s(x_k) = \sqrt{\dfrac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}

s(\bar{x}) = \dfrac{s(x_k)}{\sqrt{n}}

s(x,y) = (1/(n-1)) \cdot \sum_{i=1}^{n} (x_i - \bar{X}) \cdot (y_i - \bar{Y})

s(\bar{x}, \bar{y}) = (1/(n \cdot (n-1))) \cdot \sum_{i=1}^{n} (x_i - \bar{X}) \cdot (y_i - \bar{Y})

rho(X,Y) = \dfrac{V(Y,X)}{sigma(Y) \cdot sigma(X)}

r(x,y) = \dfrac{s(x,y)}{s(x) \cdot s(y)}

Y = f(X1, X2, ..., XN)

y = f(x1, x2, ..., xN)

P = \dfrac{V^2}{R0 \cdot [1 + alpha \cdot (t - t0)]}

Y = X1 - X2

Y = X

y = (1/n) \cdot \sum_{k=1}^{n} f(x_{1k}, x_{2k}, ..., x_{Nk})

y = f(mean(x1), mean(x2), ..., mean(xN))

u_A = u(mean(x)) = s(mean(x)) = \dfrac{s(x)}{\sqrt{n}}

mean(x) = (1/n) \cdot \sum_{i=1}^{n} x_i

s^2(x_k) = (1/(n-1)) \cdot \sum_{i=1}^{n} (x_i - mean(x))^2

s(x_k) = \sqrt{(1/(n-1)) \cdot \sum_{i=1}^{n} (x_i - mean(x))^2}

u_A(mean(x)) = s(mean(x)) = \dfrac{s(x_k)}{\sqrt{n}}

s(x_k) = \dfrac{R}{C}

u_A(mean(x)) = s(mean(x)) = \dfrac{s(x_k)}{\sqrt{n}} = \dfrac{R}{C \cdot \sqrt{n}}

s(x) = s_p = \sqrt{\dfrac{\sum_{j=1}^{m} nu_j \cdot s_j^2}{\sum_{j=1}^{m} nu_j}}

s_p = \sqrt{\dfrac{\sum_{j=1}^{m} s_j^2}{m}}

u_A(x) = u(mean(x)) = \dfrac{s_p}{\sqrt{n}}

\sqrt{(1/(m \cdot (n-1))) \cdot \sum_{i=1}^{m} \sum_{j=1}^{n} (x_{ij} - mean(x_i))^2}

\sqrt{(1/m) \cdot \sum_{i=1}^{m} s_i^2}

\sqrt{(1/\sum_{i=1}^{m} nu_i) \cdot \sum_{i=1}^{m} nu_i \cdot s_i^2}

\dfrac{s_p(x_k)}{\sqrt{n'}}

\dfrac{s(x)}{\sqrt{n'}}

\dfrac{a}{k}

\dfrac{delta_x}{2 \cdot \sqrt{3}}

0.5 \cdot (Delta[u(x_i)]/u(x_i))^(-2)

u_c(y) = \sqrt{\sum_{i=1}^N [ (∂f/∂x_i)^2 \cdot u^2(x_i) ] + 2 \cdot \sum_{i=1}^{N-1} \sum_{j=i+1}^N (∂f/∂x_i) \cdot (∂f/∂x_j) \cdot r(x_i,x_j) \cdot u(x_i) \cdot u(x_j)}

u_c(y) = \sqrt{\sum_{i=1}^N [ (∂f/∂x_i)^2 \cdot u^2(x_i) ]}

u_c(y) = \sqrt{\sum_{i=1}^N [ (∂f/∂x_i)^2 \cdot u^2(x_i) ] + \sum_{i=1}^N \sum_{j=1}^N [ (1/2) \cdot (∂^{2f}/(∂x_i∂x_j))^2 + (∂f/∂x_i) \cdot (∂^{3f}/(∂x_i∂x_j^2)) ] \cdot u^2(x_i) \cdot u^2(x_j)}

