第1次测量(j = 1)

平行样1 (i = 1)

平行样2 (i = 2)

平行样3 (i = 3)

最佳估计值(算术平均值)

醋酸浓度A_i1

$A_{11}=6.0\text{uL}/\text{L}$

$A_{21}=6.5\text{uL}/\text{L}$

$A_{31}=7.0\text{uL}/\text{L}$

$\overline{A_1}=6.5\text{uL}/\text{L}$

$A_{i1}-\overline{A_1}$

−0.5

0

0.5

n = 3

实验标准偏差

自由度

$s_1\left(A_1\right)=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(A_{i1}-\overline{A_1}\right)}^2}}=0.5\text{uL}/\text{L}$

$v_1=n-1=2$

第2次测量(j = 2)

平行样1 (i = 1)

平行样2 (i = 2)

平行样3 (i = 3)

最佳估计值(算术平均值)

醋酸浓度A_i2

$A_{12}=6.0\text{uL}/\text{L}$

$A_{22}=5.0\text{uL}/\text{L}$

$A_{32}=5.5\text{uL}/\text{L}$

$\overline{A_2}=5.5\text{uL}/\text{L}$

$A_{i2}-\overline{A_2}$

0.5

-0.5

0

n=3

实验标准偏差

自由度

$s_2\left(A_2\right)=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(A_{i2}-\overline{A_2}\right)}^2}}=0.5\text{uL}/\text{L}$

$v_2=n-1=2$

第3次测量(j = 3)

平行样1 (i = 1)

平行样2 (i = 2)

平行样3 ( i= 3)

最佳估计值(算术平均值)

醋酸浓度A_i3

$A_{13}=6.5\text{uL}/\text{L}$

$A_{23}=6.0\text{uL}/\text{L}$

$A_{33}=6.5\text{uL}/\text{L}$

$\overline{A_3}=\text{6}.3333\text{uL}/\text{L}$

$A_{i3}-\overline{A_3}$

0.1667

−0.3333

0.1667

n = 3

实验标准偏差

自由度

$s_3\left(A_3\right)=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(A_{i3}-\overline{A_3}\right)}^2}}=0.2887\text{uL}/\text{L}$

$v_3=n-1=2$

合并样本标准偏差

$s\left(A\right)=s_P\left(A\right)=\sqrt{\frac{{\displaystyle \sum _{j=1}^m{s}_j^2}}{m}}=\sqrt{\frac{{\left[s_1\left(A_1\right)\right]}^2+{\left[s_2\left(A_2\right)\right]}^2+{\left[s_3\left(A_3\right)\right]}^2}{3}}=0.4410\text{uL}/\text{L}$

自由度

$v={\displaystyle \sum _{j=1}^m{v}_j}=6$

测量结果A的A类标准不确定度

$u_A\left(A\right)=u\left(\overline{A}\right)=s_P\left(A\right)/\sqrt{n}=0.2546\text{uL}/\text{L}$

A的算术平均值

$\overline{A}=\frac{1}{m}{\displaystyle \sum _{j=1}^m{A}_j}=\frac{1}{3}\left(6.5+5.5+6.33\right)=6.11\text{uL}/\text{L}$

A的A类相对标准不确定度

$u_{r,A}\left(A\right)=\frac{u_A\left(A\right)}{\overline{A}}=\frac{0.2546}{6.11}=0.04167$