Error Analysis: Significant Figures Calculated Quantities (Other (Not Listed) Sample)

Error analysis
Student’s name
Institutional Affiliation
Lab Report
Objectives
The main objectives of the experiment are:
To learn how to determine the proper number of significant figures in calculated quantities.
To understand how errors propagate in calculations.
To develop an appreciation for the importance of proper error analysis in determining the accuracy and precision of final results.
Introduction and Theory
Error analysis is the study of uncertainties. The process of error analysis has two main functions: it acts as a guide in determining and reducing measurement uncertainties, and it provides mathematical methods for assessing the uncertainty of a quantity calculated using values obtained through measurement.
We know from experiment #1 that any measurement will have an associated uncertainty (or “error”). The uncertainty in any measured quantity depends upon: 1. the precision of the measuring instrument, 2. the skill and technique of the measurer, and 3. the consistency or steadiness of the quantity being measured. In practice, the measurement uncertainty is limited by whichever of the three sources listed above is dominant. In other words, if any one of these three factors is very uncertain or poor, then the overall measuring precision will be correspondingly reduced, even if the other two factors are very precise (Jasak & H., 1996). When making a measurement, professional will make every attempt to reduce these uncertainties and to make reliable estimates of how large they actually are.
This laboratory experiment introduces a method of error analysis which should be used throughout the course. Both written examples are presented which demonstrate the propagation of uncertainties, and an actual measurement experiment demonstrates that increased precision of a measuring instrument can radically affect the uncertainties in calculated results (Taylor & J., 1997). Before learning how to account for uncertainties in calculations, let’s consider the concept of significant figures.
1 Significant Figures
* Writing Numbers
A significant figure is a digit which has physical significance. It is essential to know how to write numbers so that the number of significant figures is properly articulated. The canon is that the digits 1upto 9 are always significant. All zeroes to the left of the decimal point and all foremost zeroes are not significant, but trailing zeroes to the right of the decimal point are significant.
NOTE: when a number is correctly written in scientific form, all the digits are significant.
Table 2.1 provides some examples:
Number

Scientific Notation

Significant Figures

6048

6.048×103

4

0.0000006

6×10-7

1

0.9000006

9,000006×10-1

7

7

7×100

1

7.0

7.0×100

2

7.00

7.00×105

3

500000

5×105

1

5000001

5.00001×105

6

500000.0

5.000000×105

7

* Significant Figures of computed Quantities
When using measured values in computations, the number of significant figures in the measured values influences the number of significant figures that should be used when reporting the computed value. Some basic rules are:
* Do not round off intermediate values determined during calculation; simply the reported value should be rounded to the right number of significant figures.
* Round uncertainties to one significant figure in a reported result. An exception is when the leading digit is a 1, in which case two significant figures are acceptable.
* The value and its uncertainty ought to have the similar accuracy.
1 Multiplication and division. The required number of significant figures in a product or quotient is equal to the least number of significant figures in any of the factors. Consider the following calculation of the area of a rectangle:
area=5.6 metres ×3.9 metres=21.84m2
The input factors have only two significant figures each, so the product can have only two significant figures. The area is therefore reported as
Area=22m2
Example:58.613.79=4.25
Example: π×5.84×2.6=48
The number of significant figures in the ultimate result is the minimum number of significant figures in any factor or divisor.1181100798830In- class Exercise: 22.8×4×42.0×7.65=29,302.56In-Class Exercise: (22.8×3.9)÷(42.0×7.65)=0.2767507003020000In- class Exercise: 22.8×4×42.0×7.65=29,302.56In-Class Exercise: (22.8×3.9)÷(42.0×7.65)=0.2767507003
2 Addition and Subtraction. The required number of significant figures in a addition or subtraction is establish by rounding off the result at the rightmost decimal place of the least accurate quantity involved in the calculation. If a measured temperature of 6°C is expressed in kelvin temperature scale where 0°C=273.16K, the accuracy is limited by the units (the “ones”) place precision of the measured temperature. The reported result is that the measured temperature is 279K; i.e. 6+ 273.16=279.16=279K
Example : 5.832+2.09+38.97+0.2+1.641=48.7
This result is accurate only to tenths place because 0.2 is accurate only to the tenths place.
Example:18.323-18.30=0.02=2×10-2
Notice that this result has only one significant figure, even though each of the values in the calculation has four and five significant figures.
center0In-Class Exercise3: 15.8 + 2.97 + 13.2 – 5.84 -18.6 = 7.53020000In-Class Exercise3: 15.8 + 2.97 + 13.2 – 5.84 -18.6 = 7.53
II. Propagation of uncertainty
An important outcome of error analysis is determining the propagation of uncertainty- the estimation of the uncertainty of quantities calculated from the experimental data. Our method of determining how uncertainty propagates through a calculation is based on the premise that, when we report a measured quantity as x±δx, we are claiming that any additional measuremens of the same quantity under the same measurement conditions will result in a measurement value between x-δx and x+δx.
1 Adding and Subtracting Data with Uncertainty.
center0If z=x+y or z=x-y, then δz=δx+δy.020000If z=x+y or z=x-y, then δz=δx+δy.
When two measured quantities are added or subtracted, the uncertainty of the result is the sum of the individual measured uncertainties (Taylor & J., 1997).
Reasoning: suppose we have two measured quantities x±δx and y±δy. If z=x+y,then the greatest value z could possibly attain is zmax=xmax+ymax=x+δx+y+δy=x+y+(δx+δy)=z +𝛿z, and the least z could be is is zmin=xmin+ymin=x-δx+y-δy=x+y-(δx+δy)=z -𝛿z,
Since the range is z-𝛿z, we have δz=δx+δy.
Using similar mathematics, if w=x-y, then δw=δx+δy
2 Multiplying and dividing data with uncertainty
center0if z=xy or z=xy,thenδz|x|+δy|y|.4000020000if z=xy or z=xy,thenδz|x|+δy|y|.
When two measured quantities are multiplied or divided, the absolute fractional uncertainty of the result is the sum of the individual measurement absolute fractional uncertainty.
3 Powers of data with uncertainty.
center0if z=xn, then δz=|nzδxx|4000020000if z=xn, then δz=|nzδxx|
The recipe is equally valid for both integer and fractional n.
4 Functions of data with uncertainty
center0if z=fx, then δz=zhigh-zlow2=fx+δx-fx-δx2020000if z=fx, then δz=zhigh-zlow2=fx+δx-fx-δx2
To determine the uncertainty when the calculation involves a function, first determine the uncertainty for the function’s argument, and then use the high/low method to determine the uncertainty of the function itself. Note the absolute value: fx+δxcould give either z max or z min.
5 Calculation with a mixture of operations.
center0When calculations involve several types of operations, the uncertainty is determined using a step to step process, where each calculation step involves one of the above uncertainty calculations.020000When calculations involve several types of operations, the uncertainty is determined using a step to step process, where each calculation step involves one of the above uncertainty calculations.
* Precision in the Final Report Result
The reported result and its associated uncertainty must be consistent in their precision
Apparatus
* Ruler
* Vernier caliper
* Micrometer
* Plastic plate
Procedure
1 First use a plastic ruler to measure the length, width and thickness of the object.
2 Use a vernier caliper to make the same measurements.
3 Use a micrometer to measure the thickness of the object
Note that all the measurements made with a given instrument have the same uncertainty.
Analysis of data
DATA

 

















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