u_c(y) = \sqrt{\sum_{i=1}^N u_i^2(y)}

u_c(y) = \sqrt{\sum_{i=1}^N u_i^2}

u_c(y) = \sqrt{\sum_{i=1}^N A_i^2 \cdot u^2(x_i)}

\dfrac{u_c(y)}{|y|} = \sqrt{\dfrac{\sum_{i=1}^N [ P_i \cdot u(x_i)}{x_i ]^2}}

\dfrac{u_c(y)}{|y|} = \sqrt{\dfrac{\sum_{i=1}^N [ u(x_i)}{x_i ]^2}}

u_c(y) = | \sum_{i=1}^N (∂f/∂x_i) \cdot u(x_i) |

u_c(y) = \sum_{i=1}^N u(x_i)

u(x_i,x_j) = (1/(n-1)) \cdot \sum_{k=1}^n (x_{ik} - x̄_i) \cdot (x_{jk} - x̄_j)

u(x_i,x_j) = (∂F/∂q) \cdot (∂G/∂q) \cdot u^2(q)

u(x_i,x_j) = \sum_{k=1}^L (∂F/∂q_k) \cdot (∂G/∂q_k) \cdot u^2(q_k)

r(x,y) = \dfrac{\sum_{i=1}^n (x_i - x̄) \cdot (y_i - ȳ)}{(n-1) \cdot s(x) \cdot s(y)}

\dfrac{r(x_i,x_j) ≈ (u(x_i) \cdot δ_j)}{u(x_j) \cdot δ_i}

nu_{eff} = \dfrac{u_c^4(y)}{\sum_{i=1}^N (u_i^4(y) / nu_i)}

nu_{eff} = \dfrac{[u_c(y)}{\dfrac{y]^4}{\sum_{i=1}^N ([P_i u(x_i)/x_i]^4 / nu_i)}}

U = k \cdot u_c

U_p = k_p \cdot u_c

k_p = t_p(nu_{eff})

u = \dfrac{a}{k}

u_{rel} = \dfrac{u}{x}

a = \dfrac{a_{plus} - a_{minus}}{2}

u_c(\bar{V}) = \sqrt{u_A^2(\bar{V}) + u_B^2(\bar{V}) + u^2(\Delta\bar{V})}

u_B(\bar{V}) = \dfrac{a}{k}

a = 14e-6 \cdot \bar{V} + 2e-6 \cdot \text{量程}

u_c^2(P) = (∂P/∂V)^2 u^2(V) + (∂P/∂R0)^2 u^2(R0) + (∂P/∂α)^2 u^2(α) + (∂P/∂t)^2 u^2(t)

\dfrac{∂P}{∂V} = \dfrac{2P}{\dfrac{V; ∂P}{∂R0}} = \dfrac{-P}{\dfrac{R0; ∂P}{∂α}} = \dfrac{-P(t-t0)}{\dfrac{(1+α(t-t0)); ∂P}{∂t}} = \dfrac{-Pα}{1+α(t-t0)}

\dfrac{u(I)}{I} = \sqrt{(u(Vs)/Vs)^2 + (u(Rs)/Rs)^2}

u(t) = \sqrt{\dfrac{4(t+t0)^2 [u^2(β)}{\dfrac{β^2 + u^2(Rs)}{Rs^2]}}}

u(I,t) = \dfrac{-2I(t+t0)}{Rs^2 \cdot u^2(Rs)}

\dfrac{u_c(P)}{P} = \sqrt{\dfrac{4u^2(I)}{\dfrac{I^2 + u^2(t)}{\dfrac{(t+t0)^2 - 4u(I,t)}{I(t+t0)}}}}

u(R_{ref}) = \sqrt{10} \cdot u(R_i)

u(R_i) = \sqrt{(Rs u(α_i))^2 + (α_i u(Rs))^2}

r(R_i, R_j) = \dfrac{u(R_i, R_j)}{u(R_i) \cdot u(R_j)}

u_c(R_{ref}) = \sum_{i=1}^{10} u(R_s) = 10 \cdot 10 mΩ = 0.10 Ω

L = L_s + d

d = L \cdot (1 + alpha \cdot theta) - L_s \cdot (1 + alpha_s \cdot theta_s)

L = (1/(1 + alpha \cdot theta)) \cdot [L_s \cdot (1 + alpha_s \cdot theta_s) + d]

