# Presentation: 666282

## Statistical Analysis

**Carl Steinhauer**
**Consultant**

## Overview

**P 401 Test Results**
**Statistical Analysis; PWL Estimate**
**Verify Production Process: Payment**

## Theory

**Assumptions**
**Normal Distribution**
**Tools: Average and Standard Deviation**
**Percent Within Limits                (PWL) Concept**

## Assumptions

**Limited # of test results****	****    Statistical Analysis****	****   Quality characteristics of large amount of material**
**Test result variability**
**	****Components:**
**	****	****	****materials**
**	****	****	****sampling-ERLPM**
**	****	****	****testing-ERLPM**
**Same Process**
**Random sampling-Lot, Sublot**
**Normal Distribution**** **

## Specific Procedures

**Sublots, Lots, Partial Lots**
**Calculations**
**Retesting**
**Outliers**

## Limit # of samples

- % Taller than 5’-5”
- % between 5’-5” and 6’-5”
- Average Height

- 	Statistical Analysis
- 		Estimate average 		  and % within limits

## n=100

**n=100**

**96 tests**

## Slide 8

## x

**  ****x+/- 1Sn=68%**

**x+/-2Sn+95%**

**x+/-3Sn+99.7%**

**x**

## L

**L**

**L=spec lower tolerance limit**
**  ****eg. Mat density 96.3 for P401**

**PWL-% of test result exceeding L**

## PWL Calculation ProceduresERLPM-page 47

**Section 110-AC 1505370-10A**
**	**Method for Computing PWL and Examples
- 	Section 110-02
**Spec. P401 Table 5-L and U Spec. Limits (page 24)**

## Given

**x****1****=2**
**x****2****=4**
**x****3****=6**
**x****4****=8**
** **
**X = **_**2+4+6+8**_** = 5**
**4**** **

## X=5
d1=2-5=-3	d12=9
d2=4-5=-1	d22=1
d3=6-5=1	d32=1
d4=8-5=3	d42=9

Sn = 	     d12+d22+d32=d42
			    n-1
Sn =	     9+1+1+9	 =		 20 
		          4-1			  3
Sn =2.58
(calculator n-1)

**d****1****=2-5=-3****	****d****1****2=9**
**d****2****=4-5=-1****	****d****2****2=1**
**d****3****=6-5=1****	****d****3****2=1**
**d****4****=8-5=3****	****d****4****2=9**
**Sn***** *****= ****	****     **_**d**_**1**_**2+d**_**2**_**2+d**_**3**_**2=d**_**4**_**2**_
**	****	****	****    ****n-1**
**Sn ****=****	****     **_**9+1+1+9**_**	**** =****	****	**** **_**20**_** **
**	****	****          ****4-1****	****	****	****  3**
**Sn ****=2.58**
**(calculator ****n****-1)**

## Roundout RulesERLPM-page 47

**Example-last digit to be kept-nearest 10****th**
**	**
**	****4.61**
**	****4.62**
**	****4.64**
**	****4.6500**
**	****4.66**
**	****4.67**
**	****4.68**
**	****4.69**
***Even Digit-same***
***Odd Digit-increase by 1***
***This case becomes 4.6***
***If it was 4.7500 it would become 4.8***

**becomes 4.7**

## MAT DensityManual Appendix E, page 4

**Sublot**
**	****	****1. = 96.0**
**	****	****2. = 97.0**
**	****	****3. = 99.0**
**	****	****4. = 100.0**
**x=98.0**
**Sn****=1.8**
**Q****L****=**_**x-L**_
**	****  ****Sn**
**Q****L****=**_**98.0-96.3**_** = .9444**
**	****	****1.8**
**Section 110-Table l, N=4**
**P****L****=82**

**Section 110-02f**

**Spec Tables 5**

**page 50 ERLPM**

## Target Density 98.0 Achieved
Sn=1.8 versus 1.3

**Sn****=1.8 versus 1.3**

- Acceptable QC Value

**18%**

**P****L ****= 82%**

**MAT Density**

**98.0**

**96.3**

## Effect of Quality Control

**90 PWL**

**82 PWL**

**Sn**** = 1.8**

**Sn**** = 1.3**
**Spec. pg 24**

**96.3%**

**98%    x**

## Air VoidsApp. D, page 3

**Sublot **
**1= 2.1**
**2= 3.2**
**3=2.5**
**4=6.0**
**X= 3.4**
**Sn****= 1.76**

- Spec. page 5 - 0.65

## QL = x-L = 3.4-2.0 = .7955

**	****Sn****     1.76**
**    ****P****L****(**_**table 1**_**) = 77%**
**	**** ****n=4**
**Q****U****= **_**U-x**_** = **_**5-3.4 **_**= .909**
**        ****Sn****       1.76**
**    ****P****U****(**_**table 1**_**) = 81%**
**	****n=4**
**PWL= P****L**** + P****U****-100**
**PWL= 77+81-100 = 58**

