Measurement And Error

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


Step 1: Calculate the Average 893 cm/sec2

936 cm/sec2

1048 cm/sec2

915 cm/sec2

933 cm/sec2

Average = 945 cm/sec2


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


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





Average Absolute Deviation: 206 cm/sec2 / 5 = 41 cm/sec2


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

Measurement And Error

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