Error, their types, their measurements Presented By:Anu BalaG.P.C. Khunimajra(Mohali)

What is an error?For example,by not using the equipment correctlyLets look at some examples.

Human errorExample 1Professor Messer is trying to measure the length of a piece of wood:Discuss what he is doing wrong.How many mistakes can you find? Six?

Human errorMeasuring from 100 end95.4 is the wrong number mm is wrong unit (cm)Hand-held object, wobblingGap between object & the ruleEnd of object not at the end of the ruleEye is not at the end of the object (parallax)He is on wrong side of the rule to see scale.Answers:How many did you find?

Human errorExample 2Reading a scale:Discuss the best position to put your eye.your eye

Human error2 is best.1 and 3 give the wrong readings.This is called a parallax error.your eyeIt is due to the gap here, between the pointer and the scale.Should the gap be wide or narrow?

Anomalous resultsWhen you are doing your practical work, you may get an odd or inconsistent or anomalous reading.This may be due to a simple mistake in reading a scale.The best way to identify an anomalous result is to draw a graph.For example . . .

Anomalous resultsLook at this graph:Which result do you think may be anomalous?A result like this should be taken again, to check it.

ERRORSIf we are making physical measurements, there is always error involved. The error is notated by using the delta, , symbol followed by the variable representing the quantity measured.For example, if we are measuring volume, the error in measuring the volume would be symbolized V.

Calculating the ErrorA simple way of looking at the error is as the difference between the true value and the approximate value.i.e: Error (e) = True value Approximate value

Example:Find the truncation error for at x= if the first 3 terms in the expansion are retained.

Sol: Error = True value Approx value

TYPE OF ERRORS

Type of errorsGross error/human ErrorsRandom ErrorsSystematic ErrorsConstant ErrorsAbsolute ErrorsRelative ErrorsPercentage Errors

Static Errors

1) Gross Errorcause by human mistakes in reading/using instrumentsmay also occur due to incorrect adjustment of the instrument and the computational mistakescannot be treated mathematicallycannot eliminate but can minimizeEg: Improper use of an instrument.This error can be minimized by taking proper care in reading and recording measurement parameter.In general, indicating instruments change ambient conditions to some extent when connected into a complete circuit.Therefore, several readings (at three readings) must be taken to minimize the effect of ambient condition changes.

TYPES OF STATIC ERROR

TYPES OF STATIC ERROR (cont)2) Systematic Error -due to shortcomings of the instrument (such as defective or worn parts, ageing or effects of the environment on the instrument)In general, systematic errors can be subdivided into static and dynamic errors.Static caused by limitations of the measuring device or the physical laws governing its behavior.Dynamic caused by the instrument not responding very fast enough to follow the changes in a measured variable.

-3 types of systematic error :-(i) Instrumental error(ii)Environmental error(iii)Observational error

TYPES OF STATIC ERROR (cont)

TYPES OF STATIC ERROR (cont)(i) Instrumental error - inherent while measuring instrument because of their mechanical structure (eg: in a DArsonval meter, friction in the bearings of various moving component, irregular spring tension, stretching of spring, etc)- error can be avoid by:(a) selecting a suitable instrument for the particular measurement application(b) apply correction factor by determining instrumental error (c) calibrate the instrument against standard

TYPES OF STATIC ERROR (cont) (ii)Environmental error- due to external condition effecting the measurement including surrounding area condition such as change in temperature, humidity, barometer pressure, etc- to avoid the error :- (a) use air conditioner (b) sealing certain component in the instruments (c) use magnetic shields

(iii) Observational error- introduce by the observer - most common : parallax error and estimation error (while reading the scale)- Eg: an observer who tend to hold his head too far to the left, while reading the position of the needle on the scale.

TYPES OF STATIC ERROR (cont)3) Random error - due to unknown causes, occur when all systematic error has accounted - accumulation of small effect, require at high degree of accuracy - can be avoid by (a) increasing number of reading (b) use statistical means to obtain best approximation of true value

What is systematic error?Systematic error is caused by any factors that systematically affect measurement of the variable across the sample. For instance, if there is loud traffic going by just outside of a classroom where students are taking a test, this noise is liable to affect all of the children's scores -- in this case, systematically lowering them. Unlike random error, systematic errors tend to be consistently either positive or negative -- because of this, systematic error is sometimes considered to be bias in measurement.

