measurement and error
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Measurement and Error
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Error in MeasurementTypes of Error
Systematic – one that always produces an error of the same sign; positive is a reading too high and negative error is a reading too low
Random – occur as variations that are due to a large number of factors each of which adds to its own contribution of the total error. These errors are a matter of chance
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Types of Systematic Error
Instrumental Error – caused by faulty, inaccurate apparatus
Personal Error – caused by some peculiarity or bias of the observer
External Error – caused by external conditions (wind, temperature, humidity)
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Random ErrorRandom errors are subject to
the laws of chance. Taking a large number of observations may lessen their effect. When al errors are random, the value having the highest probability of being correct is the arithmetic mean or average.
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Propagation of Error Scientific measurements will always
contain some degree of uncertainty. This uncertainty will depend on:
1. The instrument(s) used to make measurements
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Propagation of Error 2. The object being measured
3. The proximity to the object being measured
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Variance The uncertainty of a measurement is
indicated showing the possible variance with a plus and minus factor.
Example: You measure the length of an object five times and record the following measurements
53.33 cm, 53.36 cm, 53.32 cm, 53.34 cm, & 53.38 cm
The average is 53.35 cm; this should be written as
53.35 ± .03 cm
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Errors in Addition and Subtraction
Example: 13.02 .04 cm23.04 .03 cm14.36 .03 cm26.89 .04 cm
77.31 .14 cm
The variance of the result is equal to the sum of all the individual variances
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Errors in Multiplication and Division
Example: 13.2 .2 cm x 23.5 .3 cm Maximum and Minimum: Maximum 13.4 cm x 23.7 cm = 319 cm2
Minimum 13.0 cm x 23.2 cm = 302 cm2
Average = 310. cm2
Answer 310. 9 cm2
The variance MUST be large enough to include both
maximum and minimum
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Accuracy The closeness of a measurement to the accepted
value for a specific physical quantity. Accuracy is indicated mathematically by a number referred to as error.
Absolute Error (EA) = (Average of observed values) – (Accepted Value) Relative Error (ER) = X 100%
lueAcceptedVarorAbsoluteEr
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Precision The agreement of several measures that have been
made in the same way. Precision is indicated mathematically by a number referred to as deviation.
Absolute Deviation (DA) = (Each observed value) – (Average of all values)
Relative Deviation (DR) = x 100%llValuesAverageofA
tionoluteDeviaAverageAbs
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Example for Measuring Error and Deviation
Measured Values: 893 cm/sec2 936 cm/sec2
1048 cm/sec2
915 cm/sec2
933 cm/sec2
Accepted Value: 981 cm/sec2
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Example for Measuring Error and Deviation
Step 1: Calculate the Average 893 cm/sec2
936 cm/sec2
1048 cm/sec2
915 cm/sec2
933 cm/sec2
Average = 945 cm/sec2
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Example for Measuring Error and Deviation
Step 2: Calculate Absolute and Relative Error Absolute Error (EA) = (Average of observed values) – (Accepted Value) EA = 945 cm/sec2 – 981 cm/sec2 = 36 cm/sec2
Relative Error (ER) = x 100 %
ER = x 100% = 3.7 %2
2
sec/981sec/36
cmcm
lueAcceptedVarorAbsoluteEr
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Example for Measuring Error and Deviation
Step 3: Calculate Absolute and Relative Deviations Absolute Deviation (DA) = (Each Observed Value) – (Average of All Values)
DA = 893 cm/sec2 – 945 cm/sec2 = 52 cm/sec2
DA = 936 cm/sec2 – 945 cm/sec2 = 9 cm/sec2
DA = 1048 cm/sec2 – 945 cm/sec2 = 103 cm/sec2
DA = 915 cm/sec2 – 945 cm/sec2 = 30 cm/sec2
DA = 933 cm/sec2 – 945 cm/sec2 = 12 cm/sec2
Average Absolute Deviation: 206 cm/sec2 / 5 = 41 cm/sec2
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Example for Measuring Error and Deviation
Relative Deviation: Relative Deviation (DR) = X
100%
DR = x 100% = 4.3%
llValuesAverageofAtionoluteDeviaAverageAbs
2
2
sec/945sec/41
cmcm
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Significant Figures Usually, you will estimate one digit
beyond the smallest division on the measuring tool if the object you are measuring has a well defined edge.
When reading a measurement that someone else has made, you must determine if the digits he/she has written down are significant to the measurement.
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Significant FiguresThose digits in an observed
quantity (measurement) that are known with certainty plus the one digit that is uncertain or estimated.
The number of significant figures in a measurement depends on:
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1. Smallest divisions on a measuring tool
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2. The size of the object being measured
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3. The difficulty in measuring a particular object