1、rx一、管制圖公式說明1.計量值公式管制圖x管制圖： n為組內樣本量， m為抽樣組數；標準偏差n2minmaxxxr估計標準偏差2dr全距平均值mrmrrrrmiim121.管制上限raxrndxucl22)3(中心線xcl管制下限raxrndxlcl22)3(其中nda223r 管制圖：r的標準偏差)(23drdr管制上限rddrdrruclr423)(33中心線rcl管制下限rddrdrruclr323)(33其中23331ddd，23431dddmxnxxxxmiin1m.32121mxxxxx.sx管制圖x管制圖：第 i 組之標準偏差1)(12nxxsniiimiisms11估計標準偏

2、差4cs管制上限saxsncxucl34)3(中心線xcl管制下限saxsncxlcl34)3(其中nca433s管制圖：管制上限sbucls4中心線scls管制下限sblcls3x-rm 管制圖rm管制圖：移動全距1iiixxmrnmrmrnii1管制上限mrducl4中心線mrcl管制下限mrdlcl3(當 n=2時，3d和4d以樣本數為 2來查表 ) 個別管制圖管制上限23dmrxucl中心線xcl管制下限23dmrxlcl(當 n=2時，2d以樣本數為 2 來查表 ) * 中位數隨著計算機技術的發展，計算已經不是困難，逐步被淘汰* 2.計數值公式不良率管制圖( p chart ) 當每

3、組之樣本數均相同時：中心線kiipkp11管制上限)1,)1(3min(npppucl管制下限)0,)1 (3max(nppplcl當各組之樣本數不相同時：中心線niiipnnp11，其中knnnn.21各組管制上下限分別為管制上限)1,)1(3min(inpppucl管制下限)0,)1(3max(inppplcl不良數管制圖( pn chart ) 中心線npcl管制上限)1 (3pppuclnn管制下限)1 (3ppplclnn其中np為各組之不合格數。np 為各組不合格數之平均值。缺點數管制圖( c chart ) 中心線ccl管制上限3 ccucl管制下限3 cclcl其中c為平均不合

4、格點數kccki 1單位缺點數管制圖( u chart ) 中心線ucl管制上限3nuuucl管制下限3nuulcl其中： n = 樣本大小(檢查的單位數 ) c = 各組的缺點數u = 各組的單位缺點數= ncncu二、統計指標說明1.制程能力指數( process capability indexes ) ),max()(lslslsluslslca；其中：平均值sl：規格標準usl：規格上限lsl：規格下限6lsl)-(uslcp；：估計標準偏差( capability sigma ) 。3)-(uslcpu；：估計標準偏差( capability sigma ) 。3)(cpllsl；

5、：估計標準偏差( capability sigma ) 。),min(cplcpucpkuslxcc)(61lsluslcpcr；：估計標準偏差( capability sigma ) 。22)(6)(sllsluslcpm；：估計標準偏差( capability sigma ) 。2.制程表現指數( process performance indexes ) 6lsl)-(uslpp；：制程標準偏差( population standard sigma )。3)-(uslppu；：制程標準偏差( population standard sigma )。3)(ppllsl；：制程標準偏差( p

6、opulation standard sigma )。),min(pplppuppk)(61lsluslpppr；：制程標準偏差( population standard sigma )。22)(6)(sllsluslppm；：制程標準偏差( population standard sigma )。3.直方圖 )-(uslf)zupper(per；：制程標準偏差( population standard sigma )。)(f)zupper(perlsl；：制程標準偏差( population standard sigma )。)-(usl.)zupper(cap；：估計標準偏差( capab

7、ility sigma ) 。)(.)zupper(caplsl；：估計標準偏差( capability sigma ) 。 skew ( 偏態) = iisxxnnn3)2)(1(；s：樣本標準偏差， n 需大于 2 且 s 0。 kurtosis ( 峰度) = )3)(2()1( 3)3)(2)(1()1(24nnnsxxnnnnnii；s：樣本標準偏差， n 需大于 2且 s 0。 chi-square = 22)1(sn；s：制程標準偏差，62lslusl。4.散布圖 回歸直線y = mx + b ； 其中 m：斜率， b：截距。22)()()(xxnyxxynm222)(xxnxy

