Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided tolerace iterval cosists of two limits betwee which a give proportio β of the populatio falls with a give cofidece level 1. A oe-sided tolerace iterval is similar, but cosists of a sigle upper or lower limit. Techical Details Let X 1, X 2,, X be a radom sample for a populatio with distributio fuctio F ( X ). A ( β 1 ) β -cotet tolerace iterval ( ) L T U A ( β 1 ) T, is defied by [ F ( T ) F( T ) β ] 1, lower, oe-sided β -cotet tolerace boudt L is defied by U L [ 1 F ( T ) β ] 1 A ( β, 1 ) upper, oe-sided β -cotet tolerace boud T U is defied by L [ F ( T ) β ] 1 Note that a oe-sided tolerace limit is the same as the oe-sided cofidece limit of the quatile of F. Distributio-Free U, two-sided The defiitio of two-sided distributio-free tolerace itervals is foud i may places. We use the formulatio give by Bury (1999). The oly distributioal assumptio made about F is that it is a cotiuous, o-decreasig, probability distributio. That is, these itervals should ot be used with discrete data. Give this, the tolerace limits are T = X, T = X L ( r ) U ( s) where r ad s are two order idices. The values of r ad s are determied usig the formula where i 2 c i i ( ) β 1 β 1 i= 0 r = c s = c +1 The value of c is foud as the largest value for which the above iequality is true. 585-1
A lower, oe-sided tolerace boud is X ( r ) where r is the largest value for with the followig iequality is true. r i i ( ) β 1 β 1 i= 0 i A upper, oe-sided tolerace boud is X ( s) where s is the largest value for with the followig iequality is true. s 1 i i i ( ) β 1 β 1 i= 0 Normal-Distributio Tolerace Iterval The limits discussed i this sectio are based o the assumptio that F is the ormal distributio. Two-Sided Limits I this case, the two-sided tolerace iterval is defied by the iterval T = x ks, T = x + ks L The costructio reduces to the determiatio of the costat k. Howe (1969) provides the followig approximatio which is early exact for all values of greater tha oe where U k = uvw u = z 1+ 1 β 1+ 2 v = 1 2 χ 1, w = + 2 3 χ 1 2 + 1 ( ) Note that origially, Howe (1969) used 2 i the above defiitio of w. But Guether (1977) gives the corrected versio usig 3 show above. Oe-Sided Boud A oe-sided tolerace boud ( boud is used istead of limit i the oe-sided case) is give by Here k is selected so that TU = x + ks ( t δ k ) 1, 2 1, = = 1 where t f,δ represets a ocetral t distributio with f degrees of freedom ad ocetrality δ = z. β 585-2
Data Structure The data are cotaied i a sigle colum. ocedure Optios This sectio describes the optios available i this procedure. To fid out more about usig a procedure, tur to the ocedures chapter. Followig is a list of the procedure s optios. Variables Tab The optios o this pael specify which variables to use. Data Variables Variables Specify a list of oe or more variables for which tolerace itervals are to be geerated. You ca double-click the field or sigle click the butto o the right of the field to brig up the Variable Selectio widow. Group Variable You ca specify a groupig variable. Whe specified, a separate set of reports is geerated for each uique value of this variable. Frequecy Variable Frequecy Variable This optioal variable specifies the umber of observatios (couts) that each row represets. Whe omitted, each row represets a sigle observatio. If your data is the result of a previous summarizatio, you may wat certai rows to represet several observatios. Note that egative values are treated as a zero cout ad are omitted. Populatio Percetages Populatio Percetages for Toleraces Specify a list of percetages for which tolerace itervals are to be calculated. Note that a tolerace iterval is a pair of umbers betwee which a specified percetage of the populatio falls. This value is that specified percetage. I the list, umbers are separated by blaks or commas. Specify sequeces with a colo, puttig the icremet iside paretheses. For example: 5:25(5) meas 5 10 15 20 25. All values i the list must be betwee 1 ad 99. Data Trasformatio Optios Expoet Occasioally, you might wat to obtai a statistical report o the square root or log of your variable. This optio lets you specify a o-the-fly trasformatio of the variable. The form of this trasformatio is X = Y A, where Y is the origial value, A is the selected expoet, ad X is the resultig value. 585-3
Additive Costat Occasioally, you might wat to obtai a statistical report o a trasformed versio of a variable. This optio lets you specify a o-the-fly trasformatio of the variable. The form of this trasformatio is X = Y+B, where Y is the origial value, B is the specified costat, ad X is the value that results. Note that if you apply both the Expoet ad the Additive Costat, the form of the trasformatio is X = ( Y + B) A. Reports Tab The optios o this pael cotrol the reports ad plots displayed. Select Reports Descriptive Statistics... Normality Tests Idicate whether to display the idicated reports. Report Optios Alpha Level This is the value of alpha for the cofidece limits ad rejectio decisios. Usually, this umber will rage from 0.1 to 0.001. The default value of 0.05 results i 95% tolerace limits. ecisio Specify the precisio of umbers i the report. A sigle-precisio umber will show seve-place accuracy, while a double-precisio umber will show thirtee-place accuracy. The double-precisio optio oly works whe the Decimals optio is set to Geeral. Note that the reports were formatted for sigle precisio. If you select double precisio, some umbers may ru ito others. Also ote that all calculatios are