Table 2.

Rules for the evaluation of standard uncertainty through functional relationships with uncorrelated variables.

Rule	Notes (below)	Function	Expression giving standard uncertainty
1		y = x1 + x2	u2(y) = u2(x1) + u2(x2)
2		y = x1 − x2	u2(y) = u2(x1) + u2(x2)
3	1	y = A + Bx1 + Cx2 … + Nxn	u2(y) = B2u2(x1) + C2u2(x2) +... + N2 u2(xn)
4		y = x1/x2	(u(y)/y)2 = [(u(x1)/x1)2 + (u(x2)/x2)2]
5	1	y = Ax1/Bx2	(u(y)/y)2 = [(u(x1)/x1)2 + (u(x2)/x2)2]
6		y = x1 × x2	(u(y)/y)2 = [(u(x1)/x1)2 + (u(x2)/x2)2]
7	1	y = Ax1 × Bx2	(u(y)/y)2 = [(u(x1)/x1)2 + (u(x2)/x2)2]
8		y = (x1 × x2) / (x3 × x4)	(u(y)/y)2 = [(u(x1)/x1)2 + (u(x2)/x2)2 + (u(x3)/x3)2 + (u(x4)/x4)2]
9	1	y = xA	(u(y) / y) = |A|(u(x) / x)
10	1	y =(x1 / x2)A	(u(y)/y)2 = A2 [(u(x1)/x1)2 + (u(x2)/x2)2]
11	1	y = (x1)A × (x2)B	(u(y)/y)2 = A2 (u(x1)/x1)2 + B2(u(x2)/x2)2
12		y = ln x	u(y) = u(x) / x
13	1	y = A + ln x	u(y) = u(x) / x
14	1	y = A + ln Bx1 + ln Cx2	u2(y) = (u(x1) / x1)2 + (u(x2) / x2)2
15	1, 2, 3	y = log10 xA = A log10 x	u(y) = |A|(u(x) / x) log10 e
16	1, 2, 3	y = A + log10 x	u(y) = (u(x) / x) log10 e
17	1, 2, 3	y = A + log10 Bx1 + log10 Cx2	u2(y) = (log10 e)2 [(u(x1) / x1)2 + (u(x2) / x2)2]
18	1, 2	y = AeBx	u(y) / y = |B|u(x)
19	4	y = xw	${\left(u\left(y\right)/y\right)}^2=w^2\left({\left[\frac{u(x)}{x}\right]}^2+{\left(\text{ln}x\right)}^2{\left[\frac{u(w)}{w}\right]}^2\right)$
20	4	y = (x1 / x2)w writing q = x1/x2 for brevity	${\left(u\left(y\right)/y\right)}^2=w^2\left({\left[\frac{u(x_1)}{x_1}\right]}^2+{\left[\frac{u(x_2)}{x_2}\right]}^2+{\left(\text{ln}q\right)}^2{\left[\frac{u(w)}{w}\right]}^2\right)$
21	5	y = f(x1, x2,..., xn) g(x1, x2,..., xn)	u2(y) = (g∂f / ∂x1 + f∂g / ∂x1)2 u2 (x1) + (g∂f / ∂x2 + f∂g / ∂x2)2 u2(x2) +...
			... + (g∂f / ∂xn + f∂g / ∂xn)2 u2(xn)
22	5	y = f(x1, x2,..., xn) / g(x1, x2,..., xn)	u2(y) = (1/ g)4 [(g∂f / ∂x1 − f∂g / ∂x1)2 u2 (x1) + (g∂f / ∂x2 − f∂g / ∂x2)2 u2(x2) +...
			... + (g∂f / ∂xn − f∂g / ∂xn)2 u2(xn)]
23		y = (ln x1 − ln x2) / (x3 − x4)	u2(y) = [1/ (x3 − x4)2][(u(x1) / x1)2 + (u(x2) / x2)2]+
			[(ln (x1 / x2))2 / (x3 − x4)4][u2(x3) + u2(x4)]
24	1, 4	y = a(bx1)D1 (cx2)D2	(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	4	y = a(bx1)w1 (cx2)w2	(u(y) / y)2 = (w1u(x1) / x1)2 + (w2u(x2)/ x2)2 + (u(a)/ a)2 + (w1u(b) / b)2 + (w2u(c) / c)2
			+ (ln bx1)2 u2(w1) + (ln cx2)2 u2(w2)

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 e^x 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, …, x_n are all uncorrelated (measured) variables with random uncertainty components.