# What are the principles underlying variance analysis?

The principles of variance analysis can be summarized in one sentence: it consists in switching from budget to actual, one assumption at a time, in the order in which these assumptions were established during the budgeting process. Let me elaborate on this.

The first key principle of variance analysis is that the best way to isolate and assess the impact on Operating Income of a departure from budget on one specific estimate is to change that estimate only and see how Operating Income is affected. If the only difference between two profit equations is one specific estimate, it makes sense to attribute the difference between the two resulting profits to the change of that specific estimate.

The second key principle of variance analysis is that, to reconcile budgeted and actual Operating income, you must switch one by one all the estimates from their budgeted to their actual values. In other words, once an estimate has been switched to its actual value, it does not switch back to its budgeted value. This is crucial to ensure that variances can be added to each other to produce the overall change in Operating Income.

However, this introduces the question of the order in which these assumptions should be switched from budget to actual. Indeed, different orders will produce different variances. By convention, the third principle of variance analysis is that estimates are switched in the order in which they were set during the budgeting process: first sales volumes, then usages of resources in production, and finally price of the resources purchased for production1. This is illustrated in the following set of equations:

\begin{aligned} \text{Volume variance} & = \color{purple}{(Q_{\color{red}A}-Q_{\color{blue}B})} && \times \quad \quad RU_{\color{blue}B} && \times \quad \quad RP_{\color{blue}B} \\ \text{Usage variance} & = \quad \quad Q_{\color{red}A} && \times \color{purple}{(RU_{\color{red}A}- RU_{\color{blue}B})} && \times \quad \quad RP_{\color{blue}B} \\ \text{Price variance} & = \quad \quad Q_{\color{red}A} && \times \quad \quad RU_{\color{red}A} && \times \color{purple}{(RP_{\color{red}A}- RP_{\color{blue}B})} \end{aligned}

where $$Q$$ refers to the volume of output, $$RU$$ to a resource usage (e.g. material usage or labor usage, i.e. the quantity of input consumed for each unit of output), $$RP$$ to a resource price (e.g. material price or labor price, i.e. the price at which each unit of input is acquired); and the indices A and B mean respectively actual value and budgeted values.

Volume variances are changes in Operating Income due to differences between actual and budgeted volumes of output (products or services).

Usage variances (sometimes also called quantity or efficiency variances) are changes in Operating Income due to differences between actual and budgeted consumption of resources (e.g. materials, labor) per unit of output (product or service).

Price variances (sometimes also called rate variances) are changes in Operating Income due to differences between actual and budgeted prices of either outputs (products or services) or resources (e.g. materials, labor).

This set of formula shows the three families of estimates in the order in which they are switched: volume, then usage (or unit consumption, or quantity), and finally price (or rate). It also shows the other principles: estimates are switched one at a time: first we look at the impact of a change in volume, then the impact of a change in usage per unit, and finally the impact of a change in prices. Finally, once an estimate has been switched from its budgeted value to its actual value, it remains at its actual value in the computation of the following variances (values on the left of a difference are always actual, values on the right of a difference always budgeted). The following video shows how these principles are applied on a detailed profit equation to produce (almost) all the detailed variances formulas we will use in this chapter:

These principles hold even in situations with multiple products and multiple resources; there are just more computations to make…

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1. Note that this order is actually the reverse of the chronological order of operations where resources are first purchased, then consumed in production, and finally sold as products. We will come back to this issue later.↩︎