Uncertainty in numbers

Transcription

1 1.03 Accuracy, Precision and Significant Figures Uncertainty in numbers Story: Taxi driver (13 years experience) points to a pyramid "...this here pyramid is exactly 4511 years old". After a quick calculation, the tourist asked, how can you be so certain that this particular pyramid was built in 2498 BC? I had a friend who when to tour Egypt in Once at Egypt, there was a taxi driver with 11 year experience, wjp offered to drive him around and take him on a personal tour for a nominal fee. My friend took up the taxi driver on his offer. Toward the end of their tour the cab driver pointed to a set of pyramid and said that the age of the pyramid were exactly 4511 years old. My friend was fascinated because he did not realize that the cab driver was very knowledgeable on the exact date of the grand opening ceremony. After quizzing the cab driver further, my friend discovered that the cab driver had been working with the cab company for six years and six years ago, he was told by the company that this particular pyramid were 4500 years old AD Significant Figures, Precisions and Accuracy

2 Uncertainty in Measurements It is important to realize that any measurement will always contain some degree of uncertainty. The uncertainty of the measurement is determine by the scale of the measuring device Significant Figures, Precisions and Accuracy

3 Precision vs. Accuracy Precision: Indication of how close individual measurements agree with each other. Accuracy: How close individual measurements agree with the true value. In general, experimental measurements are taken numerous times to improve precision; In general more precise e more accurate... but not necessarily true Significant Figures, Precisions and Accuracy

4 Measurements in the lab Accuracy and precision. Each dot represents one attempt at measuring a person s height. a) shows high precision and great accuracy; the dots are tightly clustered by the true value. b) Shows high precision (tightly clustered dots) but poor accuracy (large error); perhaps the meter stick was not made correctly. c) Shows poor precision (dots are not tightly clustered) but, by accident, high accuracy; the average of the measurements would be close to the true value. d) Here there is neither precision and nor accuracy a) Precise Accurate (laudable) - Promote the analyst b) Precise Inaccurate (avoidable) - Repair the instrument c) Imprecise Accurate (by accident) -Train the analyst d) Imprecise Inaccurate (lamentable) - Hire a new analyst Significant Figures, Precisions and Accuracy

5 High Precision Low Accuracy High precision - grouping is tight. Low Accuracy - but the marks miss the target Significant Figures, Precisions and Accuracy

6 Low Precision High Accuracy High Accuracy - by accident marks are averaged around the target. Low Precision - grouping is scattered Significant Figures, Precisions and Accuracy

7 Low Precision Low Accuracy Low Precision - grouping is scattered. Low Accuracy - marks miss the target Significant Figures, Precisions and Accuracy

8 High Precision High Accuracy High Accuracy - marks are averaged around the target. High precision - grouping is tight Significant Figures, Precisions and Accuracy

9 Making a measurement In general, the uncertainty of a measurement is determined by the precision of the measuring device. A 10-mL pipet with a graduation of 0.1mL with give an uncertainty of i.e., 1.7x In this example the volume is calibrated to the tenth of a place and the measurement is uncertain to ml. In other words, if a measurement is recorded at 1.75 ml, then the actual volume might be assumed to be 1.74mL or 1.76 ml in which the 4 or 6 is the uncertain digit. The ml is arrived at by placing a 1 in the position of the last certain digit (the tenth place) and dividing by 10 (0.1 ml / 10 = 0.01 ml). A 100-mL graduated cylinder with 1-mL graduation will have an uncertainty of + 0.1mL. That is a measurement of 53.4 ml using this graduated cylinder could be read 53.3 or 53.4mL. For calibration of a 50-mL graduated cylinder with calibration of 0.2 ml, the uncertainty is +0.05mL. Devices that measure the volume of liquids. Several devices are used to measure the volume of liquids. Considerations include ease of use as well as accuracy and precision. From left to right; the blue liquid is in a 10-mL graduated pipet that can disperse liquids with an accuracy of ml; the reddish liquid is in a volumetric flask that will contain ml when filled to the mark; the yellow liquid is in a 100mL graduated cylinder that measures volumes to +0.4 ml; the green liquid is in a 10-mL graduated cylinder that measures volumes to +0.06mL; the red liquid at the right is in a 10-mL transfer pipet that delivers the liquid with an accuracy of ml Significant Figures, Precisions and Accuracy

