含义Meaning
判断平均值与标准值、或两组平均值之间的差异是否显著（而非偶然误差）。F 检验先判断两组精密度是否一致，以决定能否合并标准偏差。
Decide whether a mean differs significantly from a reference, or two means from each other. The F-test first checks whether precisions agree, so standard deviations can be pooled.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\mu_0\) | 标准值（如标准物质证书值）reference value | |
| \(s_p\) | 合并标准偏差pooled standard deviation |
推导Derivation
零假设 H₀：无显著差异。计算统计量 t。
Null hypothesis: no real difference; compute t.
若 t 大于自由度 n − 1（或 n₁ + n₂ − 2）下的临界值，则拒绝 H₀，差异显著。
If t exceeds the critical value, reject H₀.
F 检验：方差比大于 F 表临界值时精密度有显著差异，此时用韦尔奇 t 检验。
If F exceeds its critical value, precisions differ; use Welch’s t-test.
适用条件与局限Assumptions & limitations
“不显著”不等于“相同”，只是证据不足；多次比较需校正（如邦费罗尼）。
‘Not significant’ is not ‘equal’; multiple comparisons need correction.
例题Worked example
标准物质证书值 10.00%，测得 x̄ = 10.12%、s = 0.05%、n = 5：t = 0.12/0.05×√5 = 5.37 > 2.78，存在显著系统误差。
Certified 10.00 %, found 10.12 % (s = 0.05, n = 5): t = 5.37 > 2.78 — significant bias.