L ≈ L_s + d + L_s \cdot (alpha_s \cdot theta_s - alpha \cdot theta)

l ≈ l_s + d - l_s \cdot (delta_{alpha} \cdot theta + alpha_s \cdot delta_{theta})

l = l_s + \bar{d}

u_c(l) = \sqrt{c1^2 \cdot u^2(l_s) + c2^2 \cdot u^2(d) + c3^2 \cdot u^2(alpha_s) + c4^2 \cdot u^2(theta) + c5^2 \cdot u^2(delta_{alpha}) + c6^2 \cdot u^2(delta_{theta})}

u_c(l) = \sqrt{u^2(l_s) + u^2(d) + l_s^2 \cdot theta^2 \cdot u^2(delta_{alpha}) + l_s^2 \cdot alpha_s^2 \cdot u^2(delta_{theta})}

u(l_s) = \dfrac{U}{k}

u(\bar{d}) = \dfrac{s(d)}{\sqrt{n}}

u_B(d) = \dfrac{a}{\sqrt{3}}

\dfrac{ν_i ≈ 1}{\dfrac{2 \cdot [Δu(x_i)}{u(x_i)]^{-2}}}

u(d) = \sqrt{u^2(\bar{d}) + u_B^2(d)}

ν_{eff}(d) = \dfrac{u^4(d)}{\dfrac{u^4(\bar{d})}{\dfrac{ν_1 + u_B^4(d)}{ν_2}}}

u(δ_α) = \dfrac{a}{\sqrt{3}}

u(δ_θ) = \dfrac{a}{\sqrt{3}}

u(θ) = \sqrt{u^2(\bar{θ}) + u^2(Δ)}

u(α_s) = \dfrac{a}{\sqrt{3}}

c_5 = -l_s \cdot θ

u_c(l) = \sqrt{c_1^2 u^2(l_s) + c_2^2 u^2(d) + c_5^2 u^2(δ_α) + c_6^2 u^2(δ_θ)}

ν_{eff}(l) = \dfrac{u_c^4(l)}{\dfrac{u^4(l_s)}{\dfrac{ν_{eff}(l_s) + u^4(d)}{\dfrac{ν_{eff}(d) + u^4(δ_α)}{\dfrac{ν(δ_α) + u^4(δ_θ)}{ν(δ_θ)}}}}}

U_{99} = k_{99} \cdot u_c(L)

l = l_s + \bar{d}

l_s \cdot u(delta_{alpha}) \cdot u(theta)

l_s \cdot u(alpha_s) \cdot u(delta_{theta})

u_c'(l) = \sqrt{32^2 + 11.7^2 + 1.7^2}

b(t) = y1 + y2 \cdot (t - t0)

Q = \sum_{k=1}^{n} [b_k - y1 - y2 \cdot (t_k - t0)]^2

y1 = \dfrac{sum(b_k) \cdot sum(theta_k^2) - sum(b_k \cdot theta_k) \cdot sum(theta_k)}{D}

y2 = \dfrac{n \cdot sum(b_k \cdot theta_k) - sum(b_k) \cdot sum(theta_k)}{D}

s^2(y1) = \dfrac{s^2 \cdot sum(theta_k^2)}{D}

s^2(y2) = \dfrac{n \cdot s^2}{D}

r(y1,y2) = \dfrac{- sum(theta_k)}{\sqrt{n \cdot sum(theta_k^2)}}

s^2 = \dfrac{sum([b_k - b(t_k)]^2)}{n - 2}

D = n \cdot sum(theta_k^2) - (sum(theta_k))^2 = n \cdot sum((theta_k - \bar{theta})^2) = n \cdot sum((t_k - \bar{t})^2)

b(t) = y1 + y2 \cdot (t - t0)

u_c^2(y) = \sum_{i=1}^N c_i^2 u^2(x_i) + 2 \cdot \sum_{i=1}^{N-1} \sum_{j=i+1}^N c_i c_j u(x_i) u(x_j) r(x_i, x_j)

u_c^2[b(t)] = u^2(y1) + (t - t0)^2 \cdot u^2(y2) + 2 \cdot (t - t0) \cdot u(y1) \cdot u(y2) \cdot r(y1, y2)

