superimpose a finely-spaced grid over the figure and count squares

Engineering Practicum Baltimore Polytechnic Institute M. Scott

Describe 3 entirely different (but practical) ways for determining the area (in cm2) of the darkened region below (design is on a piece of

paper) to within 0.1%.

1. Superimpose a finely-spaced grid over the figure and count squares.

2. Cut out figure and weigh it. Compare that weight to that of piece of paper. If too light, transfer image to another uniformly-dense material.

3. Divide figure into local regions that can be integrated numerically.

4. Computer scan image and count pixels.

5. Build a container whose cross-section is that of the darkened figure. Fill with 1000cc water and measure level.


Describe 3 entirely different (but practical) ways for determining the area (in cm2) of the darkened region below (design is on a piece of

paper) to within 0.1%.

6. Use a “polar planimeter” – gadget that mechanically integrates the area defined by a close curve.

7. “Throw darts.” Draw rectangle (of calculable area) that encloses image. Pick random points within the rectangle and count which ones fall within the darkened figure. The ratio can be used to estimate area. (Monte Carlo Integration)


Error and UncertaintyObjectives

Students will understand the significance of and be able to quantify the following:1. Significant Figures2. Accuracy, Precision, Error, Uncertainty3. Mean Values4. Average Deviation of the Mean5. Relative and % Uncertainty6. Precision of Computed Results


Source of Lecture Information

• Bellevue College Science Division• No author or date noted• http://scidiv.bellevuecollege.edu/physics/measure&s

igfigs/Measure&sigfigsintro.html

http://scidiv.bellevuecollege.edu/physics/measure&sigfigs/Measure&sigfigsintro.html

http://scidiv.bellevuecollege.edu/physics/measure&sigfigs/Measure&sigfigsintro.html


Significant Figures

• Digits that are:– Actual Measured Values– Defined Numbers:• Unit conversions, e.g. 2.54 cm in one inch• Pi• e, base of natural logarithms• Integers, e.g. counting, what calendar year• Rational fractions, e.g. 2/5

– Computed Results using Significant Figures


Significant Figures

• Digits that are the result of actual measurements• How many significant digits in each measurement

taken with a meter stick?

Three:40.0 cm41.2 cm42.4 cm

http://serc.carleton.edu/quantskills/methods/quantlit/DeepTime.html



Significant Figures

• Be clear in your communication• Which is it?– 40 cm– 40.0 cm– 4 x 101 cm




Significant Figures

• State the number of significant figures:

5280 30.35 2

0.00307 3204100 4180.00 5


Significant Figures

• State the number of significant figures for the number described in each phrase below:

My mattress is 180 inches long 3My car gets twenty miles per gallon 25280 feet per mile 4There are ten cars in that train 2I am going to the Seven-Eleven 0


273.92 rounded to 4 digits is 273.91.97 rounded to 2 digits is 2.02.55 rounded to 2 digits is 2.64.45 rounded to 2 digits is 4.4

Significant Figures

• Rounding:If you do not round after a computation, you imply a greater accuracy than you actually measured

1. Determine how many digits you will keep2. Look at the first rejected digit3. If digit is less than 5, round down4. If digit is more than 5, round up5. If digit is 5, round up or down in order to leave an

even number as your last significant figure


Significant Figures

Rounding after math operations:• Multiplication or Division

# of sig figs in result is equal to the # of sig figs in least accurate value used in the computation

273.92 x 3.25 = 890.24; Result is rounded to 890

1/3 x 5.20 = 1.73333; Result is rounded to 1.73

1.97 x 2 = 3.94; Result is rounded to 4

2.0 x Pi = 6.28318...; Result is rounded to 6.3


Significant Figures

Rounding after math operations:• Addition or Subtraction

Place of last sig fig is important

What’s the problem here?


Significant Figures

Multiple Calculations• The least error will come from combining all terms

algebraically, then computing all at once.• If you need to show intermediate steps to a reader,

calculate sig figs at every step.


Accuracy vs. Precision

• Accuracy refers to the agreement between a measurement and the true or accepted value– Cannot be discussed meaningfully unless the true

value is known or knowable– The true value is not usually known (i.e. can never be

known)– We generally have an estimate of the true value

• Precision refers to the repeatability of measurement– Does not require us to know the true value



• You are on the equator during the Spring Equinox (Mar 20) at midday (Sun is directly overhead)

• Your watch reads 12:00pm– Is your watch accurate?– Is your watch precise?– How many significant figures does your watch

communicate?

http://benkolstad.net/?p=2003

Yes – measurement agrees with true value

Don’t know – wait until sun overhead tomorrow

4; If you have a seconds hand, then 6

http://benkolstad.net/?p=2003



• Rate the level of Accuracy and Precision as high/low


http://www.shmula.com/2092/precision-accuracy-measurement-system

http://www.shmula.com/2092/precision-accuracy-measurement-system


Accuracy vs. PrecisionRate the data precision and accuracy low or high

High Precision

Low Accuracy

Low Precision

High Accuracy

Low Precision

Low Accuracy


Error vs. Uncertainty

• Error refers to the disagreement between a measurement and the true or accepted valueIn “real” science, it’s not very important (science studies

new things where there are no true/accepted values)• Scientists design experiments and assume no error.

