Capability Analysis. Chapter 255. Introduction. Capability Analysis

Transcription

1 Chapter 55 Introducton Ths procedure summarzes the performance of a process based on user-specfed specfcaton lmts. The observed performance as well as the performance relatve to the Normal dstrbuton are output. Process capablty ratos such as C p and C p are produced. C pm and C pm may also be generated f a specfcaton target s entered. A capablty hstogram wth specfcaton lmt lnes may also be produced n ths procedure. Normalty Tests are also gven n ths procedure. Subgroup data or ndvdual values may be used. Capablty analyss, or process capablty analyss, s the comparson of the dstrbuton of sample values to the specfcaton lmts, and possbly also the specfcaton target. One basc measure of the capablty of the process s the proporton of values fallng nsde (or outsde) the specfcaton lmts. Another measure of capablty s the proporton of values that would fall nsde (or outsde) the specfcaton lmts f the data are assumed to follow the normal dstrbuton. Several capablty ratos, or capablty ndces, have been developed to summarze how well the process yelds measurements wthn the specfcaton lmts. Those produced n ths procedure are C p, C p, C pm, and C pm. C pm and C pm addtonally tae nto account the nearness of the process to the specfcaton target. Process data are typcally gathered as samples or ndvdual measurements taen from the process at gven tmes (hours, shfts, days, wees, months, etc.). If more than one value s taen at a tme, the measurements of the samples at a gven tme consttute a subgroup. 55-1

2 Typcally, an ntal seres of subgroups or ndvdual values s used to estmate the mean and standard devaton of a process. The mean and standard devaton can then be used to estmate the capablty of the process. Because the assumpton of normalty s ntegral to the usefulness of the summares, an mportant part of capablty analyss s determnng whether the data follow a Normal dstrbuton, at least approxmately. Normalty tests and the capablty hstogram can be useful for nvestgatng ths assumpton. Other Procedures for Process Capablty Some of the other procedures n NCSS that may be useful for analyzng process capablty are X-bar and R (or s) charts, IM-R Charts, Descrptve Statstcs, Stem-and-Leaf Plots (for smaller samples), Normalty Tests, Outler Tests, Dstrbuton Fttng, Box-Cox Transformaton, and the Data Smulaton Tool. Process Capablty Formulas The formulas for estmatng the mean and sgma depend on whether the data s subgroup data or ndvdual value data. Estmatng the Mean Subgroup Data Suppose we have subgroups, each of sze n. Let x j represent the measurement n the j th sample of the th subgroup. The th subgroup mean s calculated usng The formula for the overall mean s x n n x j j= = 1, x n = 1 j= 1 = If the subgroups are of equal sze, the above equaton for the grand mean reduces to = 1 n x j. x = x = 1 x1 + x + + x =. Estmatng the Mean Indvdual Values Data Suppose we have ndvdual values. The estmate of the overall mean s gven by x x = =

3 Estmatng Sgma Subgroup Data In ths procedure, sgma can be entered drectly, or there are three optons for estmatng sgma from subgroup data: sample ranges, sample standard devatons, and the weghted approach. Suppose we have subgroups, each of sze n. Let x j represent the measurement n the j th sample of the th subgroup. Estmatng Sgma Subgroup Data Sample Ranges If the standard devaton (sgma) s to be estmated from the ranges, R, t s estmated as where R R = = 1 ( R) ˆ σ = E µ R d = = σ σ The calculaton of E(R) requres the nowledge of the underlyng dstrbuton of the x j s. Mang the assumpton that the x j s follow the normal dstrbuton wth constant mean and varance, the values for d are derved through the use of numercal ntegraton. It s mportant to note that the normalty assumpton s used and that the accuracy of ths estmate requres that ths assumpton be vald. In the procedure, ths calculaton s performed f Sgma Estmaton s set to From Data R-bar or s-bar Estmate, and Range or SD Estmaton s set to Range. Estmatng Sgma Subgroup Data Sample Standard Devatons If the standard devaton (sgma) s to be estmated from the standard devatons, t s estmated as where s s = = 1 ( s) ˆ σ = E µ s c 4 = = σ σ The calculaton of E(s) requres the nowledge of the underlyng dstrbuton of the x j s. Mang the assumpton that the x j s follow the normal dstrbuton wth constant mean and varance, the values for c 4 are obtaned from c 4 = R d s c 4 n Γ n 1 n 1 Γ In the procedure, ths calculaton s performed f Sgma Estmaton s set to From Data R-bar or s-bar Estmate, and Range or SD Estmaton s set to SD. 55-3