10 Reading a temperature measurement The number of significant figures in a measurement depends on the measuring device. Two thermometers measuring the temperature of the same object are shown with expanded views. The thermometer on the right is graduated in 1 C and reads 32.3 C; the one on the left is graduated to 0.1 C and reads C. Therefore, a reading with more precision (more certainty) and therefore with more significant figures can be made with the thermometer on the left which is the more precise thermometer C 32.3 C Significant Figures, Precisions and Accuracy

11 When reading a measurement of length, remember that in general- The more accurate the measurement the smaller the error Accurate number e small errors The more precise the measurement, the lower the uncertainty. Precise number e small uncertainty cm +.1 D = 1cm error is tenth 1/ Significant Figures, Precisions and Accuracy

12 When reading a measurement of length remember that in general The more accurate the measurement the smaller the error Accurate number e small errors The more precise the measurement, the lower the uncertainty. Precise number e small uncertainty cm +.1 D = 1cm error is tenth 1/ D = 0.1cm error is tenth 1/ cm +.01 More precise measurement Significant Figures, Precisions and Accuracy

13 So why are significant figures (SF) so important? SF is an alternative method of measuring uncertainty. Scientist have found that it is useful to tell the degree of certainty of a measured number using SF. Merely writing down all the digits that are certain and not writing down any that are not certain expresses the error in the measurement Significant Figures, Precisions and Accuracy

14 Significant Figures vs. Precision How many significant figures do these numbers contain? Significant Figures 5?? 3? 4? 2 Rules of Significant Figures: Nonzero integers - always count as significant figures. Zeros Leading zeros - are zeros that precede all of the nonzero digits. They never count as significant figures. Captive zeros - are zeros that fall between nonzero digits. They all count as significant figures. Trailing zeros - are zeros at the right end of the number. They are significant only if the number contains a decimal point. Exact numbers - such as tallies or conversion factors have unlimited number of significant figures Significant Figures, Precisions and Accuracy Chp1: Measurements

15 To determine the number of Significant Figures Pacific-Atlantic Rule Decimal Present vs. Decimal Absent Place the number in the center of the map. Depending if the number has a decimal point or no decimal point determines if the analysis starts at the Pacific or the Atlantic part of US. Decimal Present Decimal Absent For numbers with a For numbers with a decimal decimal point present, point absent, draw a line draw a line starting from the Pacific to the first P Place Number here starting from the Atlantic (right) to the first non-zero non-zero number, all digits shown including the non- P A number, all digits shown including the non-zero zero number are number are significant. significant. i.e., i.e., P A 5 Significant Figures 3 Significant Figures Significant Figures, Precisions and Accuracy Chp1: Measurements

16 Significant Figures example- Now determine the number of significant figures for the examples we had earlier Significant Figures 5?? 3? 4 2? five significant figures four significant figures four significant figures 2500 two significant figures Significant Figures, Precisions and Accuracy Chp1: Measurements

17 More on significant digits and the different type of uncertainties i i i i 2500 Number of significant figures Implied uncertainty Relative uncertainty % relative uncertainty % % % % Significant Figures, Precisions and Accuracy

18 Manipulation of Significant Figures: (add and subtract) Wrong Answer = Significant Figures, Precisions and Accuracy

19 Math operation: Addition-subtraction Example - keep exponent the same in addition and subtraction operation and round off answer to least precise of the data. Consider the following example: Three individual who each gave their loose change to the Salvation Army collection pan. The first had two dollar bills a quarter and two pennies. The second donated three dollar bills and five nickels and one penny. The third gave two quarters and a few pennies. What is the total amount that was collected? $ $ ~ $0.80 = Significant Figures, Precisions and Accuracy

