D1p3 - Basic Analytical Methods

GEOLOGY, RAW MATERIALS AND RAW MIX Process Engineering Certification Program April 7 - 8, 2003 Quality Assurance

Session 1: Basic Analytical Methods

2

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Quality Assurance Definition

All those planned and systematic actions necessary to provide adequate confidence that a product or service will satisfy given requirement for quality. (ISO 8402)

3

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Quality Assurance

Quality Assurance

Quality

Quality

Quality

Planning

Control

Audit

4

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Quality Planning Strategy and Tactics

External

ƒ Professional Organizations ƒ Environmental Legislation ƒ Standard Committees ƒ Technical Customer Service ƒ Market ƒ Material Resources Control Concept ƒ Product Design

Internal

ƒ Mix Design ƒ Exploitation Concept ƒ Control Concept ƒ Education/Training

5

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Quality Control ƒ Control Plan ƒ Procedures/Methods ƒ Personnel Organization ƒ Routine Control (Raw Materials

Cement)

ƒ Non-Routine Testing (Raw Materials, Refractories, Cement

Application) ƒ Data Processing/Communication ƒ Cost Control

6

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Quality Audit

7

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Significance of Quality Control

8

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Statistical Thinking the Basis of Quality Control ƒ 1. Hypothesize - Problem definition, Variable ID, Plan changes

ƒ 2. Collect Data - Conduct trials, Do on small scale, Use DOE ƒ 3. Analyze Data - Apply statistics, Significance?, Predict? ƒ 4. Take Action - Return to step 1, Verify results, Implement

9

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Shewhart Method - Scientific Method

Does the theory need revision? Change the process Compare data to theory Summarize what was learned

Act

Check

Plan

Do

Question/theory phase Plan a change or test Plan data collection

Carry out the change or test on a small scale Observe the data Analyze the data

10

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Cycle of Indecision

Plan

Plan

Plan

Plan

11

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Do - Do - Do- Do Cycle (Do-Do Happens!)

Do

Do

Do

Do

12

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

What is a statistic? ƒ Definition: A statistic is a function of the observations in a sample

ƒ The computation of a statistic does not depend upon any unknown

constants called parameters! ƒ “If the method used to obtain a statistic is good, then the resulting

numbers will be meaningful, interpretable, and relevant to the problem or system being studied.” Dennis Keller ƒ The object of taking data and computing statistics is to provide a

basis for action! Those responsible and accountable for taking action must understand statistical methods in order to justify the actions they recommend. 13

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Some Key Points to Remember ƒ 1. Use meaningful statistics to summarize data in order to explore,

understand, describe, and quantify the underlying distribution or population. (Enumerative Study) ƒ 2. Use meaningful statistics to summarize data in order to explore,

quantify, and predict future performance of a system. (Analytical Study) ƒ Why use statistics?

A. Separate “good” from “bad” data B. Understand systems and processes C. Improve systems and processes D. Optimize the performance of systems and processes 14

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

What is an enumerative study? ƒ Consider the following Raw Meal C3S GammaMetrics data taken at

different 3-minute time intervals: Interval A B 1 130 169 2 155 168 3 166 166

C 162 158 154

D 150 146 143

ƒ Are there any difference between the C3S measurements? What

types of possible differences are there?

ƒ If there is a difference, how confident are we that this difference is

meaningful? Let’s take a vote on this? How many say there is a difference? How many say there is not a difference? 15

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

What is an analytical study? Decide what the future C3S values will be based on the following results. Raw Meal C3S via GammaMetrics

C3S

180 160 140 120 100 80 60 40 20 0 0

50

100

150

200

Time (minutes)

16

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Summarizing Data ƒ Why summarize data?

- Characterize the distribution of the data in Enumerative Studies - Quantify features of the data in Analytical Studies ƒ Measures of central tendency

- Sample average - Sample median ƒ Measures of dispersion

- Sample range - Sample variance - Sample standard deviation - Coefficient of variation

17

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Central Tendency ƒ Sample Mean or Arithmetic Average

- Data: y1, y2, y3, y4, y5, …, yn - Σ yj /n ƒ In Excel® enter the following instructions

- select the cell for the average - suppose the data are in B2 to B362 - type =AVERAGE(B2:B362) - hit the “enter” key ƒ The sample mean or arithmetic average is the “center of gravity” of

the data. ƒ Exercise 1: What are the average C3S values for the 4 time

periods? What can you say about these four averages?

