A.0 SF s-uncertainty-accuracy-precision Objectives: Determine the #SF s in a measurement Round a calculated answer to the correct #SF s Round a calculated answer to the correct decimal place Calculate % error Describe the difference between precision and accuracy Know how accuracy is quantified Know how precision is quantified
Scientific Notation Scientific notation is a format for writing very large and very small numbers. Numbers written in scientific notation have two parts: The decimal part The exponential part 6.78 x 10-8 Numbers not in scientific notation are in standard notation. Examples: 43, 000,000,000 0.00567
Scientific Notation A number is expressed in scientific notation when it is in the form: a x 10 n where a is between 1 and 10 and n is an integer. Both a and n can be negative or positive Sci-Not: 6.3 x 10 6 7.930 x 10-4 - 5 x 10 24 Not in Sci-Not: - 63 x 10 9 0.62 x 10-8
When changing scientific notation to standard notation, the exponent tells how to move the decimal: With a positive exponent (big number), move the decimal to the right (you re multiplying by 1000): 4.08 x 10 3 = 4080
When changing scientific notation to standard notation, the exponent tells how to move the decimal: With a negative exponent (small number), move the decimal to the left (you re multiplying by 0.0001): 8.93 x 10-3 = 0.00893
When changing standard notation to scientific notation follow these steps: Dry erase board example: 68,940,000 0.000569 1) Move the decimal point in the number to obtain a number between 1 and 10 (the decimal part). 2) Write the decimal part multiplied by 10 raised to the number of places you moved the decimal. 3) The exponent is positive if you moved the decimal point to the left (it is a big number) and negative if you moved the decimal point to the right (it is a small number)
Standard notation to Sci-Not 835000 =? 8.35 x 10 5 Positive exponent means big number move decimal to the right. 0.000893 =? 8.93 x 10-4 Negative exponent means small number move decimal to the left.
Scientific Notation Practice Change to Sci-Not: 1) 68700 2) 0.0043 Change to Standard Notation: 3) 5.78 x 10-5 4) 1.6 x 10 6
Answers 1)6.87 x 10 4 2)4.3 x 10-3 3)0.0000578 4)1 600 000
Uncertainty and Significant Figures SF s-why Are They Important? All measuring devices have limitations and therefore measurements always involve some uncertainty. Significant figures are used to report all precisely known numbers + one estimated digit The # of SF s indicates the precision of the instrument.
Measurement
The number of significant figures in a measurement.
Uncertainty What is the diameter of a quarter? The answer depends on the precision of the ruler used.
Uncertainty The top ruler is more precise (and more expensive). In the measurement 2.33 cm, the last digit is estimated as a 3 by using the calibration marks. The other two digits are certain. In the measurement 2.3 cm, the last digit is estimated as a 3 by using the calibration marks. The other digit is certain.
Measurement and Uncertainty When measuring an object, the last digit should always be estimated (between two calibration marks). Consider the line below. What is the length in cm? When the measurement falls on a calibration mark, the last digit is always zero. Length = 1.70 cm
Measurement and Uncertainty Consider the line below. What is the length in cm? Length = 4.00 cm
Measurement and Uncertainty What is the difference in the following lengths? They represent the length of the same line. 4.23 cm 4.2 cm 4 cm
Measurement and Uncertainty What is the difference in the following lengths? They represent the length of the same line. 4.23 cm--scientist used a fairly precise ruler 4.2 cm--scientist used a less precise ruler-he must have run out of $$ 4 cm--they cheaped-out! This ruler would have no calibration marks! 0 cm 10 cm
Why do we care about Significant Figures (SF s)? There are 2 kinds of numbers: Exact: the amount of money in your account, Metric unit equalities. These are known with certainty. Approximate: weight, height anything MEASURED, has a limit to the certainty, depending on the instrument used.
Why do we care about Significant Figures (SF s)? 3.50 inch to a scientist means the measurement is accurate to within one hundredth of an inch. Every measurement is a reflection of the precision of the measuring instrument.