Subsequent results may show error.• In school, we assume error and compare our results

with the accepted values.• Don’t discuss error until a correct data analysis is done


Error vs. Uncertainty

• Uncertainty is an interval around a value such that any repetition will produce a new result that lies within that interval.

• Value + Uncertainty (e.g. L = 1.20 + 0.15 m, or L = 1.20 m + 0.15 m)

• It is always possible to construct a completely certain sentence.


Situation: A class of students measures the length of a metal rod in centimeters.

1. Which group has the most accurate measurement? Don’t Know2. Which group has the greatest error? Don’t Know3. Which group has the most precise measurement? Group C4. Which group has the greatest uncertainty? Group D

Student Group Student 1 Student 2 Student 3 Student 4 Student 5 Average Average

Deviation

Group A 10.1 10.4 9.6 9.9 10.8 10.16 0.352

Group B 10.135 10.227 10.201 10.011 10.155 10.1458 0.0582

Group C 12.14 12.17 12.15 12.14 12.18 12.156 0.02

Group D 10.05 10.82 8.01 11.50 10.77 10.230 0.96

Group E 10 11 10 10 10 10.2 0.32


1. Which group has the least accurate measurement? Group C2. Which group has the smallest error? Group A3. Which group has the least precise measurement? Group D4. Which group has the smallest uncertainty? Group C

Update: The correct length of the rod is 10.160 cm


Deviation

Group A 10.1 10.4 9.6 9.9 10.8 10.16 0.352

Group B 10.135 10.227 10.201 10.011 10.155 10.1458 0.0582

Group C 12.14 12.17 12.15 12.14 12.18 12.156 0.02

Group D 10.05 10.82 8.01 11.50 10.77 10.230 0.96

Group E 10 11 10 10 10 10.2 0.32


• Error – difference between an observed/measured value and a true value.– We usually don’t know the true value– We usually do have an estimate

• Systematic Errors– Faulty calibration, incorrect use of instrument– User bias– Change in conditions – e.g., temperature rise

• Random Errors– Statistical variation– Small errors of measurement– Mechanical vibrations in apparatus

Error


Accuracy and Estimation

• Which type of error, systematic or random?

Systematic Low ErrorRandom

Random



Deviation

Group A 10.1 10.4 9.6 9.9 10.8 10.16 0.352

Group B 10.135 10.227 10.201 10.011 10.155 10.1458 0.0582

Group C 12.14 12.17 12.15 12.14 12.18 12.156 0.02

Group D 10.05 10.82 8.01 11.50 10.77 10.230 0.96

Group E 10 11 10 10 10 10.2 0.32

Error vs. UncertaintyAccuracy vs. Precision


Error

• Percent Error

• Relative Error


Estimating and Accuracy

• Measurements often don’t fit the gradations of scales• Two options:– Estimate with a single reading– Independently measure several times and take an average

http://scidiv.bellevuecollege.edu/physics/measure&sigfigs/C-Uncert-Estimate.html



Estimating and Accuracy

• Simplest estimate, and common practice, is to assign ½ the most precise value on the scale:

3.5 + 0.5mm




Precision of the Device• Some differences are always a part of any manufacturing

process.• The uncertainty is assumed to be ½ the smallest scale

division shown on the device.• This assumed uncertainty should be the case over the

entire length of the device.• Unless the manufacturer provides the precision




Make Multiple Measurements

• Highly recommended when concerned with accuracy of results

• Uncertainty is estimated using average deviation• Try to make each trial independent of the previous

trial – different ruler, different observer, etc.


Mean and Avg. Deviation

• When taking the mean of multiple measurements, the value for the mean should have 1 more sig fig than in the original observations


Deviation

Group A 10.1 10.4 9.6 9.9 10.8 10.16 0.352

Group B 10.135 10.227 10.201 10.011 10.155 10.1458 0.0582

Group C 12.14 12.17 12.15 12.14 12.18 12.156 0.02

Group D 10.05 10.82 8.01 11.50 10.77 10.230 0.96

Group E 10 11 10 10 10 10.2 0.32


Length (cm) Dev. from Mean15.39 0.01215.37 0.00815.37 0.00815.39 0.01215.38 0.00215.37 0.00815.37 0.00815.38 0.002

15.378 (mean) 0.008 (A.D.)




• Average Deviation is always rounded to one sig fig.• Mean can be subsequently be adjusted to contain

appropriate # sig figs.



