SPC-VI-Training[1].pptx

Classified - Internal Use Only1Chi-Square- Test 检验F-TestF 检验T-TestT 检验2Classified - Internal Use Only2 Null hypothesis. 原假设原假设 ( (无差异假设无差异假设) )Alternative hypothesis (Experimental hypothesis).备择假设备择假设 ( (调查假设调查假设) ) All hypothesis testing is done under the assumption that the null hypothesis is true.无差异推定无差异推定Goodness-of-fit test (frequency distribution test).吻合性检验吻合性检验 ( (频率分布检验频率分布检验) ) There is no enough evidence to indicate that .没有充分证据表明没有充分证据表明 . . There is evidence to indicate that .有证据表明有证据表明 Classified - Internal Use Only3Degrees of freedom v = n - 1自由度自由度: :v = v = 自由度自由度n = n = 样品数或类型数样品数或类型数 Confidence levelConfidence level(1 a)(1 a) a = a = 假警率假警率置信度置信度 Mere Chance CoincidenceMere Chance Coincidence随随机符合机符合 If your test does not result in a significant difference you can not say the subjects tested are “equal”, they are just not different. 如检验统计值未显示显著差异，可认为受检验对象无差异，如检验统计值未显示显著差异，可认为受检验对象无差异，但不能说但不能说 “等同等同”Classified - Internal Use Only4Number of Flowers (or Digits) in 10 Tosses of a CoinNumber of Heads (or Digits) ProbabilityClassified - Internal Use Only5 检验的应用检验的应用The Application Of Chi-square Test 2Classified - Internal Use Only6Chi-Square Procedures For The Analysis Of Categorical Frequency data (goodness-of-fit test) 方法用于频率分布检验2Classified - Internal Use Only7鱼1鱼2鱼3总数测得频率8929.7%12040.0%9130.3%300100%预期频率10033.3%10033.3%10033.3%33.3%300100%O = Observed Frequency (测得频率)E = Expected Frequency (预期频率)鱼1 =鱼2 =鱼3 = = 1.21 +4.0 + 0.81 = 6.022Classified - Internal Use Only8Empirical Approximation Of A Sampling Distribution重复采样得出的分布重复采样得出的分布222样本值样本值2样本百分率样本百分率Our null hypothesis: Three species of fish inhabit the river in identical proportions of one third each. What does the left distribution tell us? is the hypothesis true in 95.01% cases. Classified - Internal Use Only9Classified - Internal Use Only10Theoretical Sampling Distribution Of Chi-square ( = 2) 自由度为 2 时的分布 22Obviously, the higher the confidence level ( 1 ) the bigger thevalue. 置信度要求越高，则值 (临界值)也越高222Classified - Internal Use Only11 鱼1鱼2鱼3O8912091E100100100 = 1.21 +4.0 + 0.81 = 6.022思考题：思考题：How Confidence Level Changes The Critical Value Of ?提示：你需要多大的把握作判断提示：你需要多大的把握作判断 ( (选择多大的置信度选择多大的置信度) )? ?2Classified - Internal Use Only12Chi-Square Distribution For different Degrees Of Freedom不同自由度的分布不同自由度的分布2Classified - Internal Use Only13ExampleExample： Judge If Consumers Taste Preference Has ChangedJudge If Consumers Taste Preference Has Changed 用 检验判断消费者口味的变化判断消费者口味的变化某厂 10 年前根据市场调查确定可乐雪碧芬达产品中消费者首选比例为消费者首选比例为 0.3/ 0.4 / 0.3 . 0.3/ 0.4 / 0.3 . 最近的问卷最近的问卷调查数据如右表请采用检验判断判断消费者的口味选择是否消费者的口味选择是否已发生变化已发生变化可乐雪碧芬达O11010090E0.30 900.401200.3 0 90 90 120 90Note: Number Of Samples: 300提示：提示：1. 1. 计算检验值计算检验值2. 2. 计算计算自由度自由度3. 3. 确定确定置信度置信度 ( (此案例中，不一定需要选择此案例中，不一定需要选择高置信度高置信度) )4. 4. 