Error, their types, their measurements

What is an error? Some are due to human error… For example, by not using the equipment correctly Let’s look at some examples.

Human error Example 1 Professor 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 error Measuring from 100 end 95.4 is the wrong number Answers: Measuring from 100 end 95.4 is the wrong number ‘mm’ is wrong unit (cm) Hand-held object, wobbling Gap between object & the rule End of object not at the end of the rule Eye is not at the end of the object (parallax) He is on wrong side of the rule to see scale. How many did you find?

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

Human error 2 is best. 1 and 3 give the wrong readings. This is called a parallax error. your eye It is due to the gap here, between the pointer and the scale. Should the gap be wide or narrow?

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

ERRORS If we are making physical measurements, there is always error involved. The error is notated by using the delta, Δ, 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 ΔV.

Calculating the Error A 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 Static Errors Type of errors Gross error/human Errors Random Errors Systematic Errors Constant Errors Absolute Errors Relative Errors Percentage Errors Static Errors

TYPES OF STATIC ERROR 1) Gross Error cause by human mistakes in reading/using instruments may also occur due to incorrect adjustment of the instrument and the computational mistakes cannot be treated mathematically cannot eliminate but can minimize Eg: 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 (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.

TYPES OF STATIC ERROR (cont) - 3 types of systematic error :- (i) Instrumental error (ii) Environmental error (iii) Observational error

TYPES OF STATIC ERROR (cont) (i) Instrumental error - inherent while measuring instrument because of their mechanical structure (eg: in a D’Arsonval 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.

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

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

Systematic errors Example 2 If you have a parallax error: with your eye always too high then you will get a systematic error All your readings will be too high.

Systematic errors A particular type of systematic error is called a zero error. Here are some examples . . .

Zero errors Example 3 A spring balance: Over 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?

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

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

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

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

Zero error, Parallax error Example 8 Look at this ammeter: It has a mirror behind the pointer, near the scale. What is it for? How can you use it to stop parallax error? 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 person’s 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 Errors Random 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 errors These 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 Error When 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 Error Three other ways of defining the error are: Absolute error Relative error Percentage error

Calculation the Absolute Error ea = |True value – Approximate value|

Calculating the Error Absolute error: ea = |True value – Approximate value| Relative error is defined as:

Absolute Error The 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 Accuracy The 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 Error Percentage error is defined as:

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

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

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

Example: True value = 122 mm expected 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

Classification of Error 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 acceptable

Range of Uncertainty 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 mm tolerance nominal value range of uncertainty

Range 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 = 122.4 tolerance nominal value range of uncertainty

PERFORMANCE CHARACTERISTICS Accuracy – 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. 48

Precision – Target 1 Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example: The best situation, the shots are tightly clustered (high precision) on the center circle (high accuracy).

Precision – Target 2 Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example: The next situation, shots are near the center (high accuracy), but not tightly clustered (low precision).  

Precision – Target 3 Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example: In the next situation, a tight cluster (high precision) is far off center (low accuracy).

Precision – Target 4 Measurement precision must be interpreted in light of measurement accuracy.  Let’s use a target practice example: Finally, widely scattered shots (low precision) appear away from the center (low accuracy).

Precision - Comparison Which is the best and which is worst? Most Insidious Why? Best Worst

Example Given expected voltage value across a resistor is 80V. The measurement is 79V. Calculate, The absolute error The % of error The relative accuracy The % 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 = No Xn 1 98 2 101 3 102 4 97 5 6 100 7 103 8 9 106 10 99

LIMITING ERROR The accuracy of measuring instrument is guaranteed within a certain percentage (%) of full scale reading E.g manufacturer may specify the instrument to be accurate at 2 % with full scale deflection For 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 = 12V Therefore, 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.) X x 2 1.98 - 1 - 1

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

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