• Report the measurements below with the uncertainty that should be reported:


Deviation

Group A 10.1 10.4 9.6 9.9 10.8 10.16 0.352

Group B 10.135 10.227 10.201 10.011 10.155 10.1458 0.0582

Group C 12.14 12.17 12.15 12.14 12.18 12.156 0.02

Group D 10.05 10.82 8.01 11.50 10.77 10.230 0.96

Group E 10 11 10 10 10 10.2 0.32



10.2 + 0.4 cm

10.15 + 0.06 cm

12.16 + 0.02 cm

10 + 1 cm

10.2 + 0.3 cm

Average AverageDeviation

10.16 0.352

10.1458 0.0582

12.156 0.02

10.230 0.96

10.2 0.32


Uncertainty – Abs, Rel, %

• Absolute Uncertainty – uncertainty in value

• Relative Uncertainty – ratio of uncertainty to value

• Percent Uncertainty – relative uncertainty x 100%

2 sig figs

2 sig figs

1 sig fig



• Determine the relative uncertainty for each of the measurements below (2 sig figs in rel/% uncertainty):

0.022, 2.2%

0.011, 1.1%

0.26, 26%

0.025, 2.5%



• Determine the absolute uncertainty for each of the measurements below (1 sig fig in uncertainty):


Calculators and significant digits:

Let the uncertain digit determine the precision to which you quote a result

Calculator: 12.6892

Estimated Error: +/- 0.07

Quote: 12.69 +/- 0.07


What is an error?

• In data analysis, engineers use– error = uncertainty – error ≠ mistake.

• Mistakes in calculation and measurements should always be corrected before calculating experimental error.

• Measured value of x = xbest x

– xbest = best estimate or measurement of x– x = uncertainty or error in the measurements


All measurements have errors• What are some sources of measurement errors?

– Instrument uncertainty (caliper vs. ruler)

• Use half the smallest division.

– Measurement error (using an instrument incorrectly)

• Measure your height - not hold ruler level.– Variations in the size of the object (spaghetti is bumpy)

• Statistical uncertainty

L = 9 ± 0.5 cm

L = 8.5 ± 0.3 cm

L = 11.8 ± 0.1 cm


If no error is given, assume half the last significant figure.

• That's why you don't write 25.367941 mm.


How do you account for errors in calculations?

• The way you combine errors depends on the math function– added or subtracted– multiplied or divide– other functions

• The sum of two lengths is Leq = L1 + L2. What is error in Leq?• The area is of a room is A = L x W. What is error in A? • A simple error calculation gives the largest probable error.


Sum or difference

• What is the error if you add or subtract numbers?

• The absolute error is the sum of the absolute errors.

xx yy zz

zyxw

boundupper zyxw


What is the error in length of molding to put around a room?

• L1 = 5.0cm 0.5cm and L2 = 6.0cm 0.3cm. • The perimeter is

• The error (upper bound) is:

cm22

cm0.6cm0.5cm0.6cm0.52121

LLLLL

cm6.1

cm3.0cm5.0cm3.0cm5.02121

LLLLL


Errors can be large when you subtract similar values.

• Weight of container = 30 ± 5 g• Weight of container plus nuts = 35 ± 5 g• Weight of nuts?

%200105Result

1055Error

53035Weight

gg

gg

gg


What is the error in the area of a room?

• L = 5.0cm 0.5cm and W = 6.0cm 0.3cm.

• What is the relative error?

• What is the absolute error?

2cm0.30cm0.6cm0.5 WLA

%1515.cm0.6

cm3.0

cm0.5

cm5.0

or

W

W

L

L

A

A

22 cm5.415.0cm0.3015.0 AA

Board Derivation


Product or quotient

• What is error if you multiply or divide?

• The relative error is the sum of the relative errors.

z

yxw

boundupper z

z

y

y

x

x

w

w

xx yy zz

zz

yyxxw

)()(


Multiply by constant

• What if you multiply a variable x by a constant B?

• The error is the constant times the absolute error.

Bxw

xBw


What is the error in the circumference of a circle?

• C = 2 π R– For R = 2.15 ± 0.08 cm

• C = 2 π (0.08 cm)= 0.50 cm


Powers and exponents

• What if you square or cube a number?

• The relative error is the exponent times the relative error.

nxw

x

xn

w

w

Board Derivation


What is the error in the volume of a sphere?

• V = 4/3 π R3 – For R = 2.15 ± 0.08 cm– V = 41.6 cm3

• V/V = 3 * (0.08 cm/2.15 cm)= 0.11

• V = 0.11 * 41.6 cm3

= 4.6 cm3

Board Derivation


What is the error in the volume of a sphere?


Lab “Calculus of Errors” Explanation

superimpose a finely-spaced grid over the figure and count squares

Documents