Accuracy and Precision Chapter 1 Computing Tools Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1

Accuracy and Precision Accuracy and precision are often used interchangeably, but have different definitions: Accuracy is a measure of how close our answer is to the actual result. In the projectile example, the assumptions made in the mathematical model affect the accuracy of the results Precision is a measure of the exactness of our calculations – important that we remember that our answer can be no more precise than our input values or of intermediate calculation results Engineering Computation: An Introduction Using MATLAB and Excel 2

Significant Digits In experimental work, where the precision of all of the input values is known, calculation rules involving the number of significant digits are followed For quantities with decimal points, the number of significant digits is defined as the number of digits between the first non-zero digit and the last digit. Examples: 1214.55 6 significant digits 1214.5513 8 significant digits 0.00012 2 significant digits 10.00012 7 significant digits Engineering Computation: An Introduction Using MATLAB and Excel 3

How Many Significant Digits? 1000.0010 8 -77.32 4 0.000103 3 3.44 X 108 100 Can't tell Engineering Computation: An Introduction Using MATLAB and Excel 4

Rules for Working with Significant Digits Addition and Subtraction: the number of digits to the right of the decimal point in the answer must be equal to the least number of digits to the right of any of the inputs. Examples: 6.778 + 3.5 = 10.278 = 10.3 (1 digit) 10.0 – 0.0012 = 9.9998 = 10.0 (1 digit) Multiplication and Division: the number of significant digits in the answer must equal the least number of significant digits of the input values. Examples: 7.553 X 5.52 = 41.69256 = 41.7 (3 digits) 1.0 / 4.5567 = 0.21945706 = 0.22 (2 digits) Engineering Computation: An Introduction Using MATLAB and Excel 5

Engineering Calculations In engineering problems, the precision (number of digits) may be expressed as: 35 degrees +/- 3 degrees 35 degrees +/- 5% 36.5 to 34.7 inclusive If no range is given: report results to 3 significant digits (some texts recommend 4 significant digits if the first significant digit is a “1”) Remember that for the final answer to be precise to this number of digits, intermediate calculations should be carried out to more digits Engineering Computation: An Introduction Using MATLAB and Excel 6

FOUL! Excessive Use of digits Reporting Answers Never report your answer to a ridiculous number of digits! If would sound wrong is you said it verbally; it is equally wrong if you report it in writing FOUL! Excessive Use of digits The answer is 5.32678329546 centimeters Engineering Computation: An Introduction Using MATLAB and Excel 7

Reporting Answers The computer performs calculations to a large number of digits, but that doesn't mean the answer is that precise How many digits do Excel and MATLAB carry through in calculations? Approximately 15 for most operations Engineering Computation: An Introduction Using MATLAB and Excel 8