Definitions of Error & Measurements

Accuracy The accuracy of an instrument is the extent to which the reading it gives might be wrong. Accuracy is often quoted as a percentage of the full-scale deflection (f.s.d.) of the instrument. For example, a one foot long school laboratory scale is 12 inch long and its smallest width between two parts is 1 millimetre. Thus we can not measure the length lesser than one millimetre. Therefore, it has full scale value of 12 inch with accuracy of ±1 millimetre. Accuracy of an instrument is equal to its least count. Lesser the least count, higher the accuracy and vice-versa. Solved Problem To measure the diameter of a wire, a Vernier Caliper with least count 0.01mm and a Screw Gauge with least count 0.001mm is used. Which is more accurate? Solution The least counts of Vernier Caliper and Screw Gauge are 0.01mm and 0.001mm respectively. The least count of Screw Gauge is lesser than that of Vernier Caliper. This is why, Screw Gauge is more accurate.

3. 5 Average A number expressing the central tendency or typical value in a set of data. There are different ways of measuring central tendency, such as mode, median or most commonly the mean (average). Average is calculated by dividing the sum of the values in a set by their counts in that set. Avg = Sum Counts Solved Problem Find average of 12, 14, 17, 24, 17 and 15. Solution Average of these values is Avg = 12 + 14 + 17 + 24 + 17 + 15 6 = 16.83 This is desired average. Solved Problem Find the average of x : x ∈ I, 0 < x < 10. Solution The given set is x : x ∈ I, 0 < x < 10; x = {1, 2, 3, 4, 5, 6, 7, 8, 9}

4. 6 Average of x is x̄ = P x 9 = 45 9 = 5 This is average of set x. Solved Problem Find the average of x : x ∈ I, 0 ≤ x < 10. Solution The given set is x : x ∈ I, 0 ≤ x < 10; x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Average of x is x̄ = P x 10 = 45 10 = 4.5 This is average of set x.

5. 7 Calibration Calibration can be defined as the process of determining the rela- tionship between the values of the quantity being measured and what the instrument indicates. Calibration of an instrument is carried out by comparing the readings of instrument with the reading of the standard instruments for the same physical quan- tity. For example, one kilogram weighing mass is calibrated with the mass of the object placed in Paris for universal weight measurement.

6. 8 Deviation Deviation is defined as the act of departing from the accepted practice or the norm. In Physics, deviation means depart from the accepted value. For example, a student measures diameter of a thin wire (true dia dr = 0.0124 mm) as do = 0.0125mm. Then we shall say that there is a deviation in student’s measured value. The relative departed value is given by d = do − dr It may be positive or negative. The magnitude of departed value (absolute deviation) is given by |d| = |do − dr| It is always positive. Remember that deviation is always deviation, whether measured value is less than or more than true value.

7. 9 Environmental Error Environmental errors are errors which can arise as a result of environmental effects which are not taken account of, e.g., a change in temperature affecting the value of a resistance.

8. 10 Error The error of a measurement is the difference between the result of the measurement and the true value of the quantity being measured. A student measures diameter of a thin wire (true dia dr = 0.0124 mm) as do = 0.0125mm. It shows, there is an error in measurement that results deviation. Error in measurement is given by e = do − dr It may be positive or negative. The magnitude of error (absolute error) is given by |e| = |do − dr| It is always positive. Remember that error is always error, whether measured value is less than or more than true value.

9. 11 Hysteresis Instruments can give different readings for the same value of measured quantity according to whether that value has been reached by a continuously increas- ing change or a continuously decreasing change. This effect is called hysteresis and it occurs as a result of such things as bearing friction and slack motion in gears in instruments.

10. 12 Instrument Error Instrument is a device that is manufactured by assembling of smaller tool parts. Thus it is not perfectly rigid. There is a tolerable looseness at the joints, hinges, sliding scale, rotors etc. Electrical instruments has also tolerance in its electrical components. These tolerances cause departure of measured value from the true value. By any means, these tolerances can not be eliminated. Therefore, during measurement, instruments add errors in measured quantity due to tolerance of their component. This error added up into the measured quantity due to instruments own manufacturing tolerance is called instrument error. Solved Problem A scale has centimetre and millimetre marks and it is 20 centimetre long. A student bends it several times that deforms its length by 6mm homogeneoulsy. He measures length of his pen as 15 centimetres by using this scale, then find what is actual length of pen? Solution True length of scale is 20 centimeters. On deformation its length is increased by 6mm. Hence true length of scale becomes 20.6 centimeters while its observation length remains 20 centimeters as there is no alteration in mark labels and mark length increases on increase of physical length of scale. Now, when observed length measured by this scale is 20 centimeters, its true length will be 20.6 centimeters.

11. 13 Now, when observed length measured by this scale is 1 centimeter, its true length will be 20.6/20 centimeters. Hence, when observed length of pen is 15 centimeters then its true length would be l = 20.6 20 × 15 = 15.45cm The true length of pen is 15.45 centimeters.