4 Estmatng Sgma Subgroup Data Weghted (SD) Approach When the sample sze s varable across subgroups, a weghted approach s recommended for estmatng sgma (Montgomery, 013): ( n 1) s = 1 ˆ σ = n = 1 In the procedure, ths calculaton s performed f Sgma Estmaton s set to From Data SD Approach. 1/ Estmatng Sgma Indvdual Values Data Suppose we have ndvdual values. There are two methods n ths procedure for estmatng sgma: movng ranges and overall sample standard devaton. Estmatng Sgma Indvdual Values Data Movng Ranges If there s only one observaton per tme pont, a movng range may be calculated by fndng the range of each value wth ts prevous value: R = x x 1 Then the standard devaton (sgma) s estmated from the ranges, R, n the same manner as for subgroup data, namely, where ˆ σ = R R = = 1 E( R) µ R d = = σ σ In the procedure, ths calculaton s performed f the data are ndvdual values data, and Sgma Estmaton s set to From Data R-bar or s-bar Estmate, and Range or SD Estmaton s set to Range. Estmatng Sgma Indvdual Values Data Overall Standard Devaton If there s only one observaton per tme pont, and the process s assumed to be n control, sgma may be estmated usng the sample standard devaton ( x x) = 1 ˆ σ = 1 In the procedure, ths calculaton s performed f the data are ndvdual values data, and f Sgma Estmaton s set to From Data SD Approach. R d 55-4

5 Process Capablty Ratos Several capablty rato formulas are presented below. Further detals may be found n Montgomery (013) and Ryan (011). Cp The process capablty rato C p s gven by USL LSL C p = 6σ where USL and LSL are the upper and lower specfcaton lmts, respectvely. An estmate of C p s produced by substtutng a sutable estmate of σ, namely σˆ. Confdence ntervals for Cp are gven as C C p lower p upper = C = C p p χ χ n 1, α / n 1 n 1,1 α / n 1 Cpl and Cpu The one-sded capablty ratos C pl and C pu are defned as and µ LSL C pl = 3σ µ C = ULS pu 3σ These are estmated by substtutng mean and standard devaton estmates. Cp C p s the lesser of C pl and C pu, or C = mn( C The lower and upper confdence lmts for C p reported n NCSS are gven by p pl, C pu ) C C = C p z1 p lower α / = C + z1 p upper p α / n 1 C p n( n 3) n 6 n 1 n 1 C p n( n 3) n 6 n

6 Cpm A capablty rato whch ncorporates the nearness to the specfcaton target s defned as C pm = USL LSL ( µ ) 6 σ + T where T refers to the specfcaton target. A sutable estmate of C pm s made by substtutng estmates of the mean and standard devaton. Cpm Smlarly to C pm, C pm also accounts for nearness to the specfcaton target: C p C pm = µ T 1 + σ where T refers to the specfcaton target. A sutable estmate of C pm s made by substtutng estmates of the mean and standard devaton. Data Structure In ths procedure, the data may be n any of three formats. The frst data structure opton s to have the data n several columns, wth one subgroup per row. Example dataset S1 S S3 S4 S

7 The second data structure opton uses one column for the response data, and ether a subgroup sze or a second column defnng the subgroups. Alternatve example dataset Response Subgroup In the alternatve example dataset, the Subgroup column s not needed f every subgroup s of sze 5 and the user specfes 5 as the subgroup sze. If there are mssng values, the Subgroup column should be used, or the structure of the frst example dataset. If there are no subgroups (ndvdual values only), the only nput needed s a sngle column of values. Response

8 Procedure Optons Ths secton descrbes the optons avalable n ths procedure. To fnd out more about usng a procedure, go to the Procedures chapter. Varables Tab Ths panel specfes the varables that wll be used n the analyss. Input Type Specfy whether the data s n a sngle response column or n multple columns wth one subgroup per row. Specfy whether the data s n a sngle response column wth an assocated subgroup column, multple columns wth one subgroup per row, or a column wth ndvdual values (no subgroups). Response Column and Subgroup Column or Subgroup Sze Response Subgroup Multple Columns wth One Subgroup Per Row X1 X X

9 Response Column wth Indvdual Values (no subgroups) Response Varables Response Column Response Varable Specfy the column wth the data values. Subgroup Specfcaton Specfy whether subgroups are defned by a Subgroup ID varable, or by a subgroup sze. If the subgroup sze s 3, then subgroups are formed by gong down the response column n groups of 3. The frst subgroup would be 5. Subgroup ID Varable Specfy the column contanng the subgroup dentfers. Response ID Varable A new subgroup s created for each change n the Subgroup ID Varable, gong down. 55-9

