This paper seeks to re-assure CEOs that approximate budget figures will provide better accuracy than expected in forecasting. The accountant or CFO may be uncomfortable with the process but a total of ‘guesstimates’ is magically accurate. The principle of “Totaling for Precision” says: in developing information, tables, data or evaluations the numbers that make them up do not have to be precise; yet, the total will be highly accurate.
- The Principle
If the quantity of numerical items in a list is long enough, the total will always be accurate to a practical level of acceptance despite the use of approximations for the individual numbers. A corollary is that the longer the list of numbers, the greater the accuracy of the total. In practical terms the list should be at least ten items in length.
- A Result by Demonstration
let us use a random list of numbers as an example:
72.39
44.97
101.11
11.20
23.87
34.32
21.21
15.92
21.12
50.67
15.83
44.44
20.09
Total: 477.14
Note: You might at this point, draw up your own list of random numbers and follow the processes herein using your information.
a) First Approximation
To start the process, let us round off the second decimal digit. (We are affecting approximately 1 part in 3 digit places, or 1 per thousand of the number.)
First Approximation Original Numbers
72.4 72.39
45.0 44.97
101.1 101.11
11.2 11.20
23.9 23.87
34.2 34.32
21.2 21.21
15.9 15.92
21.1 21.12
50.7 50.67
15.8 15.83
44.4 44.44
20.1 20.09
Total: 477.0 477.14
Accuracy to original total: 99.97%
Conclusion: Although the number 44.44, for example, has been rounded to 44.4 and its change is 0.04 in 44.44 or 0.09%, the total is affected by (100.00 – 99.97 =) 0.03%. That is, the total offers a 3 times improvement in precision over one of its constituents (0.09% vs. 0.03%).
We continue with four more levels of progressively lower accuracy lists, b. through e, to demonstrate the point, in Appendix A in order to arrive at the summary below.
- Summary of the Examples
(Please see Appendix A for examples, b to e.)
Method Addition Accuracy Variance
of Total of Total
a) Rounding 1st decimal 477.0 99.97% 0.03%
b) Truncating 1st decimal 476.6 99.89% 0.1%
c) Rounding at 1st digit 476 99.76% 0.2%
d) Truncating at 1st digit 471 98.71% 1.3%
e) Rounding at tens 460 96.41% 3.6%
That is, from example totals of ‘a’ to ‘e’ we move the accuracy of the individual numbers from 1/100 of a percent to the tens digits, it represents a shift of accuracy of the individual numbers by an order of 25%. However, the accuracy of the total has shifted by less than 4%. Even with a change of order of 25% in individual values, the total changes only 4%!
As stated before, to verify the validity of the above, choose your own group of numbers of any size or accuracy.
Conclusion: The principle has been demonstrated that a group of numbers gives us leverage on accuracy by the very fact of its being a group.
- Why does it happen?
The answer will be found in the theory of numbers. In general however, a series of approximate numbers will have errors that vary both positively and negatively. When the approximate numbers are added together, a large enough group of them will balance the variances, producing a net variance less than we would intuitively expect.
- PAVF Attitudes
This approach is valuable to, and applauded by, P’s and V’s who are in a hurry to get results. Strong A’s, however, will dismiss it as inaccurate, verging on chaos. On the other hand extreme F’s will request that a committee discuss it rather than consider one person’s view.
- Applications
The power of this knowledge is that it lets us assemble information that is only approximate to and we can seem to magically convert it to come up with a result that is accurate. This allows us to get on with procedures and processes even in the absence of accurate information. We do not have to wait for the precise information to be developed; we will still get a good representation of the final result, far greater than the accuracy (i.e. “inaccuracy”) of the component numbers. While this knowledge has many uses, I will demonstrate its more common application to budgeting below. As well, its application is demonstrated with assessments of types of automobiles (see Appendix B to this paper).
- Application to Budgets
While developing a budget, especially a new one, take the liberty of wildly guessing ((guesstimating)) those budget items that remain unknown. For greater safety, ask the assembled group of people to collectively estimate a number. But if you’re alone, just take a guess.
Monthly Overhead Budget for ABC Company
Half the budget numbers are known and half are estimated.
Guess Stage Accurate Stage, Later Error
Telephone (known) 892 892 0
Salaries 21,309 21,309 0
New salaries 12,000 13,201 10%
Paper supplies 300 342 14%
Postage 467 467 0
Rent 1,233 1,233 0
New amortization 400 799 100%
Old amortization 1,111 1,111 0
Equip rental 788 788 0
Equip rent new 400 322 19%
Travel 1,300 1,506 16%
Profess fees 900 603 51%
Training 400 331 34%
Internet/cellular 500 250 50%
Promotion 2,310 2,310 0
Advertising 2,600 2,350 10%
Total 46,910 47,814
Average error of individual guesses: 43%
Error of total numbers (46,910 vs. 47,814): 1.9%
The point is that later, when the ‘guesses’ become ‘known’, the difference will be only 1 or 2% in the totals. In the meantime we have representative numbers we can work with and plan around. Of course, using approximations may drive carefully thorough accountants crazy.
Some years ago, as President of a company, I used to guesstimate each month’s financial statements this way, calculating my results on the last day of the month – i.e. with no time lag. When my controller would appear with his precise statements ten days later, we would compare results which were always close. On some occasions we were within $1 on $800,000-per- month statements. Of course that was coincidence on my part. He used to jokingly ask: “Why do you need me?” to which I responded. “You provide real accuracy and you provide credibility, not only to our bankers but to the employees here.”
- Conclusions
a) A list of imprecise numbers will give an increasingly more precise total if the list is long enough. Use this knowledge to your advantage.
b) Be willing to guess at numbers to expedite problem solving or evaluation. Precision by virtue of the length of the list will ensure a reasonable accuracy to move forward.
Bill Caswell