Significant Figures and Uncertainty Applications



Significant Figures and Uncertainty Applications – Summary by Mrs. Vital

Significant Figures (also known as Significant Digits) – Indicate the precision of an instrument. In order to determine the number of sig. figs. in a number, do the following:

1. Go to the right end of a number that HAS A DECIMAL POINT in it;

2. Count LEFT including the last digit until EITHER:

• You run out of numbers; or,

• You count and include the last non-zero digit.

3. For numbers lacking decimal points, ONLY the non-zero digits are significant and any zeros sandwiched in between them.

Examples: 0.00125 = _________ 125.00 = __________ 3.245000 = _______ 300____

0.000000000090 = __________ 6.02 X1023 =___________ 2.54 = __________

1200 = ______ 1.200 X 103 = _________ 1.20 X 103 = _________ 1.2 X 103 = _________

501000 = ______ 101=______ 33=_______ 0.56 = ________ 0.003 = _________

*** Note: All non-zero digits are always significant.***

Multiplication/Division: The number of sig. figs. in a final answer for both operations is the SAME number of sig. figs. as in the LEAST number of digits in any piece of data.

(23.4 mol)(1.2 g·mol-1) = ___________ (4.1384 m)/(298 m·s-1) = ___________

NOTE: EXACT NUMBERS DO NOT HAVE SIG. FIGS. They are conversions treated as having INFINITE sig. figs. MEASURED and REPORTED numbers have sig. figs.

Addition/Subtraction: The answer is reported to the same position as the piece of data with the least precise POSITION.

(21.3545 m + 31.2 m + 25. m) = ______________ 76.4354 m – 22.13 m = _______________

Sig. Figs. coupled with an assigned Uncertainty indicate the precision and accuracy of an instrument. In the BEST CASE, an assigned uncertainty is 0.2 of the smallest division on the instrument. What are some factors that can determine the assigned uncertainty that is needed on a measurement?

1. How well the instrument is suited to the measuring task; ruler for a marble’s diameter – poor choice!

2. How defined the edges of the object are; ruler for a shoe length – best you can do.

3. Skill of the experimenter – reading at the eye’s level as opposed to above or below the object.

4. The sophistication of the instrument – What is the smallest division on it (precision)?

5. The calibration of the instrument to ensure no systematic errors.

Uncertainty – the plus or minus value ( + ) that MUST be associated with every written and measured value during all experiments from now on. You assign it while considering the factors listed on the front. The + value gives you the range in which the exact, correct value will lie. It is NOT a percent value! It is a 1 significant digit value in the LAST position of the data is modifies. JUST as important as assigning uncertainty values is being able to correctly propagate your assigned values throughout a series of calculations. Over(

Absolute Uncertainty- the + value that is assigned from the instrument that was used to make the measurement combined with the conditions of the measurement as explained on the front. If the instrument is digital, often the value is + 1 in the final decimal position displayed on the device screen.

Relative Uncertainty- is called “relative” because it is a percent and it will increase and decrease with the size of the piece of data that it describes. 56m + 5% means 56 + 2.8m; 84m + 5% means 84 + 4.2m … SEE??? The size of the absolute uncertainty that is associated with a relative uncertainty changes with the data it is associated with; the same percent will produce a larger absolute uncertainty on a larger piece of data as in the example with 56m and 84m. Relative uncertainties do not have to be rounded to 1 sig. fig. as you will USE them to FIND the absolute uncertainty on an answer. 3 or 4 digits is usually sufficient; 2 is too few with percents and 5 is excessive.

For FINAL absolute uncertainties on answers that are the result of Addition/Subtraction operations, the + value on the answer is the SUM of the Absolute Uncertainties that are associated with each of the data, provided all the data are reported or converted into the same units.

For FINAL absolute uncertainties on answers that are the result of Multiplication/Division operations, do the following:

1. Determine the percent each absolute value is of its associated piece of data;

2. Add all the percents to give you the relative uncertainty on the final answer; and, since you are to only ever report final answers with their correct, associated absolute uncertainties, you must;

3. Multiply the answer by the percent value and divide by 100. THIS value IS the + value that is to be written in association with your answer.

Uncertainty SUMMARY: You always ADD!!! ADD, ADD, ADD!!! Either you are adding absolute uncertainties or you are adding percent uncertainties but you are always ADDING.

