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Linear regression

The use of linear regression (least squares method) is the most accurate method in segregating total costs into fixed and variable components. Fixed costs and variable costs are determined mathematically through a series of computations.

What's in Here

In this lesson, we will take a look at the least squares method, its formula, and illustrate how to use it in segregating mixed costs.

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;

x = level of activity

∑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.

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 |

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 |

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 = $26.67 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 least squares method is:

y = $11,887.68 + $26.67x.

More on Cost Behavior and Analysis

- 1Fixed and Variable Costs
- 2High-Low Method
- 3Scatter Graph Method
- 4Least Squares Method

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