Chapters 3 Uncertainty January 30, 2007 Lec_3

Outline Homework Chapter 1 Chapter 3 Start Chapter 4 Experimental Error “keeping track of uncertainty” Start Chapter 4 Statistics

Chapter 1 – “Solutions and Dilutions” Homework Chapter 1 – “Solutions and Dilutions” Questions: 15, 16, 19, 20, 29, 31, 34

Experimental Error And propagation of uncertainty Chapter 3 Experimental Error And propagation of uncertainty

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

Significant Figures (cont’d) The last measured digit always has some uncertainty.

3-1 Significant Figures What is meant by significant figures?

Examples How many sig. figs in: 3.0130 meters 6.8 days 0.00104 pounds 350 miles 9 students

“Rules” All non-zero digits are significant Zeros: Leading Zeros are not significant Captive Zeros are significant Trailing Zeros are significant 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 to the nearest tenth of a division.

3-2 Significant 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!

3-2 Significant 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 + +

3-2 Significant 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 = 0.51 ? 0.505765

3-2 Logarithms and Antilogarithms From math class: log(100) = 2 Or log(102) = 2 But what about significant figures?

3-2 Logarithms 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) =

3-2 Logarithms and Antilogarithms Try the following: Antilog 4.37 =

“Rules” Logarithms and antilogs 1. 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 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 Uncertainty 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 Uncertainty Relative uncertainty compares the size of the absolute uncertainty with its 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

3-5. Estimating Random Error (absolute uncertainty) Consider the summation: + 0.50 (+ 0.02) +4.10 (+ 0.03) -1.97 (+ 0.05) 2.63 (+ ?)

3-5. Estimating Random Error Consider the following operation:

Try this one

3-5. Estimating Random Error For exponents

3-5. Estimating Random Error Logarithms antilogs

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

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

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 What is its uncertainty? =4.1 +0.2 g/ml

The minute paper Please answer each question in 1 or 2 sentences What was the most useful or meaningful thing you learned during this session? What question(s) remain uppermost in your mind as we end this session?

Chapter 4 Statistics

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.

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.

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.

Frequency distribution

The minute paper Please answer each question in 1 or 2 sentences What was the most useful or meaningful thing you learned during this session? What question(s) remain uppermost in your mind as we end this session?