18

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Central Tendency (continued) ƒ Sample Median or “Middle Value” of the data

- Data: y1, y2, y3, y4, y5, …, yn - Order Statistic: y(1)< y(2)< y(3)< y(4)< y(5)< …
- select the cell for the median - suppose the data are in B2 to B362 - type =MEDIAN(B2:B362) - hit the “enter” key ƒ The sample median is the 50th percentile of the data. ƒ Exercise 2: What are the median C3S values for the 4 time periods?

What can you say about these four statistics?

19

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Central Tendency (continued) ƒ Sample Mode or the most frequently occurring value in the data. ƒ Referring to the C3S data presented earlier, there are no repeat

values. Thus rounding the data to the nearest tens the files can be summarized as follows: Interval 1 2 3

A 130 160 170

B 170 170 170

C 160 160 150

D 150 150 140

ƒ Note that there is no mode for the A interval of time. But the mode for

the remaining three time intervals are 170, 160, and 150 respectively. ƒ The mode is not a very useful measure of central tendency so it will

not be considered in this presentation. 20

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Which measure of central tendency to use? Average Pro Best if distribution is symmetric

Median Approximates average if distribution is symmetric Pro Basis for most Not sensitive decision to outliers making Con Influenced by Slightly more outliers difficult to calculate Con Limited use in decision making 21

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Dispersion ƒ Sample Range

- Data: y1, y2, y3, y4, y5, …, yn - Order Statistic: y(1)< y(2)< y(3)< y(4)< y(5)< …
- select the cell for the range - suppose the data are in B2 to B362 - type =Max(B2:B362)-Min(B2:B362) - hit the “enter” key ƒ Exercise 3: What are the ranges of the C3S values for the 4 time

periods? What can you say about these four values?

22

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Dispersion (continued) ƒ Sample Variance

- Data: y1, y2, y3, y4, y5, …, yn - Mean-Square Deviations from the sample mean - S2 =Σ [yj - yavg ]2 / (n-1) ƒ In Excel® enter the following instructions

- select the cell for the Sample Variance - suppose the data are in B2 to B362 - type =(STDEV(B2:B362))^2 - hit the “enter” key ƒ Sample Standard Deviation:

- S ={Σ [yj - yavg ]2 / (n-1)}½ - Root-Mean-Square Deviation from the average ƒ Exercise 4: What are the standard deviations of the C3S values for

the 4 time periods? What can you say about these results?

23

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Dispersion (continued) ƒ Process Capability Standard Deviation

- Data in Time Order: y1, y2, y3, y4, y5, …, yn - Calculate the moving ranges MRj = | yj - yj-1 | for j =2, …, n - Spcl = Median (MRj) / 0.954 ƒ Typically the Root-Mean-Square standard deviation is larger than

the Process Capability standard deviation. ƒ Exercise 5: Calculate the process capability standard deviation for

the C3S values assuming the time intervals are consecutive and the values within a time period are in time order. Data: 131, 155, 166, 170, 169, 166, 162, 158, 154, 151, 147, 143 How does this standard deviation compare with the individual standard deviations from the four time intervals? 24

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Measures of Dispersion (continued)

ƒ Coefficient of Variation (CV)

- CV = S / Yavg - The standard deviation used here can be the RootMean-Square or the Process Capability standard deviations ƒ This statistic is used when a two-way plot of standard deviations

versus averages indicates that the two statistics are positively linearly correlated. ƒ Sometimes the CV is expressed as the percent change in a

property rather than the fractional change. ƒ Exercise 6: What is the percentage coefficient of variation for the

C3S values assuming that the data are in time order? Using the averages and standard deviations from the four time periods should the standard deviation be expressed as a coefficient of variation? 25

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Which measure of dispersion to use? R Pro Easy to calc.

