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**Accuracy and Precision**

With Significant Figures

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Accuracy Accuracy – how closely a measurement agrees with an accepted value. For example: The accepted density of zinc is 7.14 g/cm3 Student A measures the density as 5.19 g/cm3 Student B measures the density as 7.01 g/cm3 Student C measures the density as 8.85 g/cm3 Which student is most accurate?

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**Error All measurements have some error.**

Scientists attempt to reduce error by taking the same measurement many times. Assuming no bias in the instruments. Bias – A systematic (built-in) error that makes all measurements wrong by a certain amount. Examples: A scale that reads “1 kg” when there is nothing on it. You always measure your height while wearing thick-soled shoes. A stopwatch takes half a second to stop after being clicked.

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**Error Error = experimental value – accepted value**

Example: The accepted value for the specific heat of water is J/gºC. Mark measures the specific heat of water as J/gºC. What is Mark’s error? Error = exp.value – acc.value Error = J/gºC – J/gºC Error = J/gºC

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**Percent Error │error │ %Error = x 100% acc.value**

Example: The accepted value for the molar mass of methane is g/mol. Jenny measures the molar mass as g/mol. What is Jenny’s percent error? First, find the error: Error = exp.value – acc.value = g/mol – g/mol Error = g/mol Percent error = │error │/ acc.value x 100% Percent error = (1.048 g/mol) / ( g/mol) x 100% Percent error = x 100% Percent error = 6.533%

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Precision Precision – describes the closeness of a set of measurements taking under the same conditions. Good precision does not mean that measurements are accurate.

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**Accuracy and Precision**

Decent accuracy, but poor precision: the average of the shots is on the bullseye, but they are widely spread out. If this were a science experiment, the methodology or equipment would need to be improved. Good precision, but poor accuracy. The shots are tightly clustered, but they aren’t near the bullseye. In an experiment this represents a bias. Good accuracy and good precision. If this were a science experiment, we would consider this data to be valid.

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**Accuracy and Precision**

Another way to think of accuracy and precision: Accuracy means telling the truth… Precision means telling the same story over and over. They aren’t always the same thing.

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**Accuracy and Precision**

Four teams (A, B, C, and D) set out to measure the radius of the Earth. Each team splits into four groups (1, 2, 3, and 4) who compile their data separately, then they get back together and compare measurements. Their data are presented below: Team A Team B Team C Team D Group 1 Group 2 Group 3 Averages km km km km km km km km km km km km km km km km km km km km

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**Which team’s data were most precise? **

Team A Team B Team C Team D Group 1 Group 2 Group 3 Averages km km km km km km km km km km km km km km km km km km km km Which team’s data were most precise? Team B’s data was most precise, because their measurements were very consistent. Which team’s data were most accurate? We can’t say yet, because we don’t know the accepted value for the radius of Earth. The accepted value is km. % Error of Team A = 3.686% % Error of Team B = 4.174% % Error of Team C = 0.130% % Error of Team D = 1.283% Team C was the most accurate team, even though their data weren’t the most precise.

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**An Easy Method to Avoid Producing Misleading Results**

Significant Figures An Easy Method to Avoid Producing Misleading Results

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A thought problem Suppose you had to find the density of a rock. Density = mass / volume You measure the rock’s mass as g You measure the rock’s volume as 9.3 cm3 When you type / 9.3 into the calculator, you get g/cm3 Should you really write all those digits in your answer, OR Is the precision of your answer limited by your measurements?

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**A thought problem The calculator’s answer is misleading.**

You don’t really know the rock’s density with that much precision. It’s scientifically dishonest to claim that you do. Your answer must be rounded to the most precise (but still justifiable) value. How do scientists round numbers to avoid giving misleading answers?

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A thought problem Scientists use the concept of significant figures to give reasonable answers. We will use sig.figs. in class to practice good science. If a scientist divided g by 9.3 cm3, he or she would report the answer as 4.9 g/cm3. Let’s find out why.

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**It’s Easy and Fast! Only two rules: One for adding and subtracting.**

One for multiplying and dividing.

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**When Adding or Subtracting**

Note the precision of the measurements Nearest 0.1? ? ? The result should have as many decimal places as the measured number with the smallest number of decimal places.

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**For Example 5.51 grams + 8.6 grams**

Round answer to nearest tenth of a gram. Calculator gives: g You write: 14.1 g

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**For Example 52.09 mL – 49 mL Round answer to nearest milliliter.**

Calculator gives: 3.09 mL You write: 3 mL

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**When Multiplying or Dividing**

You must count significant figures (sig.figs.). The result should have as many sig.figs. as the measured number with the least number of sig.figs.

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**Counting Sig. Figs. All digits are significant EXCEPT:**

Zeroes preceding a decimal fraction and Zeroes at the end of a number that has no decimal point.

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**For Example 0.0045 has 2 significant figures, BUT**

Can you see the difference?

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**For Example 45.50 has 4 sig.figs. while 45.5000 has 6 sig.figs.**

but has only 1 sig.fig.

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**Numbers With No Decimals are Ambiguous**

Does 5000 mL mean exactly 5000? Maybe...maybe not. So 5000, 500, 50, and 5 are all assumed to have one significant figure. If a writer means exactly 5000 mL, he or she must write mL or 5.000x103 mL

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**How many sig.figs. in each number?**

2000 mL 0.2 mL 20.00 mL 20 mL 52.50 mL mL mL mL 4.0 cm 40 mm 40. mm mm

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**Now let’s do some math! 5.0033 g + 1.55 g**

Answer rounded to nearest hundredth of a gram. Answer: 6.55 g Do you need to count sig.figs.? No. Not in this problem.

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Try this one... 4.80 mL – mL Answer rounded to nearest hundredth of a milliliter. Answer: 4.80 mL You might say that mL is insignificant compared to 4.80 mL

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**Another one... 5.0033 g / 5.0 mL Did you have to count sig.figs.?**

Answer must have 2 sig.figs. Answer: 1.0 g/mL Did you have to count sig.figs.? Yes. Because you are dividing, you must count sig.figs.

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**One more... 50.0 cm x 0.04000 cm Did you have to count sig.figs.?**

Answer must have 3 sig.figs. Answer: 2.00 cm2 Did you have to count sig.figs.? Yes!

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**A few special cases How many minutes are in 3.55 hours?**

1 hour = 60 minutes, so... 3.55 hours = 3.55 x 60 minutes = ??? How many sig.figs. in answer? Conversion factors do not limit sig.figs. There are exactly 60 minutes in 1 hour. Only instruments and equipment do! Answer = 213 minutes

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**A few special cases How many sig.figs. are in the number 4.50x103?**

Answer: 3 sig.figs. In scientific notation, the 10 and the exponent are not considered significant. But all of the digits in the base are sig.figs.

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**What do you do in this situation?**

400. m x 50.0 m Answer should have 3 sig.figs. Calculator gives: m2 You write: ??? Can’t write m2 Only 1 sig.fig. Can’t write m2 Has 5 sig.figs. Can’t write Not the right answer! Solution: Either write the number in scientific notation: 2.00x104 m2 Or write the number with a bar over the last sig.fig.: m3

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