Accuracy and Precision SIGNIFICANT FIGURES (Sig Figs)

1. Accuracy and Precision SIGNIFICANT FIGURES (Sig Figs)

2. Uncertainties in Scientific Measurement • All measurements are subject to error. • Types of Errors: • 1)  Systematic error: arise because to some extent, measuring instruments have built-in, or inherent, errors. • Random errors: they arise from intrinsic limitation in the sensitivity of the instrument and inability of observer to read a scientific instrument and give results that may be either too high or too low. • In talking about the degree of uncertainty in a measurement, we use the words accuracy and precision.

3. Accuracy and precision Accuracy: nearness to the actual value **based on the accuracy of the measuring device** Example: How many students are in the front row today? Swami Willets predicts there will be 8 students. How close I am reflects the accuracy of my measurement (or here, prediction) and the accuracy of the measuring device. Precision: reproducibility of a measurement Example: My 8 year old cousin sucks at bowling. She puts the ball in the gutter every time. Her aim may not be accurate but she has great precision!

4. Accuracy and precision: the target example Precise, but not accurate

5. Accuracy and precision: the target example Accurate, not precise

6. Accuracy and precision: the target example Neither accurate, nor precise

7. Accuracy and precision: the target example Accurate and precise

8. Beware of Parallax – the apparent shift in position when viewed at a different angle.

9. Significant Figures • The term Significant figures refers to digits that were measured. • When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers.

10. Significant Figures The rules for determining significant figures (sig. fig.). 1) Zeros in the middle of a numbers are significant figures. E.g. 4023 mL has 4 significant figures. 2) Zeros at the beginning of a number are not significant; they act only to locate the decimal point. E.g. 0.00206L has 3 significant figures. The zeros to the left of 2 are not sig. fig. 3) Zeros at the end of a number and after decimal point are always significant. E. g. 2.200 g has 4 sig. fig. 4) Zeros at the end of number and before the decimal point may or may not be significant figures

11. 5) A useful rule of thumb to use for determining whether or not zeros are significant figures is that zeros are not sig. fig. if the zeros disappear when scientific notation is used. E.g.0.0197 = 1.97 x10-2 the zeros are not sig. fig.; 0.01090 = 1.090 x10-2 the first two zeros are not significant and the second two are. The number 0.004004500 has 7 sig. Fig.

12. Examples • 32445 = 5 sig figs • 0.23435 = 5 sig figs • 2348.23 = 6 sig figs • 0.0023 = 2 sig figgs • 0.02300 = 4 sig figs • 1.009 = 4 sig figs • 230,004 = 6 sig figs • 100 = 1 sig fig • 1.00 x 102 = 3 sig figs • 100. = 3 sig figs

13. EXACT NUMBERS These are non-measured numbers (8 atoms), numbers in formulas (2r2), and numbers from definitions (1 in=2.54 cm). Exact numbers have an infinite number of sig figs. • 10 molecules = infinite sig figs • 4/3r3 = infinite sig figs •  = infinite sig figs

14. Significant Figures in Numerical Calculations 3) Addition and Subtraction: • The result of addition or subtraction must be expressed with the same digits beyond the decimal point as the quantity carries the smallest number of such digits. E.g.3.18 + 0.01315 3.19 • When use scientific notion, All numbers are converted to the same power of 10, and the digit terms are added or subtracted. E.g. (4.215 x 10-2) + (3.2 x 10-4) = (4.215 x 10-2) + (0.032 x 10-2) = 4.247 x 10-2 2)Exact numbers can be considered to have an unlimited number of sig. fig. There are two situations when a quantity appearing in a calculation may be exact. • By definition such as 1 in = 2.54 cm • Counting such as six faces on a cube or two hydrogen atoms in a water molecule.

15. Addition Subtraction

16. Significant Figures in Numerical Calculations 1) Multiplication or Division The result of multiplication or division, may contain only as many sig. fig. as the least precisely known quantity in the calculation. E.g. 14.79cm x 12.11cm x 5.05cm = 904cm (4 sig.fig) (4 sig. fig) (3 sig. fig) (3 sig fig.) If use scientific notion, E.g. (3.4 x 106)(4.2 x 103) = (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 1010 (to 2 significant figures) the digit terms are multiplied in the normal way and the exponents are added. The end result is changed so that there is only one nonzero digit to the left of the decimal.

17. Multiplication 3 sigfigs 2 sigfigs 2 sigfigs Division 4 sigfigs 1 sigfig 1 sigfig

18. Rounding Numbers *In a series of calculations, carry the extra digits through to the* final result, THEN round 3.05 x 5.555 x 3.0 = 50.82825 = 51 NOT 3.1 x 5.6 x 3.0 = 52.08 = 52 1) If the first digit you remove is less than 5, round down by drop it and all following digits. E.g. 5.664525 5.66 when rounded to three sig. fig. 2) If the first digit you remove is greater than 5 or 5 followed by nonzero, round up by adding 1 to the digit on the left. E.g. 5.664525 5.7 when rounded to two sig. fig.

19. Rounding Numbers 3) What happens if there is a 5 or 5 following by zeros? There is an arbitrary rule “round 5 to even”: If the number before the 5 is odd, round up. If the number before the 5 is even, let it be. The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out. E.g. Round the following number to 2 significant figures: (a) 2.35 x 102 (b) 2.45 x 102 (Answer: 2.4 x 102)