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Using analytic geometry to quantify the period-to-period changes in an array of values

Mark J. Nigrini (Department of Accounting, West Virginia University, Morgantown, West Virginia, USA)
William Karstens (Department of Physics, Saint Michael’s College, Colchester, Vermont, USA)

Managerial Auditing Journal

ISSN: 0268-6902

Article publication date: 23 July 2019

Issue publication date: 9 April 2021



This paper develops a vector variation score that quantifies the change in an array of data points from period-to-period. The array could be the amounts reported on an income tax return, the closing stock prices for a set of listed companies, the monthly sales amounts for retail locations or the monthly balances in general ledger accounts.


The score is grounded in analytic geometry. The angle θ measures whether the changes were uniformly spread across the line items. The item(s) with the largest contribution(s) to the score can be identified. Line items can be weighted such that they contribute less than fully to the score.


The method can identify tax returns with large year-on-year changes. The method can identify the fact that the price movements during earnings season are less dependent than is usually the case. The method can identify anomalies in reported sales amounts. The method should be able to identify ledger accounts’ large abnormal changes.

Research limitations/implications

Auditors will need to be trained to interpret the results and to reduce the number of false positives.

Practical implications

The score could be used in both external and internal audit applications where auditors want to quantify and rank period-on-period changes in a search for outliers.


The change score is normalized to the [0, 1] range. The results can be plotted as a polar plot for display on an auditing dashboard. The contribution of a single line item can be calculated and line items can be weighted to prevent them from having an undue influence on the results.



The authors wish to thank the editor Theo Stratopoulos and the reviewers for their constructive comments and suggestions. The authors also thank Peter Gillett, Carlo Lisi, Dick Riley, Lloyd Simons and Trevor Stewart for their guidance and comments. Thanks are also owing to Zhenhua (Jimmy) Xie for his assistance with the Python code, to Chen Zhao for her assistance with the SAS code and to Brian George for proofreading the manuscript. Earlier versions of this paper were presented at the 16th World Continuous Auditing and Reporting Symposium at Rutgers Business School – Newark, the International Conference on XBRL at the University of Kansas, the 2015 annual meeting of the American Accounting Association held in Chicago and a finance seminar at West Virginia University.


Nigrini, M.J. and Karstens, W. (2021), "Using analytic geometry to quantify the period-to-period changes in an array of values", Managerial Auditing Journal, Vol. 36 No. 1, pp. 17-39.



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