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【正文】 Normal Distribution常態分佈 “ 常態 ” 分佈是具有某些一致屬性的資料的分佈 這些屬性對理解基礎過程 （ 資料從該過程中收集 ） 的特徵非常有用 . 大多數自然現象和人爲過程都符合常態分配 ， 可以用常態分配表示 , 故大部份統計都假設是常態分佈 。 即使在資料不完全符合常態分配時 ， 分析結果也很接近 。 特別不正常的分佈若假設為常態而去分析則有可能得到誤導結果 。 有 數學 技術可 將 其 轉變成常態分佈 來作分析 。 6s 27 A Normal probability plot is a cumulative distribution plot where the vertical scale is changed in such a way that data from a Normal distribution will form a straight line: Histogram Cumulative Distribution Normal Probability Plot 常態概率圖 Normal Distribution常態分佈 6s 28 第一個屬性 : 只要知道下面兩項就可以完全描述常態分配 : 均值 標準差 常態分配 的好處 簡化 第一個分佈 第二個分佈 第三個分佈 這三個分佈有什麼不同 ? 6s 29 常態曲線和其概率 4 3 2 1 0 1 2 3 4 40% 30% 20% 10% 0% % 第二個屬性 : 曲線下方的面積可以用於估計某“事件”發生的累積概率 95% 68% 樣本值的概率 距離均值的標準偏差數 得到兩值之間的值的累積概率 6s 30 練習 6s 31 常態概率圖 1 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0 7 0 6 0 3 0 0 2 0 0 1 0 0 0 C 2 常態概率圖 頻率 1 1 0 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 5 0 0 C 1 常態概率圖 頻率 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 3 0 0 2 0 0 1 0 0 0 C 3 常態概率圖 頻率 1 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0 7 0 6 0 . 9 9 9 . 9 9 . 9 5 . 8 0 . 5 0 . 2 0 . 0 5 . 0 1 . 0 0 1 平均： 70 標準偏差： 10 資料個數： 500 AndersonDarling常態測試 A平方 : P值 : 正偏斜分佈 概率 正偏斜 1 0 6 9 6 8 6 7 6 6 6 5 6 4 6 3 6 2 6 . 9 9 9 . 9 9 . 9 5 . 8 0 . 5 0 . 2 0 . 0 5 . 0 1 . 0 0 1 常態分配 常態 概率 平均值： 70 標準偏差 :10 資料個數： 500 AndersonDarling常態測試 A平方 : P值 : 我們可以用常態概率圖檢驗一組給定的資料是否可以描述爲“常態” 如果一個分佈接近常態分配，則常態概率圖將爲一條直線。 8 07 06 05 04 03 02 0100.9 9 9. 9 9. 9 5. 8 0. 5 0. 2 0. 0 5.0 1. 0 0 1負偏斜分佈負偏斜平均： 70標準偏差： 10資料個數： 500Anderson Darling 常態測試A 平方： P 值 :概率負偏斜分佈負偏斜平均：標準偏差：資料個數：常態測試平方：值概率6s 32 神秘分佈 以寸 爲的神秘變數畫 的 常態概率圖 。 你的結論是什麼 ? 這是常態分配嗎 ? 1 5 0 1 0 0 5 0 . 9 9 9 . 9 9 . 9 5 . 8 0 . 5 0 . 2 0 . 0 5 . 0 1 . 0 0 1 神秘分佈 常態分配 A平方： P值 : 神秘 概率 平均值： 100 標準偏差： 資料個數： 500 6s 33 1 8 0 1 3 0 8 0 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0 Stat Basic Statistics Display Descriptive Statistics Graphs Graphical Summary A2 描述性統計 圖形分析總結 變數：神秘 中值的 95%信賴區間 181。的 95%信賴區間 AndersonDarling常態測試 P值 均 值 標準偏差 變異數 偏度 峰度 資料量 最小值 第一象限 中值 第三象限 最大值 181。的 95%信賴區間 s的 95%信賴區間 中值的 95%信賴區間 6s 34 資料收集時的重點 How the data are collected affects the statistical appropriateness and analysis of a data set(資料如何收集可影響統計的適切性 ). Conclusions from properly collected data can be applied more generally to the process and output. Inappropriately collected data CANNOT be used to draw valid conclusions about a process. Some aspects of proper data collection that must be accounted for are: The manufacturing environment(製程環境 )from which the data are collected. When products are manufactured in batches or lots, the data must be collected from several batches or lots. Randomization(隨機 ). When the data collection is not randomized, statistical analysis may lead to faulty conclusions. 6s 35 Continuous Manufacturing (連續 )occurs when an operation is performed on one unit of product at a time. An assembly line is typical of a continuous manufacturing environment, where each unit of product is worked on individually and a continuous stream of finished products roll off the line. The automotive industry is one example of Continuous Manufacturing. Other examples of continuously manufactured product are: ? television sets, ? fast food hamburgers, ? puters. Lot/Batch Manufacturing (批次 ) occurs occurs when operations are performed on products in batches, groups, or lots. The final product es off the line in lots, instead of a stream of individual parts. Product within the same lot are processed together, and receive the same treatment while inprocess. Lot/Batch Manufacturing is typical of the semiconductor industry and many of its suppliers. Other examples of lot/batch manufactured product include: ? chemicals, ? semiconductor packages, ? cookies. Manufacturing Environment製造環境 6s 36 In Continuous Manufacturing the most important variation is between parts In Lot/Batch Manufacturing, the variation can occur between the parts in a lot and between the lots: ? Product within the same lot is manufactured together. ? Product from different lots are manufactured separately. Because of this, each lot has a different distribution. This is important because Continuous Manufacturing is a basic assumption for many of the standard statistical methods found in most textbooks or QC handbooks. These methods are not appropriate for Lot/Batch Manufacturing. Different statistical methods need to be used to take into account the several sources of variation in Lot/Batch Manufacturing. 要注意 : 連續和批量生產所用的統計方法有些不同 6s 37 With Lot/Batch Manufacturing, each lot has a different mean. Due to random processing fluctuations, these lots will vary even though the process may be stable. This results in several “levels” of distributions, each level with its own variance and mean: ? A distribution of units of product within the same lot. ? A distribution of the means of different lots. ? The total distribution of all units of product across all lots. Lot X 1 2 3 4 5 * * * * * * * * * * Distribution of Individual Lot Distribution of Lot Means Overall Distribution of Combined Lots Variation Within Each Lot Variation Between Lots Total Variation 6s 38 The different variances of a Lot/Batch Manufacturing process form a hierarchy