查表得查表得临界值比较比较检验值与临界值2222Classified - Internal Use Only14 Example:Example: Judge If Consumers Taste Preference Has ChangedJudge If Consumers Taste Preference Has Changed 用 检验判断消费者口味的变化判断消费者口味的变化 0.500.300.200.0510.4551.0741.6423.84121.3862.4083.2195.99132.3663.6654.6427.81543.3574.8785.9899.488可乐雪碧芬达总和O11010090300E9012090300E90= 4.444120= 3.33390= 0 = 7.777 = 2 )(2EO )120100(22)90110(2)9090(2 检验值大于临界值 ( = 0.05 ): 7.777 5.991. 222我们有我们有 95%95%的把握相信，的把握相信，消费者的首选发生了变化消费者的首选发生了变化Classified - Internal Use Only152CHI-SQUARE TABLE 表表左侧为自由度 v表上行为假警率 阴影部分为假警率2Classified - Internal Use Only16Chi-Square Procedures For Comparing The Variability Of A Process With The Standard OrSated Or Implied Variability Of The Process 2Classified - Internal Use Only17Formula for Testing Process Variance222)(1(sn 22 s22Classified - Internal Use Only18 应用验证过程能力应用验证过程能力 Make Decisions About Process Variability Using22过程方差可能是一个规定的标准或要求，也可能是设备供应商承诺的规格。A variance（变异） can be a standard or a requirement, or it can be derived from（源自于） a suppliers statement（声明）.Interpretation（解释） is sometimes necessary for a suppliers statement of accuracy. For example, when a supplier states its equipment will deliver product with an accuracy of 2%, it may actually mean the delivered values will be within 2% of the target. 供应商承诺所隐涵的过程能供应商承诺所隐涵的过程能力力供应商的承诺有时需要作正确的“译释”例如：供应商所说的准确度 2%, 可能是指其设备的输出值与设定目标的偏离不超出 2% (如右)Classified - Internal Use Only19 一个合理的前提是 2% 应在 3 的范围内，即：4% = 6, = 4% 6 = 0.6667%It appears that the supplier did not really mean accuracy when stating the process performance of 2%.The statement appears to imply the range of values expected from the process. A value of can be derived from the statement.2 test should then be performed to verify the statement用 检验验证供应商承诺22Classified - Internal Use Only20Performing A -test For Process Variance -检验分析过程方差的步骤222Step 1Determine the value of for the process确定过程的标准偏差Step 2Randomly sample from the process and calculate the 3 key pieces of information.对过程随机采样，计算个过程数据Step 3Calculate test statistic. 计算的统计数据Step 4Calculate appropriate degrees of freedom.取合适的自由度Step 5Decide on an acceptable level of (false alarm).确定值Step 6Determine the critical value of from the tables.查表确定临界值Step 7Evaluate the results.评价结果222Classified - Internal Use Only21A manufacturer states that its proportioner（比例调节器，定量器，配合加料器（漏斗） can deliver sugar product with Brix varying（变更，变化） no more than 1.5% of the target.制造商声称其混比器能做到白利度偏差不超过目标值的 1.5%. In order to verify suppliers stated process performance the plant has randomly taken 20 samples from the process with two days and tested the values. The target Brix is 10.91. 为了验证制造商声称的过程性能，工厂随机取了 20 个样，并测得白利度值 (左表）.白利度目标值为 10.91.The plant has analyzed the data according to the - test procedure.工厂按 -检验的步骤对左表数据进行了分析22Classified - Internal Use Only22Target: 10.91 The stated performance: within 1.5%The Brix variation: (10.91) X ( 0.015) = 0.16365It stands for a variation range: 2 x 0.16365 = 0.3273The variation range should reasonably cover 3 (6 ) Therefore, The standard deviation: = 0.3273 6 = 0.05455白利度目标值： 10.91制造商声称的过程性能： 1.5% 白利度变化：(10.91) X ( 0.015) = 0.16365 白利度变化范围：2 x 0.16365 = 0.3273 对于 Cpk = 1.0 的过程： = 0.3273 6 = 0.05455Classified - Internal Use Only23Sampling and calculation was already done by the plant that is represented here.Based on the data sheet the plant Obtained the 3 pieces of data.Number of observations: n = 20Sample average: = 10.94Estimate of standard deviation: S = 0.155根据工厂取样及测试的结果，计算出以下三个数据： 样品数：n = 20白利度平均值： = 10.94样本标准偏差：s = 0.155xxClassified - Internal Use Only24Step 3Calculate test statistic 计算 的统计数据22 2Step 4 Calculate degrees of freedom Classified - Internal Use Only25 Critical = 38.582 = 0.005查表确定临界查表确定临界 值值An often chosen false alarm rate:22Classified - Internal Use Only26Step 7 Evaluate the results222 153.4 38.5822criticalThere is evidence to indicate that the variability measured in the process is greater than the manufacturers implied precision22Classified - Internal Use Only27STEP 1白利度目标值： 10.91制造商声称的过程性能： 1.5% 白利度变化：(10.91) X ( 0.015) = 0.16365 白利度变化范围：2 x 0.16365 = 0.3273 对于 Cpk = 1.0 的过程： = 0.3273 6 = 0.05455SUMMARYSUMMARYSTEP 2根据工厂取样及测试的结果，计算出以下三个数据： 样品数： n = 20白利度平均值： = 10.94样本标准偏差：s = 0.155STEP 6 = 38.58STEP 5 = 0.005STEP 7 153.4 38.5822CRITICAL22CRITICAL Classified - Internal Use Only28F F- -检验的应用检验的应用The Application Of F-Test Classified - Internal Use Only29 F-Test For Comparing Two Estimates Of Process VarianceClassified - Internal Use Only30F-distribution is similar to Chi-Squared distribution F-分布与 -分布相似.2 -Test is to compare an estimate of process variability to a standard value for process variability.F-Test is to compare an estimate of process variability to another estimate of process variability. - 检验用于将一个过程的变检验用于将一个过程的变异度的测定值与一个过程变异异度的测定值与一个过程变异度的标准值进行比较度的标准值进行比较F-F-检验用于将一个过程的变异检验用于将一个过程的变异度的测定值与另一个过程变异度的测定值与另一个过程变异度的测定值进行比较度的测定值进行比较22F-值概率分布密度 = 0.05Classified - Internal Use Only31F-distribution Formula SSF22211s2测定值测定值 (较大值较大值)测定值测定值 (较小值较小值) For degree of freedom degree of freedom221s21s22s21s22s21s22Classified - Internal Use Only32概率分布密度F-值 = 0.05 (2.98) And F临界值 The relation between false alarm rate and FcriticalClassified - Internal Use Only33F Varies With Degrees Of FreedomV = 2小大大Classified - Internal Use Only34表 1 F-table For A False alarm Rate Of 25%分母自由分母自由度度分子自由度分子自由度行自由度与列自由度的相交值即为行自由度与列自由度的相交值即为F F- 临界值临界值Classified - Internal Use Only35表2 F-table For A False alarm Rate Of 5%分子自由度分子自由度分母自由度分母自由度行自由度与列自由度的相交值即为行自由度与列自由度的相交值即为 F- F- 临界值临界值Classified - Internal Use Only36The Hypothesis For The F-Test : There is no difference between the two variances : Larger variance is significantly different than the smaller varianceH0HAClassified - Internal Use Only37Comparing Process Variances Using F-Test 用用F-F-检验比较两过程的方差检验比较两过程的方差Step 1 由两受检过程随机采样 Randomly sample from two processes under investigation. Step 2 计算每个过程的个过程数据 Calculate the 3 pieces of information for each process. Step 3 计算 F 统计值. Calculate F statistic. Step 4 计算自由度 Calculate the appropriate degrees of freedom.Step 5 确定恰当的 值 Decide on an appropriate false alarm rate.Step 6 Step 6 查表确定 F-临界值临界值. Determine the critical value of F from the table.Step 7 评价结果 Evaluate the results.Classified - Internal Use Only38Example: Comparing Two Process Estimates Of VariabilityMarket survey shows low Brix can Sprite, while QC report shows an average Brix of 10.93. The plant manager becomes curious about his can line capability. A process capability study was than performed. The results are listed on the left table. 