分析化学-chap7-notes-误差分析

最新推荐文章于 2025-03-19 17:58:54 发布

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最新推荐文章于 2025-03-19 17:58:54 发布

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本文探讨了误差分析中的置信区间（CI）和置信水平（CL），介绍了如何根据样本大小计算CI，并区分系统误差与随机误差。通过z和t分布进行假设检验，用于判断实验结果与已知值的差异是否显著。此外，还提到了F检验和Q、Grubbs测试在评估数据质量中的应用。

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confidence interval(CI) 置信度(区间)
certain probability,true mean,the range of values
the probability is called confidence level (CL)

$\text { CI for } \mu=\bar{x} \pm \frac{z \sigma}{\sqrt{N}}$

we might say that it is 99% probable that the true population mean…
Confidence level: the probability that true mean lies in the given range
Significant level: the probability that a result is outside the $\mathrm{CL}$
Confidence interval for the mean: the range of values within which the population mean is expected to lie with a certain probability

To estimate the true mean from
large sample: z, $\sigma$
small sample: t,s

Single measurement:
$\text { CI for } \mu=x \pm z \sigma$
Mean of N measurements (large sample):
$\text { CI for } \mu=\bar{x} \pm \frac{z \sigma}{\sqrt{N}}$
Mean of N measurements (small sample):

$\text { CI for } \mu=\bar{x} \pm \frac{t s}{\sqrt{N}}$

z 的值：
$\begin{array}{cc} \hline \text { Confidence Levels for Various } \\ \text { Values of } z \\ \hline \text { Confidence Level, \% } & \boldsymbol{z} \\ \hline 50 & 0.67 \\ 68 & 1.00 \\ 80 & 1.28 \\ 90 & 1.64 \\ 95 & 1.96 \\ 95.4 & 2.00 \\ 99 & 2.58 \\ 99.7 & 3.00 \\ 99.9 & 3.29 \\ \hline \end{array}$

Finding the Confidence Interval When σ Is Unknown:
$\text { CI for } \mu=\bar{x} \pm \frac{t s}{\sqrt{N}}$

$t=\frac{\bar{x}-\mu}{s / \sqrt{N}}$

评价-精密度：
Systematic Error or Random Error?

Comparing an Experimental Mean with a Known Value (z/t Test)
$z=\frac{\bar{x}-\mu_{0}}{\sigma / \sqrt{N}}$

$t=\frac{\bar{x}-\mu_{0}}{s / \sqrt{N}}$

If $z>z_{\text {crit }}$ or $t>t_{\text {crit}}$ , systematic error exists and the analysis method needs improvements, otherwise retain it!

$\begin{aligned} &\text { Values of } t \text { for Various Levels of Probability }\\ &\begin{array}{cccccc} \begin{array}{ccccc} \text { Degrees of } \\ \text { Freedom } \end{array} & \mathbf{8 0 \%} & \mathbf{9 0 \%} & \mathbf{9 5 \%} & \mathbf{9 9 \%} & \mathbf{9 9 . 9 \%} \\ \hline 1 & 3.08 & 6.31 & 12.7 & 63.7 & 637 \\ 2 & 1.89 & 2.92 & 4.30 & 9.92 & 31.6 \\ 3 & 1.64 & 2.35 & 3.18 & 5.84 & 12.9 \\ 4 & 1.53 & 2.13 & 2.78 & 4.60 & 8.61 \\ 5 & 1.48 & 2.02 & 2.57 & 4.03 & 6.87 \\ 6 & 1.44 & 1.94 & 2.45 & 3.71 & 5.96 \\ 7 & 1.42 & 1.90 & 2.36 & 3.50 & 5.41 \\ 8 & 1.40 & 1.86 & 2.31 & 3.36 & 5.04 \\ 9 & 1.38 & 1.83 & 2.26 & 3.25 & 4.78 \\ 10 & 1.37 & 1.81 & 2.23 & 3.17 & 4.59 \\ 15 & 1.34 & 1.75 & 2.13 & 2.95 & 4.07 \\ 20 & 1.32 & 1.73 & 2.09 & 2.84 & 3.85 \\ 40 & 1.30 & 1.68 & 2.02 & 2.70 & 3.55 \\ 60 & 1.30 & 1.67 & 2.00 & 2.62 & 3.46 \\ \infty & 1.28 & 1.64 & 1.96 & 2.58 & 3.29 \end{array} \end{aligned}$