20 Difference between Significant figures Precision What is the difference between significant figures and precision. For example in the example shown, which number has the fewest significant figures and which is the least precise?. Fewest Significant figures Least precise a) 123 vs 1.2 b) 1.23 vs c) vs d) 30 vs The number with the fewest significant figures is the number which contains the lower amount of digits (significant digits). The least precise number, on the other hand, is the number with the largest uncertainty. For example 0.1 (or 1/10) has more uncertainty than 0.01 (or 1/100) Significant Figures, Precisions and Accuracy

21 Difference between Significant figures Precision What is the difference between significant figures and precision. For example in the example shown, which number has the fewest significant figures and which is the least precise?. Fewest Significant figures Least precise a) 123 vs b) 1.23 vs c) vs d) 30 vs The number with the fewest significant figures is the number which contains the lower amount of digits (significant digits). The least precise number, on the other hand, is the number with the largest uncertainty. For example 0.1 (or 1/10) has more uncertainty than 0.01 (or 1/100) Significant Figures, Precisions and Accuracy

22 Manipulation of Significant Figures: (add and subtract) Least precise Addition and Subtraction: Uncertainty of answer (Significant figures of answer) is limited to the value with the least precise value (number with fewest digit after decimal place - the number 83 in Correct Answer our example) Significant Figures, Precisions and Accuracy

23 Try this at your desk: = a) b) c) d) 1520 e) No right answer Correct Answer (c) Significant Figures, Precisions and Accuracy

24 Manipulation of Significant Figures: Fewest number of significant figures (mult and divide) S. F. 2 S. F. 2 S. F. Answer with 2 s.f. Multiplication and division: Uncertainty of answer (Significant. Figure) is limited to the value with the fewest significant figures. In our example, 0.17 limits the value Significant Figures, Precisions and Accuracy

25 Try this at your desk: ( * 23) / ( * 30.2) = a) b) 1.76 c) 2 d) 1.8 ( * 23) e) No right answer = ( *30.2) Correct Answer (d) Significant Figures, Precisions and Accuracy

26 Arithmetic Examples: Combinations using addition-subtraction / multiplication - division Carry out addition / subtraction operation first before the multiplication or division operation. 1* ( ) = = 7.8 Working it out 2* ( ) 0.12 ( ) = Lower layer 3 ( ) + ( ) = = * ( ) + ( ) = = Significant Figures, Precisions and Accuracy

27 Arithmetic Examples: Combinations using addition / subtraction and multiplication / division Carry out addition / subtraction operation first before the multiplication or division operation ( ) = = ( ) 0.12 ( ) = = ( ) + ( ) = = 4.55 = Significant Figures, Precisions and Accuracy

28 Exception to significant figure rules: These type of numbers contain unlimited significant figures (do not influence the number of significant figures in the final answer). Number of Tallies, i.e., 5 fingers, 176 students. *Definition of numbers - i.e., Exactly 1 m = 100 cm, or 1 in = 2.54 cm Power of 10 in exponential notation i.e., 10 6 but practical to express numbers as (exponential calc) *Define conversion versus measured conversion Significant Figures, Precisions and Accuracy

29 Conversion factors: Measured versus defined One last important note: Conversion factor comes in two forms, the first are conversion factors from definitions. 60 min = 1 hr, 100cm = 1m, 5280 ft = 1 mile, 1 gal = L, 100 pennies = $1.00. Other conversion factors are based on measured values. 65 mph (65 mi = 1 hr), $10/hr (10 $ = 1 hr), 0.76Euro/$ (0.76 euro = 1.00$) Measured conversion factors do have significant figures unlike defined conversion factors that have infinite number of significant figures. Thus in the problem; If a runner for minutes at 11 mph, how far will the runner travel in miles? The answer is rounded to two significant figures min 1 hr 60 min 11 mi 1 hr = mi = 26 mi Significant Figures, Precisions and Accuracy

30 II Summary: Significant Figures Written digits of results must have right number of significant figures. Can determine number of significant figures for any number using U.S. map analogy. Addition/Subtraction Least precise number in the data determines number of significant figures in the final answer. Multiplication/Division data with fewest significant figures determines number of significant figures in the answer. * Remember the exception to the rules for significant figures Significant Figures, Precisions and Accuracy