市场调查表明拉罐雪碧白利市场调查表明拉罐雪碧白利度偏低，而品控报告显示白度偏低，而品控报告显示白利度均值为利度均值为 10.93. 10.93. 厂长决厂长决定对罐线进行过程能力分定对罐线进行过程能力分析所得数据列于左表析所得数据列于左表维修前维修前Classified - Internal Use Only39Example: Comparing Two Process Estimates Of VariabilityThe QC Manager uses The data from the capability study to calculate the process limits and the Variability that are listed on the right. 3S = 10.93 3 x 0.079Process Limits = 10.69 11.173S = 0.237xObviously, the process is too variable, and not very accurately targeted.显然，Classified - Internal Use Only40Example: Comparing Two Process Estimates Of VariabilityThe Plant Manager has ordered the Maintenance people from Engineering to investigate the can line, repair and adjustment were done. A new capability study is performed. Data are listed on the left table. In doing this, the step 1 and step 2 of the procedures are completed.维修后维修后Classified - Internal Use Only41Example: Comparing Two Process Estimates Of VariabilityStep 3 Calculate F statistic:计算 F统计值 (或称检验值):F = F = ssafterbefore22)0486. 0()079. 0(22F统计值 = 2.54Step 4 Calculate the degrees of freedom:计算自由度计算自由度: 的自由度的自由度= n1 1 = 15 的自由度的自由度 = n2 1 = 15 Sbefore2Safter212= 15Classified - Internal Use Only42Example: Comparing Two Process Estimates Of VariabilityStep 5 Decide on a false alarm rate:确定恰当的确定恰当的 值值 = 0.05Step 6 Determine the critical value of F from the table:查查 表表2 确定确定 F-临界值临界值: .F-临界值临界值 = 2.40Step 7 Evaluate the results by compare F statistical with F critical:评价结果评价结果: :2.54 2.40; F statistical F critical is rejected. We can conclude that variability is reduced.我们可以有我们可以有 95% 95% 的把握认为维修后的罐线过程能力获得了提高的把握认为维修后的罐线过程能力获得了提高H0Classified - Internal Use Only43STEP 1Randomly sample from two processes under investigation. In this example, samples are taken from the process before maintenance and that after maintenance.由两受检过程随机采样由两受检过程随机采样. 此案例中分此案例中分别为维修前及维修后的过程别为维修前及维修后的过程SUMMARYSUMMARYSTEP 2Calculate 3 pieces of information for each process.计算出二过程的以下三个数据： 样品数： n = n2 = 162 = 16过程均值： = 10.93 = 10.91样本标准偏差：s1 = 0.079 s2 = 0.0486STEP 6F 临界值 = 2.40STEP 5 = 0.05STEP 7 F 临界值 2.40 原原维修后的罐线过程能力获得了提高维修后的罐线过程能力获得了提高x2x1Classified - Internal Use Only44T- -检验的应用检验的应用The Application Of T-Test Classified - Internal Use Only45 T-Test For Comparing Two Estimates Of Process Averages Classified - Internal Use Only46Normal Distribution with = 3, = 18Z- Do you still remember Z-table and the definition of Z-value?还记得 Z-值的定义和 Z-值表吗? xz Classified - Internal Use Only47 xzClassified - Internal Use Only48Sampling Distribution Of Sample Means样本均值的采样分布样本均值的采样分布If we randomly take vast number of samples from a source population, each sample of size N=10. We then lay all the means out in the form of frequency distribution, a sampling distribution will be resulted.