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Chapter -3 Experimental Error & Data Handling

https://www.youtube.com/watch?v=5XCTFTPkhk4&list=PLLilrKdbjXDEgE97_TnAZJ0-XZrLlO1Pn&index=3

SLRC : for help 9- 5:00 pm

Chapter Outline

• Section 3-1 Significant Figures

• Section 3-2 Significant Figures in Arithmetic

• Section 3-3 Types of Error

• Section 3-4/5 Propagation of Uncertainty from Random Error & from Systematic Error

Experimental Error & Data Handling Introduction

1.) There is error or uncertainty associated with every measurement ( using different tools and instruments).

(i) except simple counting

2.) To evaluate the validity of a measurement, it is necessary to evaluate its error or uncertainty

http://www.healthcareitnews.com/news/deaths-by-medical-mistakes-hit-records "Deaths by medical mistakes hit records” t's a chilling reality – one often overlooked in annual mortality statistics: Preventable medical errors persist as the No. 3 killer in the U.S. – third only to heart disease and cancer – claiming the lives of some 400,000 people each year. At a Senate hearing Thursday, patient safety officials put their best ideas forward on how to solve the crisis, with IT often at the center of discussions.

Hearing members, who spoke before the Subcommittee on Primary Health and Aging, not only underscored the devastating loss of human life – more than 1,000 people each day – but also called attention to the fact that these medical errors cost the nation a colossal $1 trillion each year.

"The tragedy that we're talking about here (is) deaths taking place that should not be taking place," said subcommittee Chair Sen. Bernie Sanders, I-Vt., in his opening remarks.

http://www.hospitalsafetyscore.org/newsroom/display/hospitalerrors-thirdleading- causeofdeathinus-improvementstooslow "Hospital Errors are the Third Leading Cause of Death in U.S., and New Hospital Safety Scores Show Improvements Are Too Slow

Washington, D.C., October 23, 2013 – New research estimates up to 440,000 Americans are dying annually from preventable hospital errors. This puts medical errors as the third leading cause of death in the United States, underscoring the need for patients to protect themselves and their families from harm, and for hospitals to make patient safety a priority."

Medical Errors

Errors Matter

More than 13 billion clinical tests are performed each year in the United States. • Some results will be wrong • Improving reliability depends on the

scientist’s ability to recognize and prevent errors . . .

• . . . and to make intelligent decisions despite uncertainty.

Both numbers have 4 significant figures

Zeros are simple place holders

Both zeros are significant figures

zero is a significant figure

Significant Figures

1.) Definition: The minimum number of digits needed to write a given value (in scientific notation) without loss of accuracy.

(i) Examples:

142.7 = 1.427 x 102

0.006302 = 6.302 x10-3

2.) Zeros are counted as significant figures only if: (i) occur between other digits in the number

9502.7 or 0.9907

(ii) occur at the end of number and to the right of the decimal point

177.930 6

Significant Figures

3.) The last significant figure in any number is the first digit with any uncertainty

(i) the minimum uncertainty is ± 1 unit in the last significant figure (ii) if the uncertainty in the last significant figure is ≥ 10 units, then one less

significant figure should be used. (iii) Example:

9.34 ± 0.02 3 significant figures

But

6.52 ± 0.12 should be 6.5 ± 0.1 2 significant figures

4.) Whenever taking a reading from an instrument, apparatus, graph, etc. always estimate the result to the nearest tenth of a division

(i) avoids losing any significant figures in the reading process

7.45 cm 7

Significant Figures

5.) Addition and Subtraction (i) use the following procedure:

 Express all numbers using the same exponent  Align all numbers with respect to the decimal point

 Add or subtract using all given digits  Round off the answer so that it has the same number of digits to

the right of the decimal as the number with the fewest decimal places

1 decimal point12.5 x 104

2.48 x 104

+ 1.235 x 104

16.215 x 104 = 16.2 x 104

12.5 x 104

2.48 x 104

+ 1.235 x 104

1.25 x 105

2.48 x 104

+ 1.235 x 104

Significant Figures

5.) Addition and Subtraction (i) use the following procedure:

 Round off the answer to the nearest digit in the least significant figure.

 Consider all digits beyond the least significant figure when rounding.

 If a number is exactly half-way between two digits, round to the nearest even digit.

- minimizes round-off errors  Examples:

3 sig. fig.: 12.534  12.5

4 sig. fig.: 11.126  11.13

4 sig. fig.: 101.250  101.2

3 sig. fig. 93.350  93.4 9

Significant Figures

6.) Multiplication and Division (i) use the following procedure:

 Express the answers in the same number of significant figures as the number of digits in the number used in the calculation which had the fewest significant figures.

 Examples:

3.261 x 10-5

x 1.78 5.80 x 10-5

3 significant figures

34.60 / 2.4287 14.05

4 significant figures

Significant Figures

Multiplication and Division (i) use the following procedure when multiplying or dividing measurements with significant figures, the result has the same number of significant figures as the measurement with the fewest number of significant figures

5.02 × 89.665 × 0.10 = 45.0118 = 45 3 sig. figs. 5 sig. figs. 2 sig. figs. 2 sig. figs.