performed i double precisio regardless of which optio you select here. This is for reportig purposes oly. Variable Names This optio lets you select whether to display oly variable ames, variable labels, or both. Value Labels This optio applies to the Group Variable. It lets you select whether to display data values, value labels, or both. Use this optio if you wat the output to automatically attach labels to the values (like 1=Yes, 2=No, etc.). See the sectio o specifyig Value Labels elsewhere i this maual. Report Optios Decimal Places Values... obabilities Decimals Specify the umber of digits after the decimal poit to display o the output of values of this type. Note that this optio i o way iflueces the accuracy with which the calculatios are doe. Eter 'Geeral' to display all digits available. The umber of digits displayed by this optio is cotrolled by whether the PRECISION optio is SINGLE or DOUBLE. 585-4
Plots Tab The optios o this pael cotrol the appearace of the histogram ad probability plot. Select Plots Histogram ad obability Plot Idicate whether to display these plots. Click the plot format butto to chage the plot settigs. Example 1 Geeratig This sectio presets a detailed example of how to geerate tolerace itervals for the Height variable i the Height dataset. To ru this example, take the followig steps (ote that step 1 is ot ecessary if the Height dataset is ope): 1 Ope the Height dataset. From the File meu of the NCSS Data widow, select Ope Example Data. Click o the file Height.NCSS. Click Ope. 2 Ope the widow. Usig the Aalysis meu or the ocedure Navigator, fid ad select the procedure. O the meus, select File, the New Template. This will fill the procedure with the default template. 3 Specify the optios o the Variables tab. O the widow, select the Variables tab. (This is the default.) Double-click i the Variables text box. This will brig up the variable selectio widow. Select Height from the list of variables ad the click Ok. Set the Populatio Percetages to 50 75 80 90 95 99. 4 Specify the optios o the Reports tab. O the widow, select the Reports tab. Set the Decimals-Values to 3. Set the Decimals-Meas to 3. Set the Decimals-obabilities to 2. 5 Ru the procedure. From the Ru meu, select Ru ocedure. Alteratively, just click the gree Ru butto. The followig reports ad charts will be displayed i the Output widow. Descriptive Statistics Stadard Stadard Cout Mea Deviatio Error Miimum Maximum Rage 20 62.100 8.441 1.887 51.000 79.000 28.000 This report was defied ad discussed i the Descriptive Statistics procedure chapter. We refer you to the Summary Sectio of that chapter for details. 585-5
Two-Sided Percet of Parametric Parametric Noparametric Noparametric Populatio Lower Upper Lower Upper Betwee Tolerace Tolerace Tolerace Tolerace Limits Limit Limit Limit Limit 50.00 54.074 70.126 52.000 73.000 75.00 48.411 75.789 51.000 79.000 80.00 46.850 77.350 90.00 42.527 81.673 95.00 38.777 85.423 99.00 31.449 92.751 This sectio gives the parametric ad oparametric two-sided tolerace itervals. Percet of Populatio Betwee Limits This is the percetage of populatio values that are cotaied i the tolerace iterval. Parametric Lower (Upper) Tolerace Limits These are the values of the limits of a tolerace iterval based o the assumptio that the populatio is ormally distributed. Noparametric Lower (Upper) Tolerace Limits These are the values of the limits of a distributio-free tolerace iterval. These itervals make o distributioal assumptio. Lower Oe-Sided Tolerace Bouds Percet of Parametric Noparametric Populatio Lower Lower Greater Tha Tolerace Tolerace Boud Boud Boud 50.00 60.264 56.000 75.00 52.254 52.000 80.00 50.524 51.000 90.00 45.842 95.00 41.875 99.00 34.285 This sectio gives the parametric ad oparametric oe-sided tolerace bouds. Percet of Populatio Greater Tha Boud This is the percetage of populatio values that are above the tolerace boud. Parametric Lower Tolerace Boud This is the lower parametric (ormal distributio) tolerace boud. Noparametric Lower (Upper) Tolerace Limits This is the lower oparametric (distributio-free) tolerace boud. Note that some values are missig because of the small sample size i this example. 585-6
Upper Oe-Sided Tolerace Bouds Percet of Parametric Noparametric Populatio Upper Upper Less Tha Tolerace Tolerace Boud Boud Boud 50.00 63.936 65.000 75.00 71.946 73.000 80.00 73.676 76.000 90.00 78.358 79.000 95.00 82.325 79.000 99.00 89.915 79.000 This sectio gives the parametric ad oparametric oe-sided tolerace bouds. Percet of Populatio Less Tha Boud This is the percetage of populatio values that are below the tolerace boud. Parametric Lower Tolerace Boud This is the upper parametric (ormal distributio) tolerace boud. Noparametric Lower (Upper) Tolerace Limits This is the upper oparametric (distributio-free) tolerace boud. Normality Test Sectio Test ob 10% Critical 5% Critical Decisio Test Name Value Level Value Value (5%) Shapiro-Wilk W 0.937 0.21 Ca t reject ormality Aderso-Darlig 0.427 0.31 Ca t reject ormality Kolmogorov-Smirov 0.148 0.176 0.192 Ca t reject ormality D'Agostio Skewess 1.037 0.30 1.645 1.960 Ca t reject ormality D'Agostio Kurtosis -0.786 0.43 1.645 1.960 Ca t reject ormality D'Agostio Omibus 1.6918 0.43 4.605 5.991 Ca t reject ormality This report was defied ad discussed i the Descriptive Statistics procedure chapter. We refer you to the Normality Test Sectio of that chapter for details. Plots Sectio The plots sectio displays a histogram ad a probability plot to allow you to assess the accuracy of the ormality assumptio. 585-7