假如从一个源总体中重复采集假如从一个源总体中重复采集大量样本，每一样本的样品数大量样本，每一样本的样品数 N=10 N=10 将样本的均值绘制频将样本的均值绘制频率分布图若样本由正态分布率分布图若样本由正态分布的源总体中随机抽取，则均值的源总体中随机抽取，则均值也呈正态分布也呈正态分布采样分布采样分布N=10 Mean = 10 = 3均值均值Classified - Internal Use Only49T Values And T DistributionT-T-值和值和T-T-分布分布Suppose we were to draw a vast number of pairs of random samples from the population, each sample of size N=10. We then calculate t-value for each pair of samples and lay all the t-values out in the form of frequency distribution, a t-distribution will be resulted.假如从一个源总体中重复假如从一个源总体中重复采集大量样本，每一样采集大量样本，每一样本的样品数本的样品数 N=10 N=10 计算每计算每对样本的对样本的 t- t-值值, , 将将t- t-值绘制频值绘制频率分布图，即得对称有序的率分布图，即得对称有序的 t- t-分布分布 自由度为自由度为 5 5 和和 40 40 T-T-分布分布sMMMMxbxatClassified - Internal Use Only50 T-Value For Comparing The Mean Of A Sample With The Mean Of A Population T-T-值的计算值的计算 ( (比较样本均值与总体均值比较样本均值与总体均值) )t-值计算公式 ( 已知已知 或或定，定， s s 未知未知) )Equation for t-test ( assumed known or stated, s unknown)n 样品组中的样品数样品组中的样品数 样品数为样品数为n n的样品组均值的样品组均值s 样品数为样品数为n n的样品组的标的样品组的标 准偏差准偏差 已知的过程均值或已知的过程均值或定值定值n n size of sample from a process size of sample from a process average of the sample group of average of the sample group of size n size nS S standard deviation of the sample standard deviation of the sample group of size n group of size n assumed known process average assumed known process average or stated average or stated average. .xxXClassified - Internal Use Only51T-Value For Comparing The Means Of Two Samples T-T-值的计算值的计算 ( (比较二个样本均值比较二个样本均值 ) ) t- t-值计算公式值计算公式 ( (二个过二个过程的程的 和和 均未知均未知) )Equation for t-test ( and unknown for both processes)= the average of = the average of n n1 1 samplesample 样本样本 n n1 1的均值的均值 = the average of = the average of n n2 2 sample sample 样本样本 n n2 2的均值的均值 n n1 1 = the sampling size of n= the sampling size of n1 1 sample sample 样本样本 n n1 1 的采集量的采集量 n n2 2 = the sampling size of = the sampling size of n n sample sample 样本样本 n n2 2 的采集量的采集量 S Sp p = = a measure of combination ofa measure of combination of the variance of the two samples the variance of the two samples 二个样本方差组合二个样本方差组合的参数的参数x1x2Classified - Internal Use Only52T-Value For Comparing The Means Of Two Samples T-T-值的计算值的计算 ( (比较二个样本均值比较二个样本均值 ) ) 续续Calculating the pooledvariance of two samplessp2 n1 = Sampling size of the first sample set 样本样本 n n1 1 的采集量的采集量 s s1 1 = Standard deviation of the first sample set= Standard deviation of the first sample set 样本 n1 的标准偏差的标准偏差 n n2 2 = Sampling size of the second sample set= Sampling size of the second sample set 样本 n2 的采集量的采集量 s s2 2 = Standard deviation of the first sample set= Standard deviation of the first sample set 样本 n2 的标准偏差的标准偏差 Classified - Internal Use Only53正态分布与正态分布与 T-T-分布分布自由度为自由度为 18 的的正态分布与正态分布与 T 分分布布T-distribution is flatter and more spread out. The larger the degrees of freedom, the more closely the t-distribution will approximate the normal distribution. There is only one normal distribution, while there is a whole