10 Subgroup Sze Specfy the number of ndvduals n each subgroup. Response If the subgroup sze s 3, then subgroups are formed by gong down the response column n groups of 3. The frst subgroup would be 5, 6, 4; the second would be 3, 7, 6; and so on. Varables Multple Columns Data Varables Specfy the columns contanng the sample responses. Each row represents a subgroup. X1 X X Specfy Rows Specfcaton Method Select whch method wll be used to specfy the rows of the data. All Rows All rows n the response column(s) wll be used. Enter Frst Row and Last Row Specfy the frst row and the last row of the data. Frst N Rows (Enter N) The data begnnng at Row 1 and endng at Row N wll be used. Last N Rows (Enter N) The last N rows of the dataset wll be used

11 Keep Rows Varable Specfy a varable and a value n that varable column that wll be used to determne whch rows are used. Remove Rows Varable Specfy a varable and a value n that varable column that wll be used to determne whch rows wll not be used. Frst Row Specfy the begnnng row to be used. Last Row Specfy the last row to be used. N Enter the number of rows to be used. Keep Rows Varable Ths varable (column) s used to specfy whch rows of the data wll be used. Keep Rows Value Ths value determnes whch rows of the Keep Rows Varable wll be used. Remove Rows Varable Ths varable (column) s used to specfy whch rows of the data wll not be used. Remove Rows Value Ths value determnes whch rows of the Remove Rows Varable wll not be used. Lmts & Estmaton Tab The optons on the Lmts & Estmaton tab are enter the specfcaton lmts, and to specfy the method by whch sgma s estmated. Specfcaton Lmts Lower Lmt Enter the lower specfcaton lmt. Ths lower specfcaton lmt s requred for calculatons of C p, C pl, C pm, and may be used n calculatons of C p and C pm. Ths lower lmt can also be dsplayed on the capablty hstogram. Upper Lmt Enter the upper specfcaton lmt. Ths upper specfcaton lmt s requred for calculatons of C p, C pl, C pm, and may be used n calculatons of C p and C pm. Ths lower lmt can also be dsplayed on the capablty hstogram. Target Specfcaton Value Enter the target value. The target value s requred for calculaton of C pm and C pm. Ths target value can be dsplayed on the capablty hstogram. Mean Value Optons Mean Value Estmaton Specfy whether the mean wll be estmated from the data, or whether t wll be specfed drectly

12 From Data Estmate sgma based on the mean range. Only the subgroups specfed for use n calculatons wll be used. Enter Mean Value Specfy the mean value drectly. Use a Varable wth Mean Value Specfy a column contanng the mean value n row 1. Mean Value Enter the value to be used for the mean n all calculatons. Mean Value Varable Specfy a column contanng the mean value n row 1. Ths value wll be used for the mean n all calculatons. Sgma Estmaton Optons Sgma Estmaton Specfy the method by whch Sgma wll be estmated for the capablty analyss calculatons. From Data R-bar or s-bar Estmate Estmate sgma based on the average of the ranges or standard devatons (whchever s specfed under Range or SD Estmaton). When there are no subgroups (ndvdual values only), sgma wll be estmated wth the ranges of each value wth the prevous value, as s-bar cannot be estmated wth only one value at each tme pont. From Data SD Approach When there are subgroups, ths method estmates sgma usng a weghted approach estmate formula that s recommended when the subgroup sze vares across subgroups. When there are no subgroups (ndvdual values only), sgma s estmated usng the common sample standard devaton formula. Enter Sgma Value In ths case the sgma value s entered drectly. Ths sgma value s used n all calculatons nvolvng sgma. Use a Varable wth Sgma Value Specfy a column contanng the sgma value n row 1. Ths sgma value s used n all calculatons nvolvng sgma. Sgma Value Enter the value to be used for sgma. Ths sgma value s used n all calculatons nvolvng sgma Sgma Varable Specfy a column contanng the sgma value n row 1. Ths sgma value s used n all calculatons nvolvng sgma. 55-1