12. 14 Insertion Error Insertion errors are errors which result from the insertion of the instrument into the position to measure a quantity affecting its value. For example, inserting an ammeter into a circuit to measure the current will change the value of the current due to the ammeter’s own resistance

13. 15 Mean Mean is one of the tools measuring central tendency of the given data. It is calculated by dividing the sum of the values in a set by their counts in that set. Mean = Sum Counts Mean can be applied with both grouped and non-grouped data. Solved Problem Find the mean of x : x ∈ I, 0 ≤ x < 7 and y : y ∈ I, 0 ≤ y < 4. Solution From the given sets, x and y set are x : x ∈ I, 0 ≤ x < 7; x = {0, 1, 2, 3, 4, 5, 6} and y : y ∈ I, 0 ≤ y < 4; y = {0, 1, 2, 3} The combined set of x and y is S = {0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6} Now, mean is S̄ = P S 11 = 27 11 = 2.4545

14. 16 This is mean of the given sets x and y.

15. 17 Primary Standards There are primary standards for mass, length, time, current, temperature and luminous intensity which are accepted by international agreements and are maintained by national establishments.

16. 18 Range The range of an instrument is the limits between which it can made readings. For example, range of one foot long scale of school lab is from one millimetre to twelve inch. Solved Problem In a one meter long scale, there are centimetre marks, inch marks and millimetre marks. Find the range of this scale. Solution The maximum length of meter scale that can be measured by it is one meter and minimum length measured by it is its least count which is one millimeter. Hence the range of this scale is 1mm to 1m. Solved Problem The least count of school Screw Gauge is 0.001mm and its main scale has marks upto 12 millimetres. Find the range of Screw Gauge. Solution The maximum length of screw gauge that can be measured by it is twelve millimeter and minimum length measured by it is its least count which is 0.001mm. Hence the range of this scale is 0.001mm to 12mm.

17. 19 Repeatability The repeatability of an instrument is its ability to display the same reading for repeated applications of the same value of the quantity being measured. Solved Problem Can a scale measure same length in summer and winter? Explain your answer with suitable reason. Solution No. A scale can not measure the same length in summer and win- ter as its length changes in summer due to elongation under effect of heat (risen temperature). Solved Problem Do a scale has repeatability when length of same piece of cloth is measured in summer and winter? Explain your answer with suitable reason. Solution No. Scale’s length differ in summer and winter due to elongation of scale under the condition of risen temperature, hence the length measurement of same piece of cloth can not be replicated in these two seasons.

18. 20 Reliability The reliability of an instrument is the probability that it will operate to an agreed level of performance under the conditions specified for its use.

19. 21 Reproducibility The reproducibility or stability of an instrument is its ability to display the same reading when it is used to measure a constant quantity over a period of time or when that quantity is measured on a number of occasions.

20. 22 Resolution The resolution or discrimination of an instrument is the smallest change in the quantity being measured that will produce an observable change in the reading of the instrument. The determination of the resolution of a measuring in- strument is depends on the how finely smallest division (some time called least count, smallest scale division etc) is made on scale.

21. 23 Sensitivity The sensitivity of an instrument is given by ratio of the change in instrument scale reading to the change in the quantity being measured. Its unit depends on the units of inputs and outputs. In other words, the sensitivity of the instrument being measured is the ratio of the percentage change in the output quantity to the percentage change in the input quantity of the measuring instrument. ξ = ∆O ∆I Solved Problem When input voltage to a voltage driven electrical machine is changed by 10V , output current through it is changed by 0.5A. Find the sensitivity of that machine. Solution Sensitivity of a system is given by ratio of the change in instrument scale reading to the change in the quantity being measured. So, Sensitivity of machine is ξ = 0.5 10 = 0.05A/V This is desired result.

22. 24 Solved Problem When input voltage to a heating element is changed by 10V , heat emitted by it is changed by 50J. Find the sensitivity of heating element. Solution Sensitivity of a system is given by ratio of the change in instrument scale reading to the change in the quantity being measured. So, Sensitivity of heating element is ξ = 50 10 = 5J/V This is desired result. Solved Problem A variation of acceleration by 4 meter per second square cause deflec- tion of pointer in accelerometer by 20◦ angle. Find the sensitivity of the accelerometer. Solution Sensitivity of a system is given by ratio of the change in instrument scale reading to the change in the quantity being measured. So, Sensitivity of ac- celerometer is ξ = 20 4 = 5◦ s2 /m This is desired result.

23. 25 Threshold The minimum quantity of measurement, that required before mea- suring instrument can response to it and able to give detectable reading, is called threshold. Solved Problem For a pair of surfaces, maximum coefficient of static friction is 0.6 and coefficient of dynamic friction is 0.15. Find the threshold coefficient of friction for not slipping of the surfaces. Solution Threshold is that value at which system undergo from one state to other state. Here, we have to find the threshold coefficient of friction of not slipping. It means, coefficient of friction at which surfaces start slipping. From the given problem, the threshold value is µ = 0.15.

24. 26 Variation Variation refers to the differences or deviations from the recognized norm or standard. It may be a modification in structure, form or function in an organism, deviating from other organisms of the same species or group.