Error Compound Calculations
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or more quantities, each with their individual uncertainties, and then combine the information from these quantities in order to come up with a final result of our experiment. How can you state your answer for the combined result of these measurements and their uncertainties scientifically? error propagation formula The answer to this fairly common question depends on how the individual measurements are combined error propagation physics in the result. We will treat each case separately: Addition of measured quantities If you have measured values for the quantities X, Y,
Error Propagation Chemistry
and Z, with uncertainties dX, dY, and dZ, and your final result, R, is the sum or difference of these quantities, then the uncertainty dR is: Here the upper equation is an approximation that can also serve
Error Propagation Square Root
as an upper bound for the error. Please note that the rule is the same for addition and subtraction of quantities. Example: Suppose we have measured the starting position as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. Then the displacement is: Dx = x2-x1 = 14.4 m - 9.3 m = 5.1 m and the error in the displacement is: (0.22 + 0.32)1/2 m = 0.36 m Multiplication of measured quantities error propagation inverse In the same way as for sums and differences, we can also state the result for the case of multiplication and division: Again the upper line is an approximation and the lower line is the exact result for independent random uncertainties in the individual variables. And again please note that for the purpose of error calculation there is no difference between multiplication and division. Example: We have measured a displacement of x = 5.1+-0.4 m during a time of t = 0.4+-0.1 s. What is the average velocity and the error in the average velocity? v = x / t = 5.1 m / 0.4 s = 12.75 m/s and the uncertainty in the velocity is: dv = |v| [ (dx/x)2 + (dt/t)2 ]1/2 = 12.75 m/s [(0.4/5.1)2 + (0.1/0.4)2]1/2 = 3.34 m/s Multiplication with a constant What if you have measured the uncertainty in an observable X, and you need to multiply it with a constant that is known exactly? What is the error then? This is easy: just multiply the error in X with the absolute value of the constant, and this will give you the error in R: If you compare this to the above rule for multiplication of two quantities, you see that this is just the special case of that rule for the uncertainty in c, dc = 0. Exa
dividing Is one result consistent with another? What if there are several measurements of the same quantity? How can one estimate the uncertainty of a slope on a graph? Uncertainty in a single measurement Bob weighs himself on his bathroom scale. The smallest divisions on the scale are 1-pound marks, so the least count of
Propagated Error Calculus
the instrument is 1 pound. Bob reads his weight as closest to the 142-pound mark. He error propagation average knows his weight must be larger than 141.5 pounds (or else it would be closer to the 141-pound mark), but smaller than 142.5 pounds (or error propagation excel else it would be closer to the 143-pound mark). So Bob's weight must be weight = 142 +/- 0.5 pounds In general, the uncertainty in a single measurement from a single instrument is half the least count of the instrument. Fractional http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm and percentage uncertainty What is the fractional uncertainty in Bob's weight? uncertainty in weight fractional uncertainty = ------------------------ value for weight 0.5 pounds = ------------- = 0.0035 142 pounds What is the uncertainty in Bob's weight, expressed as a percentage of his weight? uncertainty in weight percentage uncertainty = ----------------------- * 100% value for weight 0.5 pounds = ------------ * 100% = 0.35% 142 pounds Combining uncertainties in several quantities: adding or subtracting When one adds or subtracts several measurements together, one http://spiff.rit.edu/classes/phys273/uncert/uncert.html simply adds together the uncertainties to find the uncertainty in the sum. Dick and Jane are acrobats. Dick is 186 +/- 2 cm tall, and Jane is 147 +/- 3 cm tall. If Jane stands on top of Dick's head, how far is her head above the ground? combined height = 186 cm + 147 cm = 333 cm uncertainty in combined height = 2 cm + 3 cm = 5 cm combined height = 333 cm +/- 5 cm Now, if all the quantities have roughly the same magnitude and uncertainty -- as in the example above -- the result makes perfect sense. But if one tries to add together very different quantities, one ends up with a funny-looking uncertainty. For example, suppose that Dick balances on his head a flea (ick!) instead of Jane. Using a pair of calipers, Dick measures the flea to have a height of 0.020 cm +/- 0.003 cm. If we follow the rules, we find combined height = 186 cm + 0.020 cm = 186.020 cm uncertainty in combined height = 2 cm + 0.003 cm = 2.003 cm ??? combined height = 186.020 cm +/- 2.003 cm ??? But wait a minute! This doesn't make any sense! If we can't tell exactly where the top of Dick's head is to within a couple of cm, what difference does it make if the flea is 0.020 cm or 0.021 cm tall? In technical terms, the number of significan
propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to https://en.wikipedia.org/wiki/Propagation_of_uncertainty the combination of variables in the function. The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, the positive square root of variance, σ2. The value of a quantity and its error are error propagation then expressed as an interval x ± u. If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is approximately a 68% probability that the error compound calculations true value lies in the region x ± σ. If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.[1] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Caveats and warnings 2.3.1 Reciprocal 2.3.2 Shifted reciprocal 3 Example formulas 4 Example calculations 4.1 Inverse tangent function 4.2 Resistance measurement 5 See also 6 References 7 Further reading 8 External links Linear combinations[edit] Let { f k ( x 1 , x 2 , … , x n ) } {\displaystyle \ ρ 5(x_ ρ 4,x_ ρ 3,\dots ,x_ ρ 2)\}} be a set of m functions which are linear combinations of n {\displaystyle n} variables x 1 , x 2 , … , x n {\displaystyle x_ σ 7,x_ σ 6,\dots ,x_ σ 5} with combination coefficients A k 1 , A k 2 , … , A k n , ( k = 1 … m ) {\displaystyle A_ σ 1,A_ σ 0,\dots ,A_ ρ 9,(k=1\dots m)} . f k = ∑ i n A k i x i or f = A x {\displaystyle f_ ρ 5=\sum _ ρ 4^ ρ 3A_ ρ 2x_ ρ 1{\text{ or }}\