Least squares method

- Introduction, cost function
- Normal equations in differential calculus
- Variable cost per unit formula, and total fixed costs formula
- Example and key takeaways

Linear regression

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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.

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.

Key Takeaways

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

Scientific calculators and spreadsheets have the capability to calculate the above, without going through the lengthy formula. But keep the above formulae just for base knowledge.

Web link

APA format

Least squares method / Linear regression (2022). Accountingverse.

https://www.accountingverse.com/managerial-accounting/cost-behavior/least-squares-method.html

https://www.accountingverse.com/managerial-accounting/cost-behavior/least-squares-method.html

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