Why do we care about Significant Figures (SF s)? To a mathematician 3.5 inches, or 3.50 inches is the same. But, to a scientist 3.5 inches and 3.50 inches is NOT the same
Pacific-Atlantic Rule P A If a decimal point is present, start on the Pacific (P) side and begin counting at the first non-zero digit all the way to the end. If a decimal is absent, start on the Atlantic (A) side and begin counting at the first non-zero digit all the way to the end.
If a Decimal P Is present then 0.0050074 5 sig figs A Is absent then 6003895400 8 sig figs
Pacific Rule Examples: 123.003 = decimal present, start on P side, begin counting # SF s = 6 Examples (Do with a study buddy): 0.00024 How many SF s? 0.453 How many SF s? 40.9 How many SF s? 43.00 How many SF s? 1.010 How many SF s? 1.50 How many SF s? 2 3 3 4 4 3
Atlantic Rule Examples: 204,000 = decimal is absent, start on A side, begin counting at first non-zero digit until you hit the end of the number. # SF s = 3 Examples (CO): 7003 How many SF s? 300 How many SF s? 27,300 How many SF s? 56 How many SF s? 4 1 3 2
Drill and Practice Underline the significant digits in each of the following numbers. 6 sig figs 7 sig figs 0.00506970 500.6790 250005 4970350000 5678493 6 sig figs 6 sig figs 7 sig figs
Scientific Notation All digits in the decimal part of in a number written in scientific notation are significant. Examples: 5.6 x 10 4 Has 2 SF s 8.90 x 10 3 Has 3 SF s
Bellwork:Use the Pacific-Atlantic Rules to determine the # of SF s a) 18.3 b) 1.83 x 10 2 c)0.00183 d) 183.0 e) 50 f) 505 g) 0.0050 h) 5000 i) 200.00 j) 0.00220 k) 22020 l) 22.20
How many Sig-Figs? a) 3 b) 3 c) 3 d) 4 e) 1 f) 3 g) 2 h) 1 i) 5 j) 3 k) 4 l) 4
Significant Numbers in Calculations A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. There are two different rules for rounding and reporting a calculated answer --Multiplying or dividing --Adding or subtracting
Sig. Fig. Math Rules Multiplication / Division: Your answer can t have more sig. figs. than the number in the problem with the least amt. of sig. figs. Example = 60.56227892 m x 35.25 m Calculator says 2134.890832 m 2 (too many SF s!) Don t forget units! How many SF s should be in final answer? 4 SF s Answer - 2135 m 2
Multiplication & Division The final answer has the same number of sig-figs as the number with the least number of sig-figs. 123m x 5.35m = 658.05m 2 = 658m 2 (3 SF s) 16cm 2 x 2cm = 32cm 3 = 30cm 3 (1 SF) 16 x 2.0g = 32g = 32g (2 SF s)
Multiplication & Division The final answer has the same number of sig-figs as the number with the least number of sig-figs. 823dm 3 4.0dm = 205.75 dm 2 = 210 dm 2 (2 SF s) 84.7 g 20 ml = 4.235 g/ml = 4 g/ml (1 SF)
Sometimes the answer must be written in scientific notation to express the correct number of SF s 25 m x 4.0 m = 100 m 2 The answer should have 2 SF s. If left as 100 m 2, it would only have 1 SF. So change it to Sci-Not: 1.0 x 10 2 m 2 now it has 2 SF s
Be Careful with Domino Rounding A. Round the following to 4 SF s: 37.4959 Answer: 37.50 B. Round the following to 3 SF s: 68.4499 Answer: 68.4
Express Result in Correct # of SF s Pay attention to units! 5.35 x 20.2 x 5.0 = 2.00 x 500 = 2.50 cm x 55.5 cm = 5.0 m x 4.000 x 10 2 m =
Multiplication Examples 5.35 x 20.2 x 5.0 = 540.35 = 540 2.00 x 500 = 1000 2.50 cm x 55.5 cm = 138.75cm 2 = 139cm 2 5.0 m x 4.000 x 10 2 m = 2000m 2 = 2.0 x 10 3 m 2 ( 2 SF s)
Express Result in Correct # of SF s 0.030 2.00 = 9101 8.8 = (3.94 x 10 7 m 2 ) (8.4 x 10-25 m) =
Express Result in Correct # of SF s 0.030 2.00 = 0.015 9101 8.8 = 1034 = 1.0 x 10 3 ** must be is Sci-Not! (3.94 x 10 7 m 2 ) (8.4 x 10-25 m) = 4.7 x 10 31 m
Addition and Subtraction When adding or subtracting, look at the decimal places. Find the measurement that is least precise (# places past the decimal). Round to the least precise place. Ex. a) 13.64 + 0.075 + 67 b) 267.8 9.36 13.64 + 0.075 267.8 + 67. 9.36 80.715 81 258.44 258.4 Round to the ones place. Round to the tenths place.