13 Reports Tab The followng optons control the format of the reports. Specfy Reports Estmaton Summary Secton Ths report gves the estmated mean and sgma to be used n the remanng calculatons. Secton Ths report gves a performance summary as well as several process capablty ratos. Normalty Tests Secton Ths report gves three normalty tests. Report Optons Precson Specfy the precson of numbers n the report. A sngle-precson number wll show seven-place accuracy, whle a double-precson number wll show thrteen-place accuracy. Note that the reports are formatted for sngle precson. If you select double precson, some numbers may run nto others. Also note that all calculatons are performed n double precson regardless of whch opton you select here. Ths s for reportng purposes only. Varable Names Ths opton lets you select whether to dsplay varable names, varable labels, or both. Page Ttle Ths opton specfes a ttle to appear at the top of each page. Plot Subttle Ths opton specfes a subttle to appear at the top of each plot. Hstogram Tab Ths panel sets the optons used to defne the appearance of the hstogram. Select Plots Capablty Hstogram Each chart s controlled by three form objects: 1. A checbox to ndcate whether the chart s dsplayed.. A format button used to call up the plot format wndow (see Hstogram Format Optons below for more formattng detals). 3. A second checbox used to ndcate whether the chart can be edted durng the run. Specfcaton Lmts on Hstogram Chec ths box to nclude the specfcaton lmt lnes on the hstogram

14 Hstogram Format Optons To learn detals regardng the format of the hstogram, go to the Hstograms chapter of the documentaton. Common optons, such as axes, labels, legends, and ttles are documented n the Graphcs Components chapter. Storage Tab The optons on ths panel control the automatc storage of the means and ranges on the current dataset. Storage Columns Store Means n Column You can automatcally store the means of each subgroup nto the column specfed here. Warnng: Any data already n ths column s replaced. Be careful not to specfy columns that contan mportant data. Store Ranges/SDs n Column You can automatcally store the range or standard devaton of each subgroup nto the column specfed here. Warnng: Any data already n ths column s replaced. Be careful not to specfy columns that contan mportant data

15 Example 1 for Subgroup Data Ths secton presents an example of how to run a capablty analyss for subgroups. The data represent 50 subgroups of sze 5 that are assumed to be n control. The specfcaton lmts for the process are 50 and 80, wth a specfcaton target of 65. The data used are n the QC dataset. We wll analyze the varables D1 through D5 of ths dataset. You may follow along here by mang the approprate entres or load the completed template Example 1 by clcng on Open Example Template from the Fle menu of the wndow. 1 Open the QC dataset. From the Fle menu of the NCSS Data wndow, select Open Example Data. Clc on the fle QC.NCSS. Clc Open. Open the wndow. Usng the Analyss menu or the Procedure Navgator, fnd and select the procedure. On the menus, select Fle, then New Template. Ths wll fll the procedure wth the default template. 3 Specfy the varables. On the wndow, select the Varables tab. Double-clc n the Data Varables text box. Ths wll brng up the varable selecton wndow. Select D1 through D5 from the lst of varables and then clc O. D1-D5 wll appear n the Data Varables box. 4 Set the Specfcaton Lmts. On the wndow, select the Lmts & Estmaton tab. Enter 50 for Lower Lmt. Enter 80 for Upper Lmt. Enter 65 for Target Specfcaton Value. 5 Run the procedure. From the Run menu, select Run Procedure. Alternatvely, just clc the green Run button. Mean Estmaton Secton Mean Estmaton Secton for Subgroups 1 to 50 Number of Subgroups 50 Estmaton Type Estmate Estmated Grand Average 67.1 Ths secton dsplays the estmated mean to be used n all calculatons. Estmated Grand Average Ths value s the average of all the observatons. If all the subgroups are of the same sze, t s also the average of all the X-bars

16 Sgma Estmaton Secton Sgma Estmaton Secton for Subgroups 1 to 50 Estmaton Estmated Estmated Type Value Sgma Ranges (R-bar)* Standard Devatons (s-bar) Weghted Approach (s-bar) * Indcates the estmaton type used n ths report. Ths report gves the estmaton of the populaton standard devaton (sgma) based on three estmaton technques. The estmaton technque used for the calculatons n ths procedure s based on the ranges. Estmaton Type Each of the formulas for estmatng sgma s shown earler n ths chapter n the Process Capablty Formulas secton. Estmated Value Ths column gves the R-bar and s-bar estmates based on the correspondng formulas. Estmated Sgma Ths column gves estmates of the populaton standard devaton (sgma) based on the correspondng estmaton type. Secton Data Summary Number of Values 50 Sgma (Estmated) Mean (Estmated) 67.1 Number of Values Ths s the number of observatons n the capablty analyss. Whle subgroups were used n the estmaton of sgma, they are no longer dstngushed n the remander of the capablty analyss. Sgma Ths s the sgma that wll be used for the capablty analyss calculatons. Mean Ths s the mean to be used for the capablty analyss calculatons. Specfcaton Summary Correspondng Specfcaton Value Z-Value Lower Lmt Upper Lmt Target Value Ths report lsts the user-specfed specfcaton values, as well as the correspondng Z-value. Specfcaton Ths column dentfes the specfcaton value type

