Microsoft PowerPoint - Chapter 7-Statistical Data Treatment and Evaluation_2018ed [相容模式]

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1 Chapter 7 tatistical Data Treatmet ad Evaluatio 熊同銘 tmhsiug@gmail.com TMHsiug 08 Chapter 07 lide of 37 Cotets i Chapter 7 7. Cofidece Iterval 7.. Probability versus Cofidece Iterval of Gaussia distributio 7.. Cofidece Iterval Applicatios for Populatio 7..3 Cofidece Iterval Applicatios for ample 7. Q Test for Outlier Rejectio 7.3 F Test for Comparig tadard Deviatio 7.4 t Test for Decidig ystematic Error 7.4. t test Methods Classificatio 7.4. Compariso a experimetal mea with a true value Compariso of two experimetal mea (upaired t test) Compariso of multiple samples (paired t test) TMHsiug 08 Chapter 07 lide of 37

2 7. Cofidece Iterval (CI) The rage of values withi which there is a specified probability (cofidece level, CL) that the true value lies. CL, e.g. 95% CL, same as * Cofidece limits are the boudary of cofidece iterval. A measuremet. 0.4 for example, the mea is. with cofidece limits of.7 ad.5, the cofidece iterval is.7 to.5. TMHsiug 08 Chapter 07 lide 3 of 37 Relative frequecy 7.. Probability versus Cofidece Iterval of Gaussia distributio % 95.46% 99.74% z ( x ) CL (probability) (z, uit: σ) CI (Area lie betwee) 50% ± % ±.9 90% ±.64 95% ±.96 99% ± % ± % ± % ± % ± % ± % ± % ± 3.50 TMHsiug 08 Chapter 07 lide 4 of 37

3 Cotiued Values of z for determiig Cofidece iterval Cofidece level, % z TMHsiug 08 Chapter 07 lide 5 of Cofidece Iterval Applicatios for Populatio If σ is kow, or > 0 (sσ): Expressio for the Expected value z x Expressio for the Measured value z x TMHsiug 08 Chapter 07 lide 6 of 37 3

4 z Expressios of the expected value ( x ) Example: What is the 95% cofidece iterval for the amout of aspiri i a sigle aalgesic tablet draw from a populatio where μ is 50 mg ad σ is 5? olutio: x = µ ±.96σ = 50 ± (.96)(5) = 50 ± 0 As: 95% of the tablets i the populatio cotai 50 mg ± 0 mg of aspiri. TMHsiug 08 Chapter 07 lide 7 of 37 z Expressios of the sigle measured value ( x ) Example: The populatio stadard deviatio for the amout of aspiri i a batch of aalgesic tablets is kow to be 7 mg of aspiri. A sigle tablet is radomly selected, aalyzed, ad foud to cotai 45 mg of aspiri. What is the 95% cofidece iterval for the populatio mea? olutio: µ = x ±.96σ = 45 ± (.96)(7) = 45 ± 4 As: At 95% cofidece level, the populatio s mea, µ, lies withi the rage of 45 mg ± 4 mg of aspiri. TMHsiug 08 Chapter 07 lide 8 of 37 4

z Expressios of the replicate measured value x Example: The populatio stadard deviatio for the amout of aspiri i a batch of aalgesic tablets is kow to be 7 mg of aspiri.

5 z Expressios of the replicate measured value x Example: The populatio stadard deviatio for the amout of aspiri i a batch of aalgesic tablets is kow to be 7 mg of aspiri. Five tablets are radomly selected, aalyzed, ad foud to cotai 45 mg of aspiri. What is the 95% cofidece iterval for the populatio mea? olutio: (.96)(7) As: At 95% cofidece level, the populatio s mea, µ, lies withi the rage of 45 mg ± 6 mg of aspiri. TMHsiug 08 Chapter 07 lide 9 of Cofidece Iterval Applicatios for ample*** Expressio for the Measured value ts x Values of studet s t Defiitio of t: x t s : t z, Ad μ x TMHsiug 08 Chapter 07 lide 0 of 37 5

6 Expressios of measured values ad σ is ukow x ts Example: What is the 95% cofidece iterval for the data i followig table? Masses of seve U peies i circulatio Pey Mass (g) TMHsiug 08 Chapter 07 lide of 37 olutio: x = 3.7 g s = 0.05 t =.45 (95% cofidece level) (.45)(0.05) g 7 As: At 95% cofidece level, the populatio s mea, μ, lies withi the rage of 3. ± 0.05 g. TMHsiug 08 Chapter 07 lide of 37 6

