In addition to statistical analysis, a variety of quantitative methods are also available to be selected by the internal auditor. These math- ematical tools are commonly used to obtain an understanding of operations and permit the drawing of conclusions in a variety of cir- cumstances through analyzing the complexities of situations. Of the many quantitative methods available to the auditor, the following sec- tions examine the most commonly used.
Trend Analysis
Trend analysis is used to evaluate the behavior of a variable such as turnover of a period of time. Such analyses can serve as evaluation criteria to determine the reasonableness of fluctuations of an extended period. Comparisons of this year’s turnover to last year’s turnover or, alternatively, this month’s turnover to the same month last year are popular.
Chi-Square Tests
Chi-square analyses are non-parametric tests capable of analyzing relationships between qualitative data. For example, do operating units in the South have particular patterns of operation different from those in the North?
Chi-square tests can check for the independence of normal classi- fications and ordinal data, and require no particular distributional pattern for the data.
Correlation Analysis
The measurement of the extent of association of one variable with another is known as correlation analysis. Two variables are said to be correlated when they move together in a detectable pattern. A direct correlation is said to exist when both variables increase or decrease
in the same time although not necessarily by the same amount. For example, one would expect inventory to decrease as sales increased. Correlation analysis is used by internal auditors to identify those factors that appeared to be related. An operational auditor, for exam- ple, may use correlation analysis to determine whether corporate per- formance is in line with industry standards by comparing the correlation of company costs of imported parts with the exchange rate fluctuations. Problems with how these statistics are computed; shortcomings in the internal auditor’s understanding of auditees’ operations, or real inefficiencies or misstatements can be pinpointed through correlation analysis.
Graphical Analysis
Graphical analysis can be useful to the internal auditor in identifying interrelationships in data, anomalies, and simple data errors.
A common form of graphical representation use by the auditor is a scatter diagram, which refers to any graph of data points. The more discernible a pattern appears in the graph, the more likely one vari- able is related to another and therefore can be used to predict the other’s value. Where no pattern can be noted, there would appear to be little, if any, correlation between the two variables.
Where a strong correlation exists, either positive or negative, the correlation value will approach 1. Where little correlation exists the correlation value will approach 0. Unfortunately, correlation values only measure linear patterns. Where there is a nonlinear relationship, correlation statistics will not disclose this. Occasionally the correla- tion value can be distorted by a single data point not conforming to the general pattern. While this can be readily seen on the graph, it may be less obvious in examining the correlation value.
Learning Curves
In conducting operational audits of performance levels of the imple- mentation of new procedures for the quality of training of new staff, a learning curve would normally be expected to be observed. As
employees gain experience with the new procedures or as the new employee becomes more experienced, the length of time taken to task should decrease.
Learning curves are evaluated by computing the time required per unit of production each time that the cumulative output is doubled. A decrease in production time per unit of 25% would result in a 75% curve. The 60% curve would result if the production time was reduced by 40%.
By measuring this curve the auditor can determine how quickly a new procedure or employee becomes productive. When a new proce- dure is recommended, calculating the initial time per unit under the old system and comparing it to a series of observations over time using the new procedures can objectively determine the impact of the revision to the procedures.
Ratio and Regression Analysis
Ratio analysis assumes a given proportional relationship between two numbers and is normally used for comparisons over time. A more advanced form of ratio analysis attempts to quantify the interrela- tionship in order to facilitate predictions in a regression analysis. Regression analysis is used to estimate the effect that a movement in one variable (the independent variable) causes a movement in the other variable (dependent variable); for example, if the sun shines, more cool drinks will be sold: but how many more? By performing the regression analysis the relationship, if any, can be identified and quantified and sales levels predicted.
Regression analysis can thus assist the auditor in understanding and quantifying data interrelationships. Unusual variations between expectations and recorded values may be noted for further investiga- tion.
Using software, the auditor can additionally conduct a multiple discriminant regression analysis relating the independent variable to a number of dependent variables simultaneously. By determining the comparative strength of the relationships, the auditor can choose the focus area to achieve greatest impact in performance improvement. Such analysis has also been used to attempt to predict bankruptcy.
As with most statistical tools, regression analysis is based on a set of underlying assumptions that must be met for its use and interpre- tations to be valid.
Linear Programming
Linear programming is an operations research tool used for the allo- cation of scarce resources or to determine optimal blends of raw materials. The constraints applicable are reduced to algebraic formu- lae, which are then solved by simultaneous equations. For example, in a production environment, machining may be capable of process- ing 100 units per machine, while finishing can handle 35 units per machine. The question of how many of each machine should be used for optimum production can be solved using linear programming.