17 Value Ths s the user-nput specfcaton value. The target value s only requred for the C pm and C pm capablty ratos. Correspondng Z-Value These are the z-values of the specfcaton lmts and target value, calculated usng the formula ˆ µ z = spec spec ˆ σ Performance Summary LL Observed Performance Normal Dstrbuton Performance - Specfcaton Value # < LL % < LL PPM < LL % < LL PPM < LL Lower Lmt (LL) 50 4/ % % UL Observed Performance Normal Dstrbuton Performance - Specfcaton Value # > UL % > UL PPM > UL % > UL PPM > UL Upper Lmt (UL) 80 11/ % % Observed Performance Normal Dstrbuton Performance - Specfcaton # Outsde % Outsde PPM Outsde % Outsde PPM Outsde Outsde Both Lmts 15/ % % Range Observed Performance Normal Dstrbuton Performance - Specfcaton (UL - LL) LL # UL LL % UL LL PPM UL LL % UL LL PPM UL Between Lmts 30 35/ % % Ths report gves the percentage of values nsde or outsde the specfcaton lmts. In ths example, the observed performance s smlar to the Normal dstrbuton (expected) performance. Specfcaton Ths dentfes the regon to be examned for performance. LL, UL, and Range The LL and the UL values are the user-specfed lower and upper specfcaton lmts. The range s the lower lmt subtracted from the upper lmt. Observed Performance # Ths gves the actual number of observed values n the correspondng regon. Observed Performance % Ths gves the percent of observed values n the correspondng regon. Observed Performance PPM Ths gves the parts per mllon number of observed values n the correspondng regon. Normal Dstrbuton Performance % If the values are assumed to follow a normal dstrbuton wth mean µˆ and standard devaton σˆ, ths s the percent of values that would fall n the correspondng regon. Ths s sometmes called the expected performance. Normal Dstrbuton Performance PPM If the values are assumed to follow a normal dstrbuton wth mean µˆ and standard devaton σˆ, ths s the parts per mllon number of values that would fall n the correspondng regon. Ths s sometmes called the expected performance

18 Process Capablty Ratos Process Capablty Ratos Confdence Level: 95.00% Capablty ---- Confdence Interval ---- Rato Value Lower Lmt Upper Lmt Cp (wth C.I.) Cp (wth C.I.) Cpl Cpu Cpm Cpm Ths report gves the values of the varous capablty ratos. Confdence ntervals are gven for C p and C p. We refer the reader to Montgomery (013) or Ryan (011) for nterpretaton detals of each rato. Capablty Rato Ths dentfes the capablty rato of each lne. Value and Confdence Interval Lmts The formulas for each of these values are gven earler n ths chapter n the Process Capablty Ratos secton under Process Capablty Formulas. 3- to 6-Sgma Lmts Lmt Type Mean Lower Lmt Upper Lmt 3-Sgma Lmts Sgma Lmts Sgma Lmts Sgma Lmts Lmts The formulas for the lmts are where m s the multpler 3, 4, 5, or 6. LL = ˆ µ m ˆ σ UL = ˆ µ + m ˆ σ Normalty Tests Secton Normalty Tests Secton User-Specfed Alpha Level: 0.05 Normalty Test Prob Test Statstc Level Concluson Shapro-Wl Do Not Reject Normalty Assumpton Anderson-Darlng Do Not Reject Normalty Assumpton Ch-Square Do Not Reject Normalty Assumpton The detals of the Shapro-Wl and Anderson-Darlng (and other) Normalty tests are dscussed n the Normalty Tests procedure. The Ch-Square goodness of ft test for normalty s obtaned by dvdng the data nto bns, and then comparng the observed counts to the expected counts for each bn usng 55-18

19 χ = = 1 ( O E ) E The ndvdual observed and expected counts are detaled n the Ch-Square Test Frequency Dstrbuton Detals secton. Ch-Square Test Frequency Dstrbuton Detals Bn Boundares Lower Upper Actual Normal Dff. Actual Normal Dff. Ch-Sqr Boundary Boundary Count Count Count Percent Percent Percent Amount Total Ths secton summarzes the contrbuton of each bn to the Ch-Square goodness of ft test statstc. Hstogram Secton Ths hstogram also dsplays the (vertcal lne) specfcaton lmts as well as the specfcaton target