7 教戰守則. Large sample use z value (A) Reportig expected value: (B) Reportig measured value: z x z x. mall sample use t value Reportig measured value: x ts ***** TMHsiug 08 Chapter 07 lide 3 of Q Test for Outlier Rejectio 一種統計檢定, 據以判斷被懷疑的數據 ( 離群值 ) 是否應予以刪除 (gross error) Procedure for Q-test (Dea ad Dixo, 95) tep : Calculate the rage (w = 最大值最小值 ) tep : Calculate questioable result (x q ) 與 the earest eighbor (x ) 之間的差值 tep 3: Calculate experimetal Q value (Q exp ) tep 4: x - x q Qexp w 查表找出 critical values for rejectio quotiet (Q crit ) TMHsiug 08 Chapter 07 lide 4 of 37 7

8 Critical values for rejectio quotiet (Q crit ) umber of Cofidece Level Observatios () 90% 95% 99% e.g. 95% cofidece level same as α= 0.05 tep 5: Q exp > Q crit : 刪除 Q exp < Q crit : 保留 TMHsiug 08 Chapter 07 lide 5 of 37 Example: The followig masses, i grams, were recorded i a experimet to determie the average mass of a U.. pey: 3.067, 3.049, 3.039,.54, 3.048, 3.079, 3.094, 3.09, 3.0 Determie if the value of.54 g is a outlier at 95% cofidece. olutio: Place the masses i order from smallest to largest:.54, 3.039, 3.048, 3.049, 3.067, 3.079, 3.094, 3.0, % cofidece, = 9, Q crit = Q exp > Q crit As:.54 is assumed a outlier, ad ca be rejected. TMHsiug 08 Chapter 07 lide 6 of 37 8

7.3 F Test for Comparig tadard Deviatio 一種統計檢定, 據以判斷兩組數據之標準偏差 ( 精密度 ) 的差異是否顯著 Procedure for F-test tep : Calculate the F exp F exp 較大者作為分子 (umerator) 較小者作為分母 (Deomiator) 所以 always F exp > tep : 查表找出

9 7.3 F Test for Comparig tadard Deviatio 一種統計檢定, 據以判斷兩組數據之標準偏差 ( 精密度 ) 的差異是否顯著 Procedure for F-test tep : Calculate the F exp F exp 較大者作為分子 (umerator) 較小者作為分母 (Deomiator) 所以 always F exp > tep : 查表找出 critical values for F (F crit ) TMHsiug 08 Chapter 07 lide 7 of 37 Critical values for F (F cric ) at 95% cofidece level ( 分子 ) ( 分母 ) tep 3: F exp > F crit : 95% 信賴度下兩組數據之標準偏差之差異顯著 F exp < F crit : 95% 信賴度下兩組數據之標準偏差之差異不顯著 TMHsiug 08 Chapter 07 lide 8 of 37 9

10 Example Degree of freedom σ or s Referece method 0. Method 0.5 Method 0. (0.) (a) Ref F.96 (0.5) 分子自由度, 分母自由度, 查表得 F crit =.30 95% 信賴度下 method 與 referece method 的標準偏差之差異不顯著 TMHsiug 08 Chapter 07 lide 9 of 37 (b) Ref (0.) F 3.06 (0.) 分子自由度, 分母自由度, 查表得 F crit =.30 95% 信賴度下 method 與 referece method 的標準偏差之差異顯著 (c) F (0.5) (0.).56 分子自由度, 分母自由度, 查表得 F crit =.69 95% 信賴度下 method 與 method 的標準偏差之差異不顯著 TMHsiug 08 Chapter 07 lide 0 of 37 0

11 7.4 t Test for Decidig ystematic Error 一種統計檢定, 據以判斷分析數據與真值的信賴區間 (cofidece iterval) 分佈的差異是否顯著, 或判斷兩組數據之高斯分佈曲線分佈的差異是否顯著兩組數據具有顯著的差異兩組數據並無顯著的差異兩組數據差異是否顯著? TMHsiug 08 Chapter 07 lide of t test Methods Classificatio () Compariso a experimetal mea with a true value () Compariso of two experimetal mea (upaired t- test) 比較兩組數據的差異, 兩組數據來自於兩個分析者或兩種分析方法或兩組樣品兩組數據的值不要求相同 (3) Compariso of multiple samples (paired t test) 個樣品, 分別以方法及方法各分析一次, 據以比較兩種方法是否具有顯著性的差異 TMHsiug 08 Chapter 07 lide of 37

12 7.4. Compariso a experimetal mea with a true value Procedure tep : tep : Calculate the t (exp) x t(exp) s 查表找出特定信賴度下, 分析數據 dof (-) 的 t (theo) tep 3: t (exp) > t (theo) : 分析值與真值差異顯著 ( 系統誤差 ) t (exp) < t (theo) : 分析值與真值差異不顯著 TMHsiug 08 Chapter 07 lide 3 of 37 Values of studet s t TMHsiug 08 Chapter 07 lide 4 of 37

