑ |
3,480 |
152,200 |
106,884,000 |
2,457,800 |

Substituting the computed values in the formula, we can compute for b.

b = | n∑xy – (∑x)(∑y) |

n∑x² – (∑x)² |

b = | (5)(106,884,000) – (3,480)(152,200) |

(5)(2,457,800) – (3,480)² |

**b = 26.6741 ≈ $26.67 per unit**

Total fixed cost (a) can then be computed by substituting the computed b.

a = | ∑y – b∑x |

n |

a = | 152,200 – (26.67)(3,480) |

5 |

**a = $11,877.68**

The cost function for this particular set using the method of least squares is:

*y = $11,887.68 + $26.67x*.