These errors cause readings to be shifted one way (or the other) from the true reading.Systematic errorsYour results will be systematically wrong.Lets look at some examples . . .

Example 1Suppose you are measuring with a ruler:Systematic errorsIf the ruler is wrongly calibrated, or if it expands,then all the readings will be too low (or all too high):

Example 2If you have a parallax error: Systematic errorswith your eye always too highthen you will get a systematic errorAll your readings will be too high.

A particular type of systematic erroris called a zero error.Systematic errorsHere are some examples . . .

Example 3A spring balance:Zero errorsOver a period of time, the spring may weaken, and so the pointer does not point to zero:What effect does this have on all the readings?

Example 4Look at this top-pan balance:Zero errorsThere is nothing on it, but it is not reading zero.What effect do you think this will have on all the readings?It has a zero error.

Example 5Look at this ammeter:Zero errorsIf you used it like this, what effect would it have on your results?

Example 6Look at this voltmeter:Zero errorsWhat is the first thing to do?Use a screwdriver here to adjust the pointer.

Example 7Look at this ammeter:Zero errorsWhat can you say?Is it a zero error?Or is it parallax?

Example 8Look at this ammeter:Zero error, Parallax errorWhat is it for?How can you use it to stop parallax error?It has a mirror behind the pointer, near the scale.When the image of the pointer in the mirror is hidden by the pointer itself, then you are looking at 90o, with no parallax.

What is random error?Caused by any factors that randomly affect measurement of the variable across the sample.Each persons mood can inflate or deflate their performance on any occasion. In a particular testing, some children may be in a good mood and others may be depressed. Mood may artificially inflate the scores for some children and artificially deflate the scores for others.Random error does not have consistent effects across the entire sample. If we could see all the random errors in a distribution, the sum would be zero. The important property of random error is that it adds variability to the data but does not affect average performance for the group.

Random ErrorsRandom errors are not inherent to the measuring process.

Frequently they are introduced by external factors that cause a scattering of the measured data.

When the scattering is distributed equally about the true value, the error can be mitigated somewhat by making multiple measurements and averaging the data.

Vibration in mechanical devices produces random errors. In electronic devices, noise produces random errors.

Random errorsThese may be due to human error,a faulty technique, or faulty equipment.To reduce the error,take a lot of readings, and then calculate the average (mean).

Constant ErrorWhen the results of observation are in error by the same amount, the error is said to be a constant error. e.g. if a scale of 15 cm actually measures 14.8 cm. Then it is measuring 0.2 cm more in every observation. This type of error will be same in all measurements done by the scale.

Another types of ErrorThree other ways of defining the error are:

Absolute errorRelative errorPercentage error

Calculation the Absolute ErrorAbsolute error. ea = |True value Approximate value|

Calculating the ErrorAbsolute error: ea = |True value Approximate value|

Relative error is defined as:

*Absolute ErrorThe difference between the measured value and the true value is referred to as the absolute error. Assume that analysis of an iron ore by some method gave 11.1% while the true value was 12.1%, the absolute error is:11.1% - 12.1% = -1.0%

*Relative Error

The relative error is the percentage of the absolute error to the true value. For the argument above we can calculate the relative error as:Relative error = (absolute error/true value)x100% = (-1.0/12.1)x100% = -8.3%

*Relative AccuracyThe percentage of the quotient of observed result to the true value is called relative accuracy.Relative accuracy = (observed value/true value)x100%For the abovementioned example:Relative accuracy = (11.1/12.1)x100% = 91.7%

Calculating the ErrorPercentage error is defined as:

ExamplesSuppose 1.414 is used as an approx to . Find the absolute, relative and percentage errors.

ExamplesSuppose 1.414 is used as an approx to . Find the absolute, relative and percentage errors.

ExamplesSuppose 1.414 is used as an approx to . Find the absolute, relative and percentage errors.