5.892 ÷ 6.10 = 0.96590 = 0.966 4 sig. figs. 3 sig. figs. 3 sig. figs.

7.) Logarithms and Antilogarithms (i) the logarithm of a number “a” is the value “b”, where:

(ii) example:

(iii) The antilogarithm of “b” is “a”

(iv) the logarithm of “a” is expressed in two parts

a = 10b or Log(a) = b

The logarithm of 100 is 2, since: 100 = 102

a = 10b

Log(339) = 2.530

character mantissa 12

Significant Figures

7.) Logarithms and Antilogarithms (v) when taking the logarithm of a number, the number of significant figures

in the resulting mantissa should be the same as the total number of significant figures in the original number “a”

(vi) Example:

Log(5.403 x 10-8) = -7.2674

(vii) when taking the antilogarithm of a number, the number of significant figures in the result should be the same as the total number of significant figures in the mantissa of the original logarithm “b”

(viii) Example:

Antilog(-3.42) = 10-3.42 =3.8 x 10-4

4 sig. fig. 4 sig. fig.

2 sig. fig. 2 sig. fig.

Types of Error

• Every measurement has some uncertainty.

• Experimental error: difference between the “true” value and the measured value

• Three types of experimental error: • Systematic • Random • Gross (blunders)

Systematic Error

Systematic error (determinate error): arises from a flaw in equipment or experiment design

• If conduct experiment again in exact manner, error is reproducible

• Can be discovered and corrected (in theory) Ways to Detect • Analyze a known sample (certified reference material). • Use different method to measure same quantity. • Different labs analyze identical samples (“round

robin”). • Analyze blank sample. If observe nonzero result,

method has systematic error.

Ways to Correct • Calibrate glassware and instruments

(Section 2-9).

• Use standard addition (Section 5-3) or internal standards (Section 5-4) to correct for matrix effects.

Random Error

Random error (indeterminate error): arises from uncontrolled variables in measurement

• Equal chance of being positive or negative • Always present; cannot be eliminated • Might be reduced with better technique

Examples of Random Error • Subjective reading of a scale (varies with individual) • Electrical noise in an instrument

Gross Error (Blunder)

Gross error (blunder): due to accidental but significant departures from procedure

• Caused by procedural, instrumental, or clerical mistakes (unrecoverable)

• Should be recorded in the lab notebook • May be so serious that data are rejected or experiment is redone

Examples of Blunders • Calculation errors • Overshooting a titration end point • Dropping, discarding, or contaminating a sample • Instrument failure

- Accuracy and Precision (i) Accuracy: refers to how close an answer is to the “true” value

 Generally, don’t know “true” value  Accuracy is related to systematic error

(ii) Precision: refers to how the results of a single measurement compares from one trial to the next

 Reproducibility  Precision is related to random error

Low accuracy, low precision Low accuracy, high precision

High accuracy, low precision High accuracy, high precision 18

Errors in Weighing: Sources

Tolerance (mg) Tolerance (mg)

Grams Class 1 Class 2 Milligrams Class 1 Class 2

500 1.2 2.5 500 0.010 0.025

200 0.5 1.0 200 0.010 0.025

100 0.25 0.50 100 0.010 0.025

50 0.12 0.25 50 0.010 0.014

20 0.074 0.10 20 0.010 0.014

10 0.050 0.074 10 0.010 0.014

5 0.034 0.054 5 0.010 0.014

2 0.034 0.054 2 0.010 0.014

1 0.034 0.054 1 0.010 0.014

(allowable deviations, Uncertainty)

Remember this is from chp1-2

Both measures of the precision associated with a given measurement. - Absolute uncertainty: margin of uncertainty associated with a measurement Example:

If a buret is calibrated to read within ± 0.02 mL, the absolute uncertainty for measuring 12.35 mL is:

Absolute Uncertainty = 12.35 ± 0.02 mL = 12.33-12.37

-Relative uncertainty: compares the size of the absolute uncertainty with the size of its associated measurement

(v) Example: For a buret reading of 12.35 ± 0.02 mL, the relative uncertainty is:

)100( ValueMeasured

yUncertaintAbsolute precenty UncertaintRelative 

0.2%0.16%(100) mL12.35

mL0.02 y(%)UncertaintRelative 

(Make sure units cancel)

1 sig. fig.

-We can usually estimate or measure the random error associated with a measurement, such as the length of an object or the temperature of a solution. The uncertainty might be based on how well we can read an instrument or on our experience with a particular method. If possible, uncertainty is expressed as the standard deviation or as a confidence interval

- Absolute and Relative Uncertainty

- Propagation of Uncertainty (i) Most experiments require arithmetic operations on several numbers, each of which has a random error.

The uncertainty of the final result is not simply the sum of individual errors. The absolute or relative uncertainty of a calculated result can be estimated

using the absolute or relative uncertainties of the values used to obtain that result.