Uncertainties Assigned when Average of Data is Taken- First, each piece of data has it’s own absolute uncertainty resulting from the precision of the instrument. So, determine if the range of the AVERAGED data is larger than the precision of the instrument. HOW?

➢ IF the range of data is LARGER than the instrument’s precision: 1. Find the average value of the data. Subtract the average or MEAN from the LARGEST contributing piece of data AND subtract the lowest piece of contributing data to that mean FROM the average. The uncertainty that you must assign is the + that is the larger of the differences from the average value. If they are not about equal, you probably have an outlier contributing to your mean so carefully consider the data. You may need more trials.

➢ IF the resulting uncertainty is not larger than the value the instrument can give alone, then to assign the + on the average you simply put the + that reflects the precision of the instrument.

➢ IF taking measurements only one time (very poor experimental practice), then if you are adding the values you take the uncertainty assigned to each measured value, square EACH uncertainty for each contributing piece of data; NEXT, add all the squared uncertainties together; FINALLY, take the square root of that sum and THIS is the uncertainty that is to be assigned to the average of the data. Same rounding rules apply; 0.5 or greater rounds UP into one sig. fig. while less than 0.5 is to round down.

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***Note: Absolute uncertainties are to have only 1 SIG. FIG.! They must always round up or down to have ONE SIGNIFICANT DIGIT (following the normal rounding rules). The position of the ONE SIG FIG of the uncertainty value is exactly the SAME position you MUST make as the last position of precision of the data it reflects as the + range in data as a result. IF you have an uncertainty with a value in the SAME position as the piece of data it goes with, THEN you round into the digit in THAT position; otherwise, you are to round to the nearest number in that least certain position of the data. So, 24.508 m2 with an uncertainty result of 0.029 m2 MUST BE rounded up to 0.03 m2 and thus 24.508 MUST become 24.51 for it’s final position of precision to reflect the 1 sig fig of the uncertainty value. The final result is then: 24.51 + 0.03 m2.

Also, you must be logical so THINK THINK THINK about your answers!

PRECISION = How many digits to the right of a decimal can be reported… more = MORE precise.

ACCURACY = How close to the TRUTH or theoretically accepted value a calculated or measured value is … closer = MORE ACCURATE.

Systematic Errors: Calibration errors that cause all measured data to be a fixed value lower or higher than the truth (Higher means the instrument is reading high, Lower if reading low). Systematic errors CANNOT be reduced by repeating measurements.

Random Errors: Human errors or instrumental incorrect usage errors. Random errors CAN be reduced by repeat measuring with the equipment.

4 trials of measurements are recommended which can enable you to omit 1 result if it is a clear outlier from the other 3. 3 reproduced measurements is a statistical minimum for certainty. The average of 3 pieces of data can enable you to compare your range of answers against the inherent uncertainty of an instrument. For example, 25.03 g, 25.06 g, 25.04 g and 26.03 g. The instrument is designed to have a + 0.01g precision. The average without the 26.03g is 25.04g so + 0.02g must be assigned. Your measuring techniques prevented the result from being the BEST the instrument can do. That is FINE, but you MUST account for it. If you include 26.03g, you would need to report as: 25.29g for the average and then the + 0.74 g to reach 26.03 and – 0.26 g to reach 25.03 g. This is a VERY asymmetric mean between the high and low from the mean; thus, the HIGHEST value must be an outlier as the + to include the high is nearly 0.5 greater than the – value. SO, remove the high, re-average and check again until the + and – are symmetric within 1 of each other in a reasonable final Sig. Fig. position.

So again: Final bit of information to consider in working with uncertainties. Unless you have taken at least 4 pieces of data, you cannot consider any as being an outlier to be removed from the data set. If only 3 pieces of data, then you must include all in the average to determine if the instrumental uncertainty is sufficient or if you must determine from the SUBTRACTION method described under the “Uncertainties Assigned When the Average of Data is Taken”.

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Problem Solving Algorithm (General Steps!)

1. Read the problem to understand the situation that is presented.

2. Identify the given or known information on your paper with a labeled diagram.

3. Identify the unknown information, or what it is that you will try to solve for on your paper.

4. Determine and write down any formula relationships or unit conversions that you perceive you will need.

5. Solve the relations for the unknown via algebra FIRST in writing showing all steps.

6. Substitute in your data and calculate your answer showing all unit cancellations and consider your answer for the sense of it; in other words, ask yourself: Does this answer make sense given all the information I have and does it answer the question that is asked?

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