measurement and error
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Measurement and Error
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
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)
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.
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
Propagation of Error 2. The object being measured
3. The proximity to the object being measured
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
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
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
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
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
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
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
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
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
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
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.
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:
1. Smallest divisions on a measuring tool
2. The size of the object being measured
3. The difficulty in measuring a particular object