Examples i) 83.25 ii) 4.02 iii) 0.2983 0.1075 + 0.001 + 1.52 83.1425 = 83.14 Round to the Hundredths place. 4.021 = 4.02 Round to the Hundredths place. 1.8183 = 1.82 Round to the Hundredths place.
Examples 42, 000 53.9 + 698 + 60 42, 698 = 113.9 = 43, 000 110 Round to the Thousands place. Round to the Tens place.
Learning Check-Study Buddy In each calculation, round the answer to the correct number of significant figures. A. 235.05 + 19.6 + 2.1 = 1) 256.75 2) 256.8 3) 257 B. 58.925-18.2 = 1) 40.725 2) 40.73 3) 40.7
Solution A. 235.05 + 19.6 + 2.1 = 2) 256.8 B. 58.925-18.2 = 3) 40.7
Sig. Figs.-Mixed Operations Significant Figures in Mixed operations (Use PEMDAS) (1.7 x 10 6 2.63 x 10 5 ) + 7.33 =??? Step 1: Divide the numbers in the parenthesis. How many sig figs? Step 2: Add the numbers. How many decimal places to keep? (6.463878327 ) + 7.33 6.463878327 + 7.33 13.7938 Step 3: Round answer to the appropriate decimal place. 13.8 or 1.38 x 10 1
Precision and Accuracy Precision --Refers to how close the measurements in a series are to each other. --Can indicate how well your tools are working, not what the tools are measuring. --Can indicatethe technique of the scientist-poor technique leads to poor precision and vice versa. Accuracy --Refers to how close each measurement is to the accepted, true, or literature value. --Checks how right or correct your answer is. --Only need one measurement and the true value.
Precision, Accuracy, and Error Precision refers to how close the measurements in a series are to each other. Accuracy refers to how close each measurement is to the actual value. Systematic error produces values that are either all higher or all lower than the actual value. This error is part of the experimental system. Random error produces values that are both higher and lower than the actual value.
Figure 1.9 Precision and accuracy in a laboratory calibration. precise and accurate precise but not accurate
Figure 1.9 continued Precision and accuracy in the laboratory. random error systematic error
Quantifying Accuracy Percent error, sometimes referred to as percentage error, is an expression of the difference between a measured value and the known or accepted value. It is often used in science to report the difference between experimental values and accepted (true) values. The formula for calculating percent error is: Absolute value bars
Quantifying Precision In this class, we will not quantify precision Take multiple measurements of the same thing under the same conditions. --Calculate the mean (average) --Calculate the standard range-highest value lowest value For a better view: --Calculate the mean absolute deviation Subtract each individual measurement from the mean and then take the average of these numbers.
Why Perform Multiple Trials? --Can show reliability and reproducibility, this can add validity to the conclusions you draw --Multiple trials can give you information about only precision. Poor technique can contribute to poor precision. Performing many trials will NOT ensure good accuracy. In fact, nothing can be determined about accuracy with just data from trials. You need a true (literature) value in order to determine accuracy,
Calculating Percent Error The literature value for the atomic mass of an isotope of nickel is 57.9 g/mol. If a laboratory experimenter determined the mass to be 56.5 g/mol, what is the percent error (rounded to the hundredths place)? See board. Answer: 2.42 %