ENGR 107 - Introduction to Engineering 1

ENGR 107 – Introduction to Engineering

Estimation,Accuracy and Precision,

andSignificant Figures

(Lecture #3)

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Estimation

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A rough calculation, often using incomplete or uncertain data, that is still close enough to

be useful.Definition courtesy of Wikipedia

Synonym: approximation

Estimation

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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

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Estimation

Exercise:

Calculate the volume of a box to the nearest cubic meter.

The dimensions of the box are:W = 3.75 mL = 1.675 m H = 2.35 m

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Estimation

Exercise:

Calculate the density of a material to the nearest kg / m3.

The mass and volume of the material are:Mass = 489.54 kgVolume = 7.5 m3

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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.

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Estimation

Calculate the volume of the classroom, using your height as a “measuring stick”.

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Accuracy and Precision

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Accuracy and Precision

In measurements, accuracy and precision have different meanings and cannot be

used interchangeably.

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The degree of closeness of a measurement to the actual or true value.

Definition courtesy of Wikipedia

Accuracy

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Precision

The degree to which repeated measurements under unchanged conditions show the

same results.Definition courtesy of Wikipedia

Also called reproducibility or repeatability.

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Accuracy vs Precision

Accurate

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Accuracy vs Precision

Precise

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Accuracy vs Precision

Accurate and Precise

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Accuracy vs Precision

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Measurements

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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.

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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.

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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.

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Measurement Estimation

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Errors

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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

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Significant Digits

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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.

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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.

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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.

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Significant Digits

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Significant Digits

Rounding:

Increase the last digit retained by 1 if the first digit dropped is greater than 5.

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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.

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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.

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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.