8、xxyb三、標準偏差公式 估計標準偏差( capability sigma ) 以r估計：2dr；( rxchart及 x-rm chart時使用 )。 以s估計：4cs；( sxchart時使用 )。 實際計算：kiikinjiijnxx1112) 1()( 制程標準偏差( population standard devitation ) )1()(12nxxnii四、參數表1.表 1:x(bar)-r 和 x(bar)-s 參數均值圖均值圖計算控制限用的系數標準差估計值的除數計算控制限用的系數標準差估計值的除數na2d2d3d4a3c4b3b421.8801.128-3.2672.6590

9、.7979-3.27631.0231.693-2.5711.9540.8862-2.56840.7292.059-2.2821.6280.9213-2.26650.5772.326-2.1141.4270.9400-2.08960.4832.543-2.0041.2870.95150.0301.97070.4192.7040.0761.9241.1820.95940.1181.88280.3732.8470.1361.8641.0990.96500.1851.81590.3372.9700.1841.8161.0320.96930.2391.761100.3083.0780.2231.7770

10、.9750.97270.2841.716110.2853.1730.2561.7440.9270.97540.3211.679120.2663.2580.2831.7170.8860.97760.3541.640130.2493.3360.3071.6930.8500.97940.3821.618140.2353.4070.3281.6720.8170.98100.4061.594150.2233.4720.3471.6530.7890.98230.4281.572160.2123.5320.3631.6370.7630.98350.4481.552170.2033.5880.3781.622

11、0.7390.98450.4461.534180.1943.6400.3911.6080.7180.98540.4821.518190.1873.6890.4031.5970.6980.98620.4971.503200.1803.7350.4151.5850.6800.98690.5101.490210.1733.7780.4251.5750.6630.98760.5231.477220.1673.8190.4341.5660.6470.98820.5341.466230.1623.8580.4431.5570.6330.98870.5451.455240.1573.8950.4511.54

12、80.6190.98920.5551.445250.1533.9310.4591.5410.6060.98960.5651.435計算控制限用的系數子組容量- 圖-s 圖控制圖的常數和公式表全距圖標準差圖計算控制限用的系數2.表 2: 中位數 和 單值 參數中位數圖單值圖na2d2d3d4e2d2d3d421.8801.128-3.2672.6601.128-3.26731.1871.693-2.5741.7721.693-2.57440.7962.059-2.2821.4572.059-2.28250.6912.326-2.1141.2902.326-2.11460.5482.534-2.0

13、041.1842.534-2.00470.5082.7040.0761.9241.1092.7040.0761.92480.4332.8470.1361.8641.0542.8470.1361.86490.4122.9700.1841.8161.0102.9700.1841.816100.3623.0780.2231.7770.9753.0780.2231.777子組容量中位數圖單值圖全距圖全距圖計算控制限用的系數標準差估計值的除數計算控制限用的系數計算控制限用的系數標準差估計值的除數計算控制限用的系數3.表 3:正態分布參數表 ppm 值: x.x0 x.x1x.x2x.x3x.x4x.x5

14、x.x6x.x7x.x8x.x94.00.000033.90.000050.000050.000040.000040.000040.000040.000040.000040.000030.000033.80.000070.000070.000070.000060.000060.000060.000060.000050.000050.000053.70.000110.000100.000100.000100.000090.000090.000080.000080.000080.000083.60.000160.000150.000150.000140.000140.000130.000130.0

15、00120.000120.000113.50.000230.000220.000220.000210.000200.000190.000190.000180.000170.000173.40.000340.000320.000310.000300.000290.000280.000270.000260.000250.000243.30.000480.000470.000450.000430.000420.000400.000390.000380.000360.000353.20.000690.000660.000640.000620.000600.000580.000560.000540.00

16、0520.000503.10.000970.000940.000900.000870.000840.000820.000790.000760.000740.000713.00.001350.001310.001260.001220.001180.001140.001110.001070.001040.001002.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.00142.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.00192.70.00350.00340.0

17、0330.00320.00310.00300.00290.00280.00270.00262.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.00362.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.00482.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.00642.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.00842.2

18、0.01390.01360.01320.01290.01250.01220.01190.01160.01130.01102.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.01432.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.01831.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.02331.80.03590.03510.03440.03360.03290.03220.03140.0307

19、0.03010.02941.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.03671.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.04551.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.05591.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.06811.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.08231.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.09851.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.11701.00.15870.15620.15390.15150.14920.1469