Table to Find and Propagate Error or Uncertainty to Output from Inputs Based on Formula (Important)

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Rules for the evaluation of standard uncertainty through functional relationships with uncorrelated variables. Note how the error for both multiplication and division (6 and 4) is propagated using RMS value.

Standard Uncertainity

Rule

Notes (below)

Function

Expression giving standard uncertainty

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

,

16

,

17

,

18

,

19

20

21

22

23

[(ln (x1 / x2))2 / (x3 − x4)4][u2(x3) + u2(x4)]

24

,

(u(y) / y)2 = (D1u(x1) / x1)2 + (D2u(x2) / x2)2 + (u(a)/a)2 + (D1u(b)/b)2 + (D2u(c) / c)2

25

Notes:

1.A, B, C, D, D1, D2 and N (all upper case) are constants with no uncertainty. They may be integers such as 2 or 3, a decimal number, a mathematical constant such as π, negative or positive.

2.e is Euler’s number and ex is the exponential function. The value of e is approximately 2.7183.

3.log10e is approximately 0.4343.

4.a, b, c, w, w1 and w2 (all lower case) are uncorrelated (measured) variables with random uncertainty components.

5.x, x1, x2, x3, …, xn are all uncorrelated (measured) variables with random uncertainty components.