2

S Unbiased

Pro Good for Best for 1
S Used in most Calc.

Best for sym. dist. Not easy to understand

Spcl Better than S if data are time ordered Not sensitive to shifts

CV Good if avg and S correlated

Easy to understand Wrong if avg and S are not correlated

Con Not good Need a if n>5 calc. Holcim (US) Inc - CTS

26

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Checking for Outliers in Small Sets of Data ƒ Statistic for Testing for Extreme Values in Data

Statistic: (y(2)- y(1)) / ( y(n) - y(1) ) or (y(n)- y(n-1)) / ( y(n) - y(1) ) ƒ Criteria for Testing for Extreme Values in Data

n 3 4 5 6 7

Exclude the value if the statistic above exceeds the tabulated value with indicated confidence 90% 95% 98% 99% .8 8 6 .9 4 1 .9 7 6 .9 8 8 .6 7 9 .7 6 5 .8 4 6 .8 8 9 .5 5 7 .6 4 2 .7 2 9 .7 8 0 .4 8 2 .5 6 0 .6 4 4 .6 9 8 .4 3 4 .5 0 7 .5 8 6 .6 3 7

ƒ Exercise 7: Are there any outliers for C3S values within the 4 time

periods?

Holcim (US) Inc - CTS

27

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Least Significant Difference (LSD) ƒ Statistic for Testing for Extreme Values between Individual

Measurements or Averages Statistic:

| Y1 - Y2 | or | Yavg-1 - Yavg-2 |

ƒ Criteria for deciding the values are different

| Y1 - Y2 | > 2S(2)½ or | Yavg-1 - Yavg-2 | > 2S[1/ n +1/ m]½ ƒ The confidence level in these decisions is 95% ƒ Exercise 8: Let S be the average of the standard deviations for the

C3S values for the 4 time periods. Are any of the 4 averages outliers? Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

28

12.12.2005 PECP02 - 2002

Checking for Outliers in Time Ordered Data ƒ Statistic for Testing for Extreme Values in Time Ordered Data

- Data in Time Order: y1, y2, y3, y4, y5, …, yn - Calculate the moving ranges MRj = | yj - yj-1 | for j =2, …, n - Calculate Average MR = Σ MRj / (n-1) - Calculate Critical Value = 3.268 Σ MRj / (n-1) ƒ Any MR that exceeds the critical value represents a shift in the time

ordered data. ƒ Once outlier MR’s have been identified, a new average MR can be

calculated and the process repeated. When more than 10% outlier MR’s are identified, excessive variation is a problem ƒ One or both of the two values in the moving range may be outliers. ƒ This method can be used with repeat tests when questionable

results are observed. Here the Range of the repeat tests replaces the Moving Range. 29

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Checking for Outliers in Time Ordered Data (cont.) ƒ Exercise 9: Considering the 12 C3S values as time ordered

observations as in the earlier exercise, determine if any of the moving ranges are outliers. Do outliers MR’s affect the estimate of the process capability standard deviation? What do outlier MR’s mean physically?

30

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Are two standard deviations significantly different? ƒ Earlier calculations from Exercise 4 showed that the standard

deviations for the C3S values for the 4 time intervals were as follows: Interval A B C D StDev 18.2 1.8 4.2 3.6 ƒ Are these standard deviations equal? In particular are the standard

deviations different for A and B? How about for B and C? ƒ In Excel® open the spreadsheet with the C3S values for the 4 time

intervals. Then click on “Tools”, “Data Analysis”, “F test: two sample for variances”.

ƒ Put the cursor in the first window, and highlight the data for interval

A. Put the cursor in the second window, and highlight the data for interval B. Accept the default risk of deciding the two standard deviations are not equal when in fact they are equal. This will give a 95% confidence for the decision you are making. 31

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Are two standard deviations significantly different? ƒ For a comparison of time intervals A and B, here is the Excel®

input:

ƒ On the next page is the output. ƒ Exercise 10: Compare the C3S standard deviations for intervals B

and C after discussing the following output on the next