【正文】 Roughly 9098% of the data are within ?2s of m. Roughly 99100% of the data are within ?3s of m. 6075% 9098% 99100% m m s m 2 s m + s m + 2 s m + 3 s m 3 s Spread(散佈 ) 統(tǒng)計(jì) 26 Jason Lee 2023/07/04 The shape of a distribution can be described by skewness 歪斜 (denoted by ?1) and by kurtosis凹擊平坦 (denoted by ?2). ?1 0 ?1 = 0 ?1 0 ?2 0 ?2 = 0 ?2 0 歪斜 凹擊平坦 Shape (形狀 ) 統(tǒng)計(jì) 27 Jason Lee 2023/07/04 N)(Xn1i2i2????msNX= 1i??Nimnx=xn1=ii?N) (X= N1=i2i??ms ? ?1 21?????nxxniis母體均值 樣本均值 母體標(biāo)準(zhǔn)偏差 樣本標(biāo)準(zhǔn)偏差 常用 計(jì)算公式 ~ 母體 變異 樣本 變異 1n)X(Xn1i2i2?????s~ 統(tǒng)計(jì) 28 Jason Lee 2023/07/04 The most important and useful distribution shape is called the Normal distribution, which is symmetric(對稱 ), unimodal(單峰 ), and free of outliers (沒有特異點(diǎn) ): Normal Distribution常態(tài)分佈 “ 常態(tài) ” 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(chǔ)過程 （ 資料從該過程中收集 ） 的特徵非常有用 . 大多數(shù)自然現(xiàn)象和人爲(wèi)過程都符合常態(tài)分配 ， 可以用常態(tài)分配表示 , 故大部份統(tǒng)計(jì)都假設(shè)是常態(tài)分佈 。 即使在資料不完全符合常態(tài)分配時 ， 分析結(jié)果也很接近 。 特別不正常的分佈若假設(shè)為常態(tài)而去分析則有可能得到誤導(dǎo)結(jié)果 。 有 數(shù)學(xué) 技術(shù)可 將 其 轉(zhuǎn)變成常態(tài)分佈 來作分析 。 統(tǒng)計(jì) 29 Jason Lee 2023/07/04 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(tài)概率圖 Normal Distribution常態(tài)分佈 統(tǒng)計(jì) 30 Jason Lee 2023/07/04 第一個屬性 : 只要知道下面兩項(xiàng)就可以完全描述常態(tài)分配 : 均值 標(biāo)準(zhǔn)差 常態(tài)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 ? 統(tǒng)計(jì) 31 Jason Lee 2023/07/04 常態(tài)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(jì)某“事件”發(fā)生的累積概率 95% 68% 樣本值的概率 距離均值的標(biāo)準(zhǔn)偏差數(shù) 得到兩值之間的值的累積概率 統(tǒng)計(jì) 32 Jason Lee 2023/07/04 常態(tài)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(tài)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(tài)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(tài)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(biāo)準(zhǔn)偏差： 10 資料個數(shù)： 500 AndersonDarling常態(tài)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(tài)分配 常態(tài) 概率 平均值： 70 標(biāo)準(zhǔn)偏差 :10 資料個數(shù)： 500 AndersonDarling常態(tài)測試 A平方 : P值 : 我們可以用常態(tài)概率圖檢驗(yàn)一組給定的資料是否可以描述爲(wèi)“常態(tài)” 如果一個分佈接近常態(tài)分配，則常態(tài)概率圖將爲(wèi)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(fù)偏斜分佈負(fù)偏斜平均： 70標(biāo)準(zhǔn)偏差： 10資料個數(shù)： 500Anderson Darling 常態(tài)測試A 平方： P 值 :概率負(fù)偏斜分佈負(fù)偏斜平均：標(biāo)準(zhǔn)偏差：資料個數(shù)：常態(tài)測試平方：值概率統(tǒng)計(jì) 33 Jason Lee 2023/07/04 資料收集時的重點(diǎn) 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(tǒng)計(jì)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(huán)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(jī) ). When the data collection is not randomized, statistical analysis may lead to faulty conclusions. 統(tǒng)計(jì) 34 Jason Lee 2023/07/04 Continuous Manufacturing (連續(xù) )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. 生產(chǎn)製造環(huán)境 統(tǒng)計(jì) 35 Jason Lee 2023/07/04 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(xù)和批量生產(chǎn)所用的統(tǒng)計(jì)方法有些不同 統(tǒng)計(jì) 36 Jason Lee 2023/07/04 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 “l(fā)evels” 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