Uncertainty and Significant Figures in Scientific Measurements Cartoon courtesy of Lab-initio.com

Uncertainty in Measurement We collect a lot of data in science. A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.

Why Is there Uncertainty in Numbers? Measurements are performed with instruments and tools No instrument or tool can read to an infinite number of decimal places Which of these balances would have the greatest uncertainty in measurement? The greatest accuracy? 1.0g 1.000g

Precision and Accuracy Accuracy: how close your value is to the true value. Precision: how close several measurements are to each other if made in the same manner. Garbage can demo Label the following as accurate, precise, neither or both. Precise ONLY not accurate BOTH Precise AND accurate Neither

Applying Accuracy and Precision to a Scientific Problem The following data was obtained by students in a laboratory experiment regarding density. The small metal object that was weighed had an actual mass of 25.11 grams. What do you think about the accuracy and precision of each data set given this information? Data Set 1 Data Set 2 Data Set 3 Data Set 4 24.06 grams 25.12 grams 23.76 grams 26.51 grams 28.09 grams 25.09 grams 23.80 grams 25.08 grams 29.56 grams 25.14 grams 23.78 grams 23.63 grams Neither Both Precision Neither

Average of readings + (Range/2) To know just how Precise measurements are (How close or how uncertain) Precision using range can be used. How precise were your measurements? Precision using range formula: COPY IT! Precision = Average of readings + (Range/2) For example: 1.5g + .02g How precise was this measurement? We use the uncertainty of +/- to determine this. *The bigger the +/- value the LOWER the precision! Close to 0 means HIGH precision!

Ex. You and your partner measure the height of a velociraptor as 106cm, 104cm. 104cm and 103cm? Let’s try it! Formula Substitute Solve! 104.25 +/- 1.5 cm *104 +/- 1 cm Is really correct with sig figs 104 +/- 1.5 cm How precise was this groups measurement? *The bigger the +/- value the LOWER the precision! Close to 0 means HIGH precision!

Accuracy: Experimental Error There is ALWAYS error in experiments. Acceptable reasons for error on labs Limitations in the measuring devices Contaminated chemicals Temperature and pressure conditions Air resistance Anything you cannot control is ok. But Never EVER Human Error or saying “I messed up”. If there is human error, you must redo the experiment without the human error.

To know just how accurate measurements were made and how much error there is we calculate Percent Error. For example, if asked “How accurate was your answer?” The best way to answer is something like “I had a percent error of 5.4%.” Percentage of Error Formula: COPY IT! *The lines mean ‘absolute value’ so no + or – usually

Accuracy and Precision WS Ex. The boiling point of ethanol was found three times and averaged to be 76.35°C. If the accepted boiling point of ethanol is 78.1°C, then what is the percent error? Let’s try it together! FSS! Formula –substitute – solve! % Accuracy and Precision WS %Error HW 2.24% Therefore how accurate was the measurement? 97.76%

Significant Figures: Are really just a way to represent the accuracy of a measurement 1 ft vs 1.125 ft 6ft 6.254 ft Your measurements can only be as accurate as your worst tool.

Rules for Counting Significant Figures - Details 1. Nonzero integers Numbers from 1-9 always count as significant figures. 123456789 3456 has 4 significant figures

Rules for Counting Significant Figures - Details 2. Zeros 1 - Leading zeros (BEFORE numbers) do not count as significant figures. 0000000011234654789 Ex. 0.0486 has 3 significant figures Alcohol water jug demo

Rules for Counting Significant Figures - Details 3. Zeros 2 - Captive zeroes ‘trapped between’ between numbers always count as significant figures. 100000007 50078900123 Ex. 16.07 has 4 significant figures

Rules for Counting Significant Figures - Details 4. Zeros 3 Trailing zeros (at the end) are significant only if the number contains a decimal point. 50000000 no and 500.00000 yes Ex. 9.300 has 4 significant figures Why else put zeroes there?

Rules for Counting Significant Figures - Details 5. Exact numbers technically have an infinite number of significant figures. So you DON’T consider them when figuring out sig figs. Ex. 1 inch = 2.54 cm, exactly 760 torr = 1013.6 Pa These have INFINITE ACCURACY! SO infinite sig figs IGNORE THEM!

Rules for Counting Significant Figures - Details 6. Scientific notation: the initial number is significant but the rest are considered trailing or leading zeroes so.. 1.30098 x10-5 Ex. 1.25 x103 has… 3 Sig Figs 1.00715 x10-7 has… 6 Sig Figs…

Sig Fig Practice #1 1.0070 m  5 sig figs 17.10 kg  4 sig figs How many significant figures in each of the following? Write down the question and your answer NO CALLING OUT! 1.0070 m  5 sig figs 17.10 kg  4 sig figs 100,890 L  5 sig figs 3.29 x 103 s  3 sig figs 0.0054 cm  2 sig figs 3,200,000  2 sig figs

Rules for Significant Figures in your final calculation: 7. Multiplication and Division: # sig figs in the result equals the number with the least accurate measurement used in the calculation. It’s like using a poor tool to measure something. IT REDUCES ACCURACY 6.38 x 2.0 = 12.76  13 (2 sig figs)

Calculator Sig Fig Practice #2 Correct Answer Calculation Calculator says: 3.24 m x 7.0 m 22.68 m2 23 m2 100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3 0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft

Rules for Significant Figures in your Final Calculation 8. Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least accurate measurement. 6.8 + 11.934 = 18.734  18.7 (3 sig figs)

Calculator Sig Fig Practice #3 Correct Answer Calculation Calculator says: 3.24 m + 7.0 m 10.24 m 10.2 m 100.0 g - 23.73 g 76.27 g 76.3 g 0.02 cm + 2.371 cm 2.391 cm 2.39 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb Practice calculations and SF

Applying Sig Fig’s to Tools 31.0 0C All tools have uncertainty. The last digit of ANY tool you use MUST be estimated. The SIG FIG guidelines are as follows. 1. Go to one place past the smallest line 2. That is your “uncertain digit” 3. Even if your measurement is exactly on a line you must use the correct digits by adding 0’s to show what the tool COULD have measured. -9.5 0C 52.8 mL 373.35 g Reading tools and SF