Chapters 3Uncertainty

January 30, 2007Lec_3

Outline

Homework Chapter 1 Chapter 3

Experimental Error“keeping track of uncertainty”

Start Chapter 4 Statistics

Homework

Chapter 1 – “Solutions and Chapter 1 – “Solutions and Dilutions”Dilutions”

Questions: 15, 16, 19, 20, 29, 31, Questions: 15, 16, 19, 20, 29, 31, 3434

Chapter 3Chapter 3Experimental ErrorExperimental Error

And propagation of And propagation of uncertaintyuncertainty

Keeping track of uncertainty

Significant Figures Propagation of Error

35.21 ml 35.21 (+ 0.04) ml

Suppose

You determine the density of some mineral by measuring its mass 4.635 + 0.002 g

And then measured its volume 1.13 + 0.05 ml

)(

)(

mlvolume

gmass

ml

g1018.4

Significant Figures (cont’d)

The last measured digit always has some uncertainty.

3-1 Significant Figures3-1 Significant Figures

What is meant by significant figures?

Significant figures: Significant figures:

Examples

How many sig. figs in:a. 3.0130 meters b. 6.8 daysc. 0.00104 poundsd. 350 milese. 9 students

“Rules”

1. All non-zero digits are significant2. Zeros:

a. Leading Zeros are not significantb. Captive Zeros are significantc. Trailing Zeros are significant

3. Exact numbers have no uncertainty

(e.g. counting numbers)

Reading a “scale”

What is the “value”?

When reading the scale of any apparatus, try to estimate When reading the scale of any apparatus, try to estimate to the nearest tenth of a division.to the nearest tenth of a division.

3-23-2Significant Figures in ArithmeticSignificant Figures in Arithmetic

We often need to estimate the uncertainty of a result that has been computed from two or more experimental data, each of which has a known sample uncertainty.

Significant figures can provide a marginally good way to express uncertainty!

3-23-2Significant Figures in ArithmeticSignificant Figures in Arithmetic

Summations: When performing addition and subtraction

report the answer to the same number of decimal places as the term with the fewest decimal places

+10.001+ 5.32+ 6.130

?

Try this one

1.632 x 105

4.107 x 103

0.984 x 106

+

0.1632 x 106

0.004107 x 106

0.984 x 106

+

3-23-2Significant Figures in ArithmeticSignificant Figures in Arithmetic

Multiplication/Division: When performing multiplication or division

report the answer to the same number of sig figs as the least precise term in the operation

16.315 x 0.031 = ?0.5057650.51

3-23-2Logarithms and AntilogarithmsLogarithms and Antilogarithms

From math class:log(100) = 2Or log(102) = 2But what about significant figures?

3-23-2Logarithms and AntilogarithmsLogarithms and Antilogarithms

Let’s consider the following:An operation requires that you take the log of 0.0000339. What is the log of this number?

log (3.39 x 10-5) = log (3.39 x 10-5) = log (3.39 x 10-5) =

3-23-2Logarithms and AntilogarithmsLogarithms and Antilogarithms

Try the following:Antilog 4.37 =

“Rules”

Logarithms and antilogs1. In a logarithm, keep as many digits to

the right of the decimal point as there are sig figs in the original number.

2. In an anti-log, keep as many digits are there are digits to the right of the decimal point in the original number.

3-4. Types of error3-4. Types of error Error – difference between your answer and

the ‘true’ one. Generally, all errors are of one of three types. Systematic (aka determinate) – problem with the

method, all errors are of the same magnitude and direction (affect accuracy)

Random – (aka indeterminate) causes data to be scattered more or less symmetrically around a mean value. (affect precision)

Gross. – occur only occasionally, and are often large.

Absolute and Relative Absolute and Relative UncertaintyUncertainty

Absolute uncertainty expresses the margin of uncertainty associated with a measurement.Consider a calibrated buret which has an uncertainty + 0.02 ml. Then, we say that the absolute uncertainty is + 0.02 ml

Absolute and Relative Absolute and Relative UncertaintyUncertainty

Relative uncertainty compares the size Relative uncertainty compares the size of the absolute uncertainty with its of the absolute uncertainty with its associated measurement. associated measurement.

Consider a calibrated buret which has an uncertainty is + 0.02 ml. Find the relative uncertainty is 12.35 + 0.02, we say that the relative uncertainty is

tmeasuremen of magnitude

yuncertaint absolutety UncertainRelative

3-5. Estimating Random 3-5. Estimating Random Error (absolute uncertainty)Error (absolute uncertainty)

Consider the summation:

+ 0.50 (+ 0.02)+4.10 (+ 0.03) -1.97 (+ 0.05)

2.63 (+ ?)

