Uncertainty in measurement & Significant figures
Difference between Precision & Accuracy-The measurement involving counting of identifiable object can be done accurately but many scientific measurements cannot be done accurately. The accuracy of any measurement depends upon (i) the accuracy of the measuring device used & (ii) the skill of the operator. So the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual (true) value. The precision of a measurement system, also called repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.
If the average value of different measurement is close to true value than the measurement is said to be accurate(the separate measurement may not close to each other) & If the values of different measurement are close to each other & close to their average value than the measurement is said to be precise. For example-let the actual length of an area is 10.5.
Four different persons done the 5 measurements follow:
|
Measurement
|
1
|
2
|
3
|
4
|
5
|
Average
|
|
A
|
10.3
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10.4
|
10.5
|
10.6
|
10.7
|
10.5
|
|
B
|
10
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10.1
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10.2
|
10.3
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10.4
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10.2
|
|
C
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10.1
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10.3
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10.5
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10.7
|
10.9
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10.5
|
|
D
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10
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10.7
|
10.9
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10.1
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10.3
|
10.8
|
Measurement by A is done with both good accuracy & good precision.
Measurement by B is done with poor accuracy but good precision.
Measurement by C is done with good accuracy but poor precision.
Measurement by D is done with poor accuracy & poor precision.
A measurement can have a good accuracy but poor precision because different measurement may give a correct average. The converse is not true. Good precision corresponding mean good accuracy.
Significant Figures-Experimental measurements have some uncertainty associated with them. One would always like the results to be precise & accurate. For example if the uncertainty length can be expressed by any experiment as 14.6+/- 0.1 cm.
A convenient method of expressing the uncertainty in measurement is to express it in terms of significant figure instead of using the notation +/-1.
Thus a measured quantity is expressed in terms of such a number which include all digits which are certain & last digit is uncertain. The total number of digits in the number is called the number of Significant figure.
The number of significant figures in a measurement is the number of figures that are known with the certainty plus one that is uncertain, beginning with the first non zero digit.
In order to determine the significant figures in a measured quantity, there are some rules which are as follow-
Rule-1 All non zero digits are significant.
For example, 165cm, 0.165,2006 & 9.05 has three, three, four, three significant figures respectively.
Rule-2 Zeros to the left of the first non-zero digit in the number are not significant. For example-0.005 g & 0.026 g has one & two significant number respectively.
Rule-3 Zeroes between non-zero digits are significant.e.g.2.05 has three significant figures.
Rule-4 Zeroes to the right of the decimal are significant.e.g.5.00 g, 0.050 g, 0.5000 g have three, two & four significant figure respectively.
Rule-5 If a number ends with zeros that are not to the right of a decimal, the zeros may or may not be significant.e.g.1500 g may have two, three or four significant figures.
The three possibilities of last rule can be removed by expressing the number in Scientific/Exponential notation. In this the number is written in the standard exponential form as N x 10n Where N=a number with a single non-zero digit to the left of the decimal point & n=an integer called exponent.
So for 1500 gm,1.5x103 g(2 significant figure),1.50 x 103g(3 significant figure),1.500 x 103 g(4 significant figure).In this exponential all the zeros to the right of the decimal point are significant.
Calculations with significant figures-While carrying out calculations with numbers which measured in quantitative analysis, the rule used is that the accuracy of the final result is limited to the least accurate measurement, In other words, final result can't be more accurate than the least accurate number involved in the calculation.
Rounding Off-When experimentally measured quantities are calculated, the final result contain many non-significant figures. When this happen, the final result are rounded off. In this the extra digits are dropped with or without minor changes in the figure retained. There are some rules for that which is as follow-
1-If the digit following the last digit is less than 5, and then the last digit is left unchanged.e.g.46.32 only two figures are to be retained as significant figures. The last digit is 6, following the figure 3 which is less than five, so 6 will remain as it is & the final result is 46.
2- If the digit following the last digit is more than 5, then the last digit is increased by one.e.g.52.87 is to be rounded off to three significant digits. The last digit is 8, following the figure 7 which is more than 5, so 8 would be increased by one & the final result is 52.9
3- If the digit following the last digit is equal to 5, then the last digit is left unchanged if it's even & it's increased by one if its odd.e.g. 1.235 & 1.225.in this final result is 1.24 1.22 respectively.
Calculation involving multiplication & division-In multiplication & division, the final result should be reported as having the same number of significant digits as the number with least number of significant digits. For example-6.26 x 5.8=36.308 rounded off to 36 as the number with least significant figure is 5.8(two significant digit).The final result is to be limited to two significant digit & is expressed as 36 after rounding off.
Similar for division 5.27/12= 0.439, final result is 0.44 as 12 has least number of significant figure two so final result is 0.44.
If in a calculation some exact number is involved, it itself is regarded to have infinite number of significant digits & the digits of the final result is limited by the other number.
e.g. 11(exact number) x 2.55 kg (3 significant figure) =28.05 kg. Rounded off to 28.0 kg (3 significant figure)
Calculation involving addition & subtraction-In addition & subtraction, the final result should be reported to the same number of decimal places as the number with the minimum number of decimal places. for example-35.52+10.3=45.82,rounded off to 45.8 as 10.3 has digits to least number of decimal places, namely one, so the answer is rounded off to only one decimal place & the final result is 45.8 only.
Similar in subtraction,3.56-0.021=3.539,rounded off to 3.54 as 3.56 has digits to least number of decimal place, two only so the result is to be limited to two decimal places & final result is 3.54.
Dimensional Analysis (Unit factor Method)-Any calculation involving the use of the dimension of the different physical quantities involved is called dimensional analysis. It is used to convert a physical quantity given in one type of unit into some other unit; the used method is called factor label method or unit factor method. It consist of following steps-
(i)First determine the unit conversion factor.
(ii)Multiply the given physical quantity with the unit conversion factor, retaining the units of physical quantity as well as that of the unit conversion factor in such a way that all the cancel out leaving behind only required units.
(iii)If the conversion involves a number of steps, each conversion factor is used in such a way that the units of preceding factor cancel out.
Original quantity (in former unit) ´conversion factor=equivalent quantity
This is based on the fact that ratio of each fundamental quantity in one unit with their equivalent quantity in other unit is equal to one
For example in case of mass
1 Kilogram / 2.205 pond = 1 = 1 Kilogram / 1000 gm
So 1 kg = 2.205 pond = 1000 gm
In this way any derived unit first expressed in dimension and each fundamental quantity like mass length time are converted in other system of desired unit to work out the conversion factor.
Similarly we can deduce other conversion factor for other quantity in different unit by the dimensional analysis method.
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