Example:True value = 122 mmexpected value = 120 mm

Then:a. absolute error = True value - expected value absolute error = 122 mm 120 mm = 2 mm Ans

b. relative error = absolute error / expected value relative error = 2 mm / 120 mm = 0.017 Ans Note: relative error has no units.

c. percent error = relative error 100% percent error = 0.017 100% = 1.7 % Ans

The following are general classifications for errors:

For consumer purposes, 5-10% error is acceptable For engineering purposes, 1% error is acceptable For scientific purposes, 0.1% error is acceptableClassification of Error

Range of uncertainty is reported as a nominal value plus or minus an amount called the tolerance. Reported value: 120 mm 1 mm = 119 mm to 121 mmRange of Uncertaintynominal valuerange of uncertainty

Range of uncertainty is also reported as a nominal value plus or min Range of uncertainty is reported as a nominal value plus or minus an amount called the tolerance us an percent tolerance. Reported value 120 mm 2% = 117.6 mm to 122.4 mm

Note: 2% of 120 = 2.4, 120 - 2.4 = 117.6, 120 +2.4 = 122.4Range of Uncertaintyrange of uncertainty

PERFORMANCE CHARACTERISTICSAccuracy the degree of exactness (closeness) of measurement compared to the expected (desired) value. Resolution the smallest change in a measurement variable to which an instrument will respond.Precision a measure of consistency or repeatability of measurement, i.e successive reading do not differ. Sensitivity ratio of change in the output (response) of instrument to a change of input or measured variable.Expected value the design value or the most probable value that expect to obtain.Error the deviation of the true value from the desired value.

Measurement precision must be interpreted in light of measurement accuracy. Lets use a target practice example:Precision Target 1

The best situation, the shots are tightly clustered (high precision) on the center circle (high accuracy).

Measurement precision must be interpreted in light of measurement accuracy. Lets use a target practice example:Precision Target 2

The next situation, shots are near the center (high accuracy), but not tightly clustered (low precision).

Measurement precision must be interpreted in light of measurement accuracy. Lets use a target practice example:Precision Target 3

In the next situation, a tight cluster (high precision) is far off center (low accuracy).

Measurement precision must be interpreted in light of measurement accuracy. Lets use a target practice example:Precision Target 4

Finally, widely scattered shots (low precision) appear away from the center (low accuracy).

Which is the best and which is worst?Precision - ComparisonBestMost InsidiousWhy?Worst

Example

Given expected voltage value across a resistor is 80V.The measurement is 79V. Calculate,The absolute errorThe % of errorThe relative accuracyThe % of accuracy

Solution (Example)

Given that , expected value = 80V measurement value = 79V

i. Absolute error, e = = 80V 79V = 1V

ii. % error = = = 1.25%

iii. Relative accuracy, = 0.9875

iv. % accuracy, a = A x 100% = 0.9875 x 100% = 98.75%

Example From the value in table 1.1 calculate Table 1.1 the precision of 6th measurement?

Solution the average of measurement value

the 6th reading Precision =

NoXn1982101310249751016100710389891061099

LIMITING ERRORThe accuracy of measuring instrument is guaranteed within a certain percentage (%) of full scale readingE.g manufacturer may specify the instrument to be accurate at 2 % with full scale deflectionFor reading less than full scale, the limiting error increases

LIMITING ERROR (cont)Example Given a 600 V voltmeter with accuracy 2% full scale.Calculate limiting error when the instrument is used to measure a voltage of 250V?

Solution

The magnitude of limiting error, 0.02 x 600 = 12VTherefore, the limiting error for 250V = 12/250 x 100 = 4.8%

LIMITING ERROR (cont)Example Given for certain measurement, a limiting error for voltmeter at 70V is 2.143% and a limiting error for ammeter at 80mA is 2.813%. Determine the limiting error of the power.

Solution The limiting error for the power = 2.143% + 2.813% = 4.956%

Example What is the relative error in the approximation X = 2.0 to x = 1.98?

Solution

Relative error = = = 0.01010 (to 5d.p.)Xx1-21.981-

In summaryHuman errors can be due to faulty technique.Systematic errors, including zero errors, will cause all your results to be wrong.Random errors can be reduced by taking many readings, and then calculating the average (mean).Parallax errors can be avoided.Anomalous results can be seen on a graph.

THE END

SWTJC STEM ENGR 1201Content Goal 13Quality Factors

Accuracy "refers to how close the reported value comes to the true value Precision "refers to the clustering of a group of measurements

Scientists and engineers express measurement quality using,

Errors in measurement can be classified in two types:

1.Systematic Errors results from a measurement that is inherently wrong. 2. Random Errors results from the effect of external factors.SWTJC STEM ENGR 1201Content Goal 13Error Types

ABSOLUTE, RELATIVE, AND PERCENT ERRORThe actual error from the true value is called the absolute error.The relative error is the absolute error divided by total quantity. In the case of volume, .The percentage error is the relative error multiplied by 100.

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