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**How Reliable Are Measurements?**

Section 2.3 How Reliable Are Measurements?

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**Objectives Define and compare accuracy and precision.**

Use significant figures and rounding to reflect the certainty of the data. Use percent error to describe the accuracy of experimental data.

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Accuracy Accuracy refers to how close a measured value is to an accepted value. Which of the following is the MOST ACCURATE measurement for the density of iron (7.87 g/cm3)? ￭ 7.90 g/cm3 ￭ 8.00 g/cm3 ￭ 7.85 g/cm3 ￭7.50 g/cm3

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Precision Precision refers to how close a series of measurements are to each other. Find p. 48, Table 3. Which student had the MOST PRECISE data? Are precise measurements always accurate? See p. 47, Fig. 10.

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**What factors account for imprecise or inaccurate data?**

Was the procedure followed consistently? Were careful measurements taken? Was the data recorded correctly? Were calculations done correctly? Was equipment free of contamination? Was equipment calibrated before use? Were all environmental conditions constant?

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Percent Error Experimental values are those measured during an experiment. Values that are considered true are called accepted values. Error is the absolute value of the difference between the experimental and accepted values. Percent Error = Expt. Value – Accepted Value x 100 Accepted value

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Practice Problems Student B measured the density of an unknown substance at 1.51 g/cm3. The substance was sucrose, with an accepted density value of 1.59 g/cm3. What is Student B’s percent error? Student C measured the density of the sucrose at 1.70 g/cm3. What is Student C’s percent error?

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Significant Figures Precision of a measurement is limited by the tools used to make that measurement. For example, unless you have a very sophisticated stopwatch, the time you read will be limited to the nearest second. This is because most watches/clocks only have a second hand.

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Significant Figures Data must be reported in the number of digits that reflects the precision of the tools used. Significant figures (also called significant digits) include all known (readable) digits plus one estimated digit. See Fig. 12on p What is the length of the rod, in the correct number of sig. figs?

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**RULES FOR RECOGNIZING SIGNIFICANT FIGURES**

Non-zero #s are always significant. (56.7 m has 3 sig. figs.) Zeros between sig. digits are significant. (102 has 3 sig. figs.) All final zeros to the right of the decimal are significant. (38.0 has 3 sig. figs.) Zeros that act as placeholders are NOT SIGNIFICANT. (0.094 has 2 SFs; 100 has 1 SF) Counting #s & defined constants have an infinite # of sig. figs. (5 atoms; 1 hr = 60 min.)

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Practice Problems Determine the number of sig. figs. in each measurement below. 807,000 cm mg 1.250 x 10-3 m mL 93.0 g s

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**Sig. Figs. In Math. Operations**

In Addition & Subtraction Your answer must have the same number of digits to the right of the decimal as the value with the fewest digits to the right of the decimal. 28.0 cm cm cm = cm The value with the fewest digits to the right of the decimal is The answer must therefore but rounded to the same number of places cm is the answer.

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**Footnote on “Rounding”**

Look to the digit immediately to the right of the last significant figure. If this number is less than 5, leave the significant figure as it is. g becomes g in 4 sig. figs. If it is 5 or greater, round up the last significant figure. g becomes 2.36 g in 3 sig. figs. and 2.4 in 2 sig. figs.

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**Sig. Figs. In Math. Operations**

In Multiplication & Division Your answer must have the same number of significant figures as the measurement with the fewest sig. figs. 3.20 cm x 3.65 cm x 2.05 cm = cm3 Each measurement has 3 sig. figs. So your answer must have 3 sig. figs. The answer is, therefore, 23.9 cm3.

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Practice Problems Complete the following problems. Round off the answers to the correct number of sig. figs. 5.236 cm – 3.14 cm 1.23 m x 2.0 m 102.4 m/51.2 s 24 m x 3.26 m/168 m 3 x (2 x 105)/8.4 x 106

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