13 Example: A studet check her procedure by aalyzig a sample kow to cotai 98.76% (w/w) a CO 3. Five replicate determiatios of the %w/w a CO 3 i the stadard were made with the followig results: 98.7%, 98.59%, 98.6%, 98.44%, 98.58% Is the mea for these five trials sigificatly differet from the accepted value at the 95% cofidece level? olutio: 95% cofidece, dof = 5 = 4, t theo =.78 t exp > t theo As: At the 95% cofidece, their differece is sigificat (may be affected by a determiate source of error) TMHsiug 08 Chapter 07 lide 5 of Compariso of two experimetal mea (upaired t-test) 通常應用於三種方式的顯著性統計 : 同一樣品同一方法, 不同兩個分析者同一樣品同一分析者, 不同兩個方法 ( 引伸 : 同一方法同一分析者, 不同兩個樣品 ) Remark: 需要先比較兩組數據是否具有顯著性的差異 (Ftest), 再使用適當的統計檢定量如 (t-test) TMHsiug 08 Chapter 07 lide 6 of 37 3

14 4 Chapter 07 TMHsiug 08 lide 7 of 37 tep : F-test 精密度無顯著差異 tep : 計算 pooled stadard deviatio ( p ) ) ( ) ( ) ( ) ( x x x x j j i i pooled Procedure Chapter 07 TMHsiug 08 lide 8 of 37 tep 3: 統計檢定量 : (exp) x x t p tep 4: 依據所定之 cofidece level (e.g., 95%) 以及 degree of freedom + - 取得 t (theo) tep 5: t (exp) > t (theo) : 顯著差異或存在系統誤差 t (exp) < t (theo) : 無顯著差異

15 Example 同一個樣品分別以 ew method 和 referece method 分析 Fe(III) 之結果 : ew method (%) Referece method (%) At 95% cofidece level,ew method 是否與 referece method 具有 sigificat differece? TMHsiug 08 Chapter 07 lide 9 of 37 olutio 先確認兩種方法之精密度是否具有顯著差異 : F cal ew method Referece method Mea D (0.673) (0.34) 4.3 分子自由度為 6-=5, 分母自由度為 5-=4, 查表得 F crit = 6.6 F cal < F crit, At 95% cofidece level, 兩種方法之精密度無顯著差異 TMHsiug 08 Chapter 07 lide 30 of 37 5

16 p ( ) ( ) = (0.673) (6 ) (0.34) (5 ) 6 5 = t (exp) x x p = % cofidece level,dof = + - = 6+5- = 9, t (theo) =.6 t (exp) < t (theo) As: At 95% cofidece level, 兩種分析方法之分析結果無顯著差異 TMHsiug 08 Chapter 07 lide 3 of Compariso of multiple samples (paired t test) 通常應用於不同來源的個樣品, 分別以已知方法分析一次, 並以另一新分析一次, 據以比較兩種方法是否具有顯著性的差異 * Paired data: Two sets of data cosistig of results obtaied usig several samples draw from differet sources. TMHsiug 08 Chapter 07 lide 3 of 37 6

17 Procedure tep : 計算差異 : d i = A i -B i t exp d s tep : 計算所有 d i 之平均值 : tep 3: 計算所有 d i 之標準偏差 : d d d i ( di d) tep 4: 計算統計檢定量 : t exp d d TMHsiug 08 Chapter 07 lide 33 of 37 tep 5: 依據所定之 cofidece level (e.g., 95%) 以及 dof - 取得 t theo tep 6: t (exp) > t (theo) : 顯著差異或存在系統誤差 t (exp) < t (theo) : 無顯著差異 TMHsiug 08 Chapter 07 lide 34 of 37 7

18 EXAMPLE 6 個檢體分別以 ew method 和 referece method 分析 blood urea itroge (BU) 之結果如下 : ample ew method, mg/dl Referece Method, mg/dl Differece, di A B C D E F At 95% cofidece level,ew method 是否與 referece method 具有 sigificat differece? TMHsiug 08 Chapter 07 lide 35 of 37 olutio d 0.8 d = 0.4 d 0.8 t(exp) d 0.4 At cofidece level 95%,dof = - = 6- = 5 t (theo) = t (0.05,5) =.57 t (exp) < t (theo) As: At 95% cofidece level, 兩組分析數據無顯著差異 TMHsiug 08 Chapter 07 lide 36 of 37 8

19 Ed of Chapter 7 TMHsiug 08 Chapter 07 lide 37 of 37 9

11.兩組資料的比較：針對母體平均數及成功率

11.兩組資料的比較：針對母體平均數及成功率 . 兩料數 ( 率 ( 料連兩數 Q : 女異女料數 { 女 } 連女女女女女女女? 3 料 (biary{ } 兩率 Q : all pass 女 all pass 例兩女料 Y {all pass, ot all pass} all pass 女女 all pass 女 all pass 4 all pass 例 S/F 女 all pass 例女 S/F