(ii) Addition and Subtraction  The absolute uncertainty of a number calculated by addition or

subtraction is obtained by using the absolute uncertainties of numbers used in the calculations as follows:

 Example:

     22value2valueAnswer .Uncert.Abs.Uncert.Abs.Uncert.Abs 1

Value Abs. Uncert. 1.76 (± 0.03)

+ 1.89 (± 0.02) - 0.59 (± 0.02) 3.06

      04.002.002.003.0.. 222 AnswerUncertAbs

Answer:

A + e 1

B + e 2

C + e 3

D + e 4

2 3

2 2

2 14 eeee 

exp-1 Look for uncertainty for Burets and pipets

- Propagation of Uncertainty  Example:  The volume delivered by a buret is the difference between final and initial

readings. If the absolute uncertainty in each reading is ±0.02 mL, what is the absolute uncertainty in the volume delivered?

-Suppose that the initial reading is 0.05 (±0.02) mL and the final reading is 17.88 (±0.02) mL. The volume delivered is the difference:

A + e 1

B + e 2

C + e 3

D + e 4

2 3

2 2

2 14 eeee 

Regardless of the initial and final readings, if the uncertainty in each one is ±0.02 mL, the uncertainty in volume delivered is ±0.03 mL.

in exp-1 Look for uncertainty for Burets and pipets

Propagation of Uncertainty (iii) Once the absolute uncertainty of the answer has been determined, its

relative uncertainty can also be calculated, as described previously.  Example (using the previous example):

 Note: To avoid round-off error, keep one digit beyond the last significant figure in all calculations.

- drop only when the final answer is obtained

%1%3.1)100( 06.3

04.0 .(%)Uncert.lRe 

Round-off errors

1 sig. fig.

)100( ValueMeasured

yUncertaintAbsolute yUncertaintRelative 

Value Abs. Uncert. 1.76 (± 0.03)

+ 1.89 (± 0.02) - 0.59 (± 0.02) 3.06

0.041

Errors - Propagation of Uncertainty

(i) Multiplication and Division For multiplication and division, first convert all uncertainties into percent relative

uncertainties. Then calculate the error of the product or quotient as follows: The relative uncertainties are used for all numbers in the calculation

 Example:

     22value2valueAnswer .Uncert.lRe.Uncert.lRe.Uncert.lRe 1

     

%64.5 02.059.0

02.089.103.076.1 e

 

 

  %4.3)100(

59.0

02.0 .

%1.1)100( 89.1

02.0

%7.1)100( 76.1

03.0

...Re

 

allforUncertl

%4.3,%1.1%,7.1.Uncert.lRe 

      %4%0.44.31.17.1.Uncert.lRe 222Answer 

3 sig. fig.

1 sig. fig.

2 3

2 2

2 14 %%%% eeee 

The answer is 5.64(±4.0)%. 24

- Propagation of Uncertainty (ii) Once the relative uncertainty of the answer has been obtained, the absolute uncertainty can also be calculated:

(iii) Example (using the previous example): convert relative uncertainty into absolute uncertainty

)100( ValueCalculated

yUncertaintAbsolute y(%)UncertaintRelative 

Rearrange:

value)d(calculate y(%)UncertaintRelative

yUncertaintAbsolute )100(



2.023.0 100

%0.4 )64.5( 

 

  

yUncertaintAbsolute 1 sig. fig.

Propagation of Uncertainty from Random Error

Addition and subtraction: use absolute uncertainty of the individual terms (include units)

     final i

e e ee e2 2 21 2 3 2 i

Multiplication and division: use percent relative uncertainty

final % % % % % i

e e e e e     2 2 2 21 2 3 i

Mixed operations: follow proper algebraic rules for mathematical manipulation

- Propagation of Uncertainty (iv) For calculations involving Both additions/subtractions and

multiplication/divisions:  Treat calculation as a series of individual steps  Calculate the answer and its uncertainty for each step  Use the answers and its uncertainty for the next calculation, etc.  Continue until the final result is obtained

Example:

First operation: differences in brackets

      

 ?.619.0 02.089.1

02.059.003.076.1 

 

      

   226 02.003.003.0

036.017.102.059.003.076.1



 3 sig. fig.

3 sig. fig.

1 sig. fig., but carry two sig. fig. through calculation

     22value2valueAnswer .Uncert.lRe.Uncert.lRe.Uncert.lRe 1 2

3 2

2 2

14 %%%% eeee 

2 3

2 2

2 14 eeee 

-step 1 subtraction

-step 2 division

- Propagation of Uncertainty (v) Example:

Second operation: Division

   

 

   21 2

%.1%.3%3%.3

%3%61.0 %.189.1

%.317.1

02.089.1

03.017.1



  

  

Convert to relative uncertainty

3 sig. fig.

1 sig. fig.

Because the uncertainty begins in the 0.01 decimal place, it is reasonable to round the result to the 0.01 decimal place:

- Propagation of Uncertainty (vi) Uncertainty of a result should be consistent with the number of significant

figures used to express the result.

(vii) Example:

1.019 (±0.002)

28.42 (±0.05)

But: 12.532 (±0.064)  too many significant figures

12.53 (±0.06)  reduce to 1 sig. fig. in uncertainty same reduction in results

Result & uncertainty match in decimal place

The first digit in the answer with any uncertainty associated with it should be the last significant figure in the number.

Try example : page 46-47

Remember : Homework ? 29