hRC = 100 \cdot (0.002 mm) - d - Δc - Δb - Δs

HRC = \dfrac{hRC}{0.002 mm}

u_c^2(h) = u^2(d) + u^2(Δc) + u^2(Δb) + u^2(Δs)

u^2(\bar{d}) = \dfrac{s_p^2(d_k)}{\dfrac{5 + δ^2}{12}}

u^2(Δc) = \dfrac{s_{av}^2(\bar{z}_{s})}{\dfrac{m + s_{av}^2(\bar{z})}{n}}

u^2(Δb) = \dfrac{(x_z')^2}{24}

u_c^2(h) = \dfrac{s^2(d_k)}{\dfrac{5 + δ^2}{\dfrac{12 + s_{av}^2(\bar{z}_{s})}{\dfrac{m + s_{av}^2(\bar{z})}{\dfrac{n + (x_z')^2}{24 + u^2(Δs)}}}}}

H_{RockwellC} = \dfrac{h_{RockwellC}}{0.002 mm} = \dfrac{[100 \cdot (0.002 mm) - \bar{d}]}{0.002 mm}

u_c^2(h) = \dfrac{[s_p(d_k)^2}{\dfrac{5 + δ^2}{\dfrac{12 + s_{av}(z_s)^2}{\dfrac{6 + s_{av}(z)^2}{\dfrac{6 + (x \cdot z')^2}{24 + u(Δ_s)^2]}}}}}

u_c(h) = \sqrt{u_c^2(h)}

h = 64.0 HRC = 0.1280 mm

M(KOH) = A_r(K) + A_r(O) + A_r(H)

ω(KOH) = \dfrac{V(HCl) \cdot c(HCl) \cdot M(KOH)}{m}

\dfrac{u_c(y)}{y} = \sqrt{\dfrac{\sum_{i=1}^{N} [ P_i \cdot u(x_i)}{x_i ]^2}}

u_r(y) = \sqrt{\sum_{i=1}^{N} [ P_i \cdot u_r(x_i) ]^2}

u_{cr}[ω(KOH)] = \sqrt{u_r^2[V(HCl)] + u_r^2[c(HCl)] + u_r^2[M(KOH)] + u_r^2[m]}

u_r[V(HCl)] = \dfrac{a}{k} = \dfrac{0.6\%}{\sqrt{3}} = 0.35\% = 3.5e-3

u_r[c(HCl)] = \dfrac{U_r}{k} = \dfrac{1e-3}{2} = 0.5e-3

u[M(KOH)] = \sqrt{u^2[A_r(K)] + u^2[A_r(O)] + u^2[A_r(H)]}

u_r[M(KOH)] = \dfrac{u[M(KOH)]}{M(KOH)} = \dfrac{0.003}{56.10024} = 5.3e-5

u_r(m) = \sqrt{(U_r/k)^2 + (s_{rep})^2} = \sqrt{(3e-4/3)^2 + (0.3e-4)^2} = 1e-4

u_{cr}[ω(KOH)] = \sqrt{(3.5e-3)^2 + (0.5e-3)^2 + (5.3e-5)^2 + (0.1e-3)^2} = 3.5e-3

U_r = k \cdot u_c

omega(KOH) = \dfrac{V(HCl) \cdot c(HCl) \cdot M(KOH)}{m}

U = omega(KOH) \cdot U_r

y = t_s + Δt_s

u(t_s) = \sqrt{u1(t_s)^2 + u2(t_s)^2}

u(Δt_s) = \dfrac{U_{95}}{k_p}

u_A = \dfrac{s(y)}{\sqrt{m}}

u_c(y) = \sqrt{u1(t_s)^2 + u2(t_s)^2 + u_A^2}

U = k \cdot u_c(y)

C = y - t

Δ = -C

t_p(ν) = \dfrac{t_p(ν2) + (t_p(ν1) - t_p(ν2)) \cdot (ν - ν2)}{ν1 - ν2}

t_p(ν) = \dfrac{t_p(ν2) + (t_p(ν1) - t_p(ν2)) \cdot (1/ν - 1/ν2)}{\dfrac{1}{\dfrac{ν1 - 1}{ν2}}}