20 Example for Indvdual Value Data The capablty analyss of ndvdual value data s nearly the same as the analyss of subgroup data. The only dfference s the way n whch sgma s estmated. Ths secton presents an example of how to run a capablty analyss for ndvdual value data. The data represent 00 part wdths of a process that s assumed to be n control. The specfcaton lmts for the process are 300 and 400, wth a specfcaton target of 350. The data used are n the Capablty dataset. We wll analyze the varable Wdth of ths dataset. You may follow along here by mang the approprate entres or load the completed template Example by clcng on Open Example Template from the Fle menu of the wndow. 1 Open the Capablty dataset. From the Fle menu of the NCSS Data wndow, select Open Example Data. Clc on the fle Capablty.NCSS. Clc Open. Open the wndow. Usng the Analyss menu or the Procedure Navgator, fnd and select the procedure. On the menus, select Fle, then New Template. Ths wll fll the procedure wth the default template. 3 Specfy the varables. On the wndow, select the Varables tab. Change the Input Type to Response Column wth Indvdual Values (no subgroups). Double-clc n the Response Varable text box. Ths wll brng up the varable selecton wndow. Select Wdth from the lst of varables and then clc O. Wdth wll appear n the Response Varable box. 4 Set the Specfcaton Lmts. On the wndow, select the Lmts & Estmaton tab. Enter 300 for Lower Lmt. Enter 400 for Upper Lmt. Enter 350 for Target Specfcaton Value. 5 Set the Sgma Estmaton Method. On the wndow, select the Lmts & Estmaton tab. Change the Sgma Estmaton to From Data SD Approach. 6 Run the procedure. From the Run menu, select Run Procedure. Alternatvely, just clc the green Run button. 55-0

21 Output Mean Estmaton Secton for All Indvduals Number of Indvduals 00 Estmaton Type Estmate Estmated Grand Average Sgma Estmaton Secton for All Indvduals Estmaton Estmated Estmated Type Value Sgma Ranges (R-bar) Standard Devaton* * Indcates the estmaton type used n ths report. Secton Data Summary Number of Values 00 Sgma (Estmated) Mean (Estmated) Specfcaton Summary Correspondng Specfcaton Value Z-Value Lower Lmt Upper Lmt Target Value Performance Summary LL Observed Performance Normal Dstrbuton Performance - Specfcaton Value # < LL % < LL PPM < LL % < LL PPM < LL Lower Lmt (LL) 300 4/ % % UL Observed Performance Normal Dstrbuton Performance - Specfcaton Value # > UL % > UL PPM > UL % > UL PPM > UL Upper Lmt (UL) 400 6/ % % Observed Performance Normal Dstrbuton Performance - Specfcaton # Outsde % Outsde PPM Outsde % Outsde PPM Outsde Outsde Both Lmts 10/ % % Range Observed Performance Normal Dstrbuton Performance - Specfcaton (UL - LL) LL # UL LL % UL LL PPM UL LL % UL LL PPM UL Between Lmts / % % Process Capablty Ratos Confdence Level: 95.00% Capablty ---- Confdence Interval ---- Rato Value Lower Lmt Upper Lmt Cp (wth C.I.) Cp (wth C.I.) Cpl Cpu Cpm Cpm to 6-Sgma Lmts Lmt Type Mean Lower Lmt Upper Lmt 3-Sgma Lmts Sgma Lmts Sgma Lmts Sgma Lmts

22 Normalty Tests Secton User-Specfed Alpha Level: 0.05 Normalty Test Prob Test Statstc Level Concluson Shapro-Wl Reject Normalty Assumpton Anderson-Darlng Do Not Reject Normalty Assumpton Ch-Square Do Not Reject Normalty Assumpton Ch-Square Test Frequency Dstrbuton Detals Bn Boundares Lower Upper Actual Normal Dff. Actual Normal Dff. Ch-Sqr Boundary Boundary Count Count Count Percent Percent Percent Amount Total Hstogram Secton for All Indvduals The output descrptons for each secton of the output are presented n Example 1. The only dfference n formulas n Example compared to Example 1 s the dfference n the calculaton of the sgma estmate. In Example, the common sample standard devaton formula usng all the ndvdual values s used to calculate the sgma estmate. 55-