...222 cbay ssss

3-5. Estimating Random 3-5. Estimating Random ErrorError

Consider the following operation:

?)(010406.0)04.0(97.1

)0001.0(0050.0)02.0(10.4

...222

c

s

b

s

a

s

y

scbay

Try this one

)4.0(3.42)5(1030)10(820

)001.0(050.0)2.0(6.11)2.0(3.14

3-5. Estimating Random 3-5. Estimating Random ErrorError

For exponents

a

sx

y

s

ay

For

ay

x

aS is ain y uncertaint

3-5. Estimating Random 3-5. Estimating Random ErrorError

Logarithms antilogs

a

ss

ay

For

ay 434.0

S is ain y uncertaint

log

a

ay sy

s

aantiy

For

303.2

S is ain y uncertaint

log

a

Question

Calculate the absolute standard deviation for a the pH of a solutions whose hydronium ion concentration is

2.00 (+ 0.02) x 10-4

a

ss

ay

ay 434.0

S is ain y uncertaint

log

a

Question

Calculate the absolute value for the hydronium ion concentration for a solution that has a pH of 7.02 (+ 0.02)

[H+] = 0.954992 (+ ?) x 10-7

ay sy

s

aantiy

303.2

S is ain y uncertaint

log

a

Suppose

You determine the density of some mineral by measuring its mass 4.635 + 0.002 g

And then measured its volume 1.13 + 0.05 ml

)(

)(

mlvolume

gmass 1018.4

What is its uncertainty?

...222

c

s

b

s

a

s

y

scbay

=4.1 +0.2 g/ml

The minute paper

Please answer each question in 1 or 2 sentences

1) What was the most useful or meaningful thing you learned during this session?

2) What question(s) remain uppermost in your mind as we end this session?

Chapter 4 Chapter 4

StatisticsStatistics

General Statistics Principles Descriptive Statistics

Used to describe a data set.

Inductive Statistics The use of descriptive statistics to accept or

reject your hypothesis, or to make a statement or prediction

Descriptive statistics are commonly reported but BOTH are needed to interpret results.

Error and Uncertainty Error – difference between your answer

and the ‘true’ one. Generally, all errors are of one of three types. Systematic (aka determinate) – problem

with the method, all errors are of the same magnitude and direction (affect accuracy).

Random – (aka indeterminate) causes data to be scattered more or less symmetrically around a mean value. (affect precision)

Gross. – occur only occasionally, and are often large. Can be treated statistically.

The Nature of Random Errors Random errors arise when a system of

measurement is extended to its maximum sensitivity.

Caused by many uncontrollable variables that are an are an inevitable part of every physical or chemical measurement.

Many contributors – none can be positively identified or measured because most are so small that they cannot be measured.

Random Error

Precision describes the closeness of data obtained in exactly the same way.

Standard deviation is usually used to describe precision

Standard Deviation Sample Standard deviation

(for use with small samples n< ~25)

Population Standard deviation (for use with samples n > 25)

U = population mean IN the absence of

systematic error, the population mean approaches the true value for the measured quantity.

1

)( 2

n

xxs i

N

xi2)(

Example

The following results were obtained in the replicate analysis of a blood sample for its lead content: 0.752, 0.756, 0.752, 0.760 ppm lead. Calculate the mean and standard deviation for the data set.

Standard deviation

0.752, 0.756, 0.752, 0.760 ppm lead.

755.0x

Distributions of Experimental Data

We find that the distribution of replicate data from most quantitative analytical measurements approaches a Gaussian curve.

Example – Consider the calibration of a pipet.

Replicate data on the calibration of a 10-ml pipet.

Trial Volume Trial Volume Trial Volume1 9.988 18 9.975 35 9.9762 9.973 19 9.980 36 9.9903 9.986 20 9.994 37 9.9884 9.980 21 9.992 38 9.9715 9.975 22 9.984 39 9.9866 9.982 23 9.981 40 9.9787 9.986 24 9.987 41 9.9868 9.982 25 9.978 42 9.9829 9.981 26 9.983 43 9.97710 9.990 27 9.982 44 9.97711 9.980 28 9.991 45 9.98612 9.989 29 9.981 46 9.97813 9.978 30 9.969 47 9.98314 9.971 31 9.985 48 9.98015 9.982 32 9.977 49 9.98316 9.983 33 9.976 50 9.97917 9.988 34 9.983

Mean 9.982 mlmedian 9.982 mlspread 0.025 mlStandard Deviation 0.0056 ml

Frequency distributionVolumeRange, mL Number in Range % in range9.969 to 9.971 3 69.982 to 9.974 1 29.975 to 9.977 7 149.978 to 9.980 9 189.981 to 9.983 13 269.984 to 9.986 7 149.987 to 9.989 5 109.990 to 9.992 4 89.993 to 9.995 1 2

9.965 9.970 9.975 9.980 9.985 9.990 9.9950

2

4

6

8

10

12

14

Nu

mb

er o

f m

ea

sure

men

ts

Range of measured values

22 2/)(

2

1

xey

The minute paper

Please answer each question in 1 or 2 sentences

1) What was the most useful or meaningful thing you learned during this session?

2) What question(s) remain uppermost in your mind as we end this session?