ENGR Introduction to Engineering1 ENGR 107 – Introduction to Engineering Estimation, Accuracy and Precision, and Significant Figures (Lecture #3)

ENGR Introduction to Engineering2 Estimation

ENGR Introduction to Engineering3 A rough calculation, often using incomplete or uncertain data, that is still close enough to be useful. Definition courtesy of Wikipedia Synonym: approximation Estimation

ENGR Introduction to Engineering4 Estimation Estimations are used when  Insufficient information is available  Available information is uncertain  Problem is too difficult to solve analytically  Problem is impossible to solve using available analysis tools. Estimations are used when  An inexact result is useful  A range (i.e. upper and lower bounds) is useful

ENGR Introduction to Engineering5 Estimation Exercise: Calculate the volume of a box to the nearest cubic meter. The dimensions of the box are: W = 3.75 m L = m H = 2.35 m

ENGR Introduction to Engineering6 Estimation Exercise: Calculate the density of a material to the nearest kg / m 3. The mass and volume of the material are: Mass = kg Volume = 7.5 m 3

ENGR Introduction to Engineering7 Estimation Exercise: Determine the number of tiles, to the nearest integer number, needed to tile a wall. Dimensions of the tile: 4.5 in. x 4.5 in. Dimensions of the wall: 7.5 ft. x 11 ft.

ENGR Introduction to Engineering8 Estimation Calculate the volume of the classroom, using your height as a “measuring stick”.

ENGR Introduction to Engineering9 Accuracy and Precision

ENGR Introduction to Engineering10 Accuracy and Precision In measurements, accuracy and precision have different meanings and cannot be used interchangeably.

ENGR Introduction to Engineering11 The degree of closeness of a measurement to the actual or true value. Definition courtesy of Wikipedia Accuracy

ENGR Introduction to Engineering12 Precision The degree to which repeated measurements under unchanged conditions show the same results. Definition courtesy of Wikipedia Also called reproducibility or repeatability.

ENGR Introduction to Engineering13 Accuracy vs Precision Accurate

ENGR Introduction to Engineering14 Accuracy vs Precision Precise

ENGR Introduction to Engineering15 Accuracy vs Precision Accurate and Precise

ENGR Introduction to Engineering16 Accuracy vs Precision

ENGR Introduction to Engineering17 Measurements

ENGR Introduction to Engineering18 Measurements Engineers must be able to measure physical quantities and express these measurements in numerical form. Engineers must have confidence that the measurements and subsequent calculations and decisions made based on the measurements are reasonable.

ENGR Introduction to Engineering19 Measurements Any physical measurement that is not a countable number will be approximate. Errors are likely to be present regardless of the precautions used when making the measurements. Significant digits are used to express, numerically, the accuracy of a measurement.

ENGR Introduction to Engineering20 Measurement Estimation There is a finite accuracy to which every engineering measurement can be made. There is a limited number of significant digits that can be included in the numerical representation of a measurement. The engineer must estimate the measurement between the smallest graduations on the instrument.

ENGR Introduction to Engineering21 Measurement Estimation

ENGR Introduction to Engineering22 Errors

ENGR Introduction to Engineering23 Errors Systematic  A bias in the measurement leading to the mean of a set of measurements differing significantly from the expected value.  Can be identified and eliminated Random  An error in the measurement leading to inconsistent values for repeated measurements of the same attribute.  Caused by unpredictable fluctuations in the readings of the measurement equipment, in the environment, etc.  Cannot be eliminated

ENGR Introduction to Engineering24 Significant Digits

ENGR Introduction to Engineering25 Numerical Values For numbers less than one, a zero is written in front of the decimal point. A space, not a comma, is used to divide numbers of three orders of magnitude or more. For very large or very small numbers, use scientific notation to reduce the unwieldy nature of these numbers.

ENGR Introduction to Engineering26 Significant Digits A significant digit, or significant figure, is defined as any digit used in writing a number, except those zeros that are used only for location of the decimal point or those zeros that do not have any nonzero digit to their left.

ENGR Introduction to Engineering27 Significant Digits Numbers 10 or larger that are not written in scientific notation and that are not counts (exact values) can cause difficulties in interpretation when zeros are present. If uncertainty results from using standard decimal notation, use scientific notation so that the reader will clearly understand your intent.

ENGR Introduction to Engineering28 Significant Digits

ENGR Introduction to Engineering29 Significant Digits Rounding: Increase the last digit retained by 1 if the first digit dropped is greater than 5.

ENGR Introduction to Engineering30 Significant Digits Multiplication and Division: The product or quotient should contain the same number of significant digits as the number with the fewest significant digits.

ENGR Introduction to Engineering31 Significant Digits Addition and Subtraction: The sum or difference should include significant digits only as far to the right as in the least precise number.

ENGR Introduction to Engineering32 Arithmetic and Significant Digits In calculator or computer applications it is not practical to perform intermediate rounding (i.e. between arithmetic operations). It is normal practice to perform the entire calculation and then report a reasonable number of significant figures.  The number of significant digits in the result cannot exceed that in the value with the fewest significant digits.  The result cannot be more precise than any of the values included in the calculation.