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Section 2.3 Uncertainty in Data • Define and compare accuracy and precision. • Describe the accuracy of experimental data using error and percent error. • Apply rules for significant figures to express uncertainty in measured and calculated values. experiment: a set of controlled observations that test a hypothesis Section 2.3 Uncertainty in Data (cont.) accuracy percent error precision significant figures error Measurements contain uncertainties that affect how a result is presented. Accuracy and Precision • Accuracy refers to how close a measured value is to an accepted value. • Precision refers to how close a series of measurements are to one another. Example: The acceleration due to gravity on Earth is 9.8 m/s2. Look at the lab data for two students. Whose data was most accurate? Whose data was most precise? Sue’s Trials 1 9.79 2 3 9.81 9.82 Fred’s Trials 1 How would you describe Fred’s data? Calculated acceleration (m/s2) 2 3 Calculated acceleration (m/s2) 8.85 8.54 9.86 Accuracy and Precision (cont.) • Error is defined as the difference between and experimental value and an accepted value. •The error equation is error = experimental value – accepted value. Example: The acceleration due to gravity on Earth is 9.81 m/s2. Look at the lab data for two students. What is the error for Sue’s trial 3? Sue’s Trials 1 9.79 2 3 9.81 9.82 Fred’s Trials What is the error for Fred’s trial 1? Calculated acceleration (m/s2) 1 2 3 Calculated acceleration (m/s2) 8.85 8.54 9.86 Accuracy and Precision (cont.) • Percent error expresses error as a percentage of the accepted value. The percent error is always expressed as an absolute value. Example: The acceleration due to gravity on Earth is 9.81 m/s2. Look at the lab data for two students. What is the % error for Sue’s trial 3? Sue’s Trials 1 9.79 2 3 9.81 9.82 Fred’s Trials What is the % error for Fred’s trial 1? Calculated acceleration (m/s2) 1 2 3 Calculated acceleration (m/s2) 8.85 8.54 9.86 • In a lab, students were to calculate the density of aluminum. The accepted value is 2.7 g/cm3. • In George’s experiment, they found the value to be 2.5 g/cm3. Find: A). George’s error: B). George’s percent error: Section 2.3 Assessment A substance has an accepted density of 2.00 g/L. You measured the density as 1.80 g/L. What is the percent error? A. 0.20 g/L A 0% D D. 0.90 g/L C C. 0.10 g/L A. A B. B C. C 0% 0% 0% D. D B B. –0.20 g/L • 3.3 Significant Figures –(Sig Figs) Significant Figures • Often, precision is limited by the tools available. • Significant figures include all known digits plus one estimated digit. Significant Figures (cont.) • Rules for significant figures – Rule 1: Nonzero numbers are always significant. – How many significant figures in each of the following: 1234 22 7.7654 Rule 2: Zeros between nonzero numbers are always significant. (sandwich rule) How many significant figures? 101 .103 3 3 5500.002 7 Rule 3: All final zeros to the right of the decimal are significant. (You shouldn’t tack a zero on after the decimal if you don’t actually measure it). How many significant digits? 123.40 103.440 .1230 Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation. In 11 000, the 3 zeros are just place-holders…they are just showing that the number is eleven-thousand, not eleven. How many significant figures? 10300 0.0012 0.11030 Rule 5: Counting numbers and defined constants have an infinite number of significant figures. What this means….if you say there are 8 people in the room, that has infinite significance. You can’t have 8.2 people…or 7.9 people. If you were to average 3 masses, 55.5 grams, 54.6 grams and 53.5 grams….how would you do that? The 3 that you divide by has infinite significance since it is a counting number. Rounding Numbers • Calculators are not aware of significant figures. • When we find the answers to problems, the significant figures in the answer will be calculated based on the significant figures in the original data. Answers will usually be rounded. Rounding Numbers (cont.) • Rules for rounding – Rule 1: If the digit to the right of the last significant figure is less than 5, do not change the last significant figure. – Rule 2: If the digit to the right of the last significant figure is greater than 5, round up to the last significant figure. – Rule 3: If the digits to the right of the last significant figure are a 5 followed by a nonzero digit, round up to the last significant figure. Rounding Numbers (cont.) • Rules for rounding (cont.) – Rule 4: If the digits to the right of the last significant figure are a 5 followed by a 0 or no other number at all, look at the last significant figure. If it is odd, round it up; if it is even, do not round up. (round to the nearest even number). – There is a reason for this rule. If you don’t do this, you are constantly skewing your lab results to the high end. Let’s look at an example. Trial # Time (sec) 1 5.5 2 3 4 5 6 6.5 7 7.5 6 7 8 8.5 What is the average? Trial # Time (sec) old rounding 1 5.5 Trial # Time (sec) New rounding 1 5.5 2 3 4 6 6.5 7 2 3 4 6 6.5 7 5 6 7 7.5 8 8.5 5 6 7 7.5 8 8.5 What is the average of the rounded values? What is the average of the rounded values? • If we consistently round up in a lab setting (like you have probably done your whole life), you are constantly skewing the results upward. • By rounding with the even rules, the round “ups” balance the round “downs” and the data is more true to what it should be. Rounding Numbers (cont.) • Rule for Addition and subtraction with significant figures – Round numbers so the answer is significant to the same column as the least significant number. – Huh???? 43.24 m + 2.2 m • Let’s look at an example: – Joe, Sally, and Fred count the money in their piggy banks. – Joe counts only the dollars. He says he has $43. in his bank. – Sally counts only the dollars and dimes. She says she has $53.4 in her bank. – Fred counts all the money in his bank. He ends up with $22.56. – If we add their totals together, we get 43 53.4 + 22.56 118.96 Is this really the total for the three banks? – If we add their totals together, we get 43 53.4 + 22.56 118.96 Is this really the total for the three banks? • Of course not….because we don’t know how many pennies Sally had…or dimes or pennies that Joe had. • The best we can do is to say that the three had a total of about $119. • In the same way, the results in an addition or subtraction can only be good to the least accurate place of any one of the numbers. • Examples: 5.514 + 22.33 22.34 + 195.63 • Rule for Multiplication and division with significant figures – Round the answer to the same number of significant figures as the original measurement with the fewest significant figures. – In other words, your answer can’t have more significant figures than the number with the fewest significant figures. • Examples: Multiply or divide the following, maintaining significance. 43.224 x 22.3 = 13.1 x 102 = 22.0 x 0.0015 = Section 2.3 Assessment Determine the number of significant figures in the following: 8,200, 723.0, and 0.01. A. 4, 4, and 3 A 0% D D. 2, 4, and 1 C C. 2, 3, and 1 A. A B. B C. C 0% 0% 0% D. D B B. 4, 3, and 3