**Air Voids**

**23%**

**58%**

**19%**

**2**

**3.4**

**5**

**PWL= P****L**** + P****U****-100**
**PWL= 77 + 81-100 =58**

## Payment

**Spec-par 401-8.1a-page 29**
**	****MAT Density PWL=82**
**	****Air Voids PWL= 58 **
**Lot Pay Factor**
**Air Voids- 1.4 x 58-12= 69.2%**
**Mat Density- 0.5 x 82+55= 96%**
**Use **_**lower**_** of 2 values- 69.2%**

**Lower  value**

## Joint DensityAppendix E, page 5

**93.3**
**95.0**
**97.0**
**96.0**
**X= 95.3**
**Sn****= 1.58**
**Q****L****= **_**(95.3-93.3)**_** = 1.2658**
**	****        ****1.58**
**	****P****L****= 93**
**	****  ****Spec. page 21 par. 401-5.2(b)(3) if < 71% there is a 5% penalty**

**Table 5**

## Partial Lotsspec page 20

**Section P-401-5.1c**

## Sample Problem

**Flow-Appendix D, page 5**
_**Partial lot situation-6 sublots**_
**8.0, 8.2, 8.5, 8.2, 8.9, 9.1**
**X= 8.5**
**Sn****= 0.44**
**	****	****	****QL= **_**x-L**_** = **_**8.5-8.0**_** = 1.1364; PL= 88**
**         ****	****        ****	****         ****Sn ****	****       0.44****	**
**	****	****	****QU= **_**U-X**_** = **_**16-8.5**_** = 18.75; PL= 100**
**      ****	****      ****	****         ****Sn****	****       0.44**
**	****	****	****PWL= 88 + 100-100= 88<90**
**Corrective Action! 401-5.2(b)(2)**

## Outliers

**Spec -pg 23****	****401-5.2d**
**	****     ****-pg 25****	****401-5.3c**
**MAT Density and Air Voids**

## Slide 26

## Slide 27

## Test for Outliers

**MAT Density**
**	****94.0**
**	****96.0**
**	****97.0**
**	****98.0**
**	****x= 96.2**
**	****Sn= 1.71**

**QL= **_**96.2-96.3**_** = -.0585**
**             ****1.71**
**   ****PL= <50%**

**ASTM E 178, par. 4**
**T****1****= (x-x****1****)/****Sn**
**T****1****= **_**96.2-94 **_**= 1.286**
**          ****1.71**

## Table 1-ASTM E 178

N=4

Upper 5% significance level 1.463

Since 1.286<1.463 the 94.0 test value is not considered an outlier and is retained!

**N=4**
**Upper 5% significance level 1.463**
**Since 1.286<1.463 the 94.0 test value is not considered an outlier and is retained!**

## Sample Problem-Outliers

_**Air Voids**_
**2.0, 4.8, 4.9, 5.0**
**X=4.2**
**Sn=1.45**
**Q****L****= **_**4.2-2.0**_** = 1.5172; PL= 100**
**          ****1.45**
**Q****U****= **_**5-4.2**_** = 0.5517; PU= 69**
**          ****1.45**
**PWL= (100+69)-100=69**
**ASTM E 178 par. 4**
**	****T****n****= (x-x****1****)/****Sn**** = **_**4.2-2.0**_** = 1.517**
**	****	****                 ****1.45**
**Table 1, ASTM E 178, N=4, 5% significance**
**T= 1.463<1.517 **
**therefore 2.0 is the outlier and it is discarded**

## Resampling401-5.3 page 25

_**MAT Density ONLY**_
**(Appendix E-pg 4)**
**Prior MAT Density- 96, 97, 99, 100**
**PWL 82**
**4 new cores 96, 96, 97, 98**
**AVG-all 8,  97.4**
**Sn= **** 1.51**
**Q****L****= **_**x-L**_** = **_**97.4-96.3**_** = .7337**
**	****     ****Sn****      1.51**
**Table 1, N= 8**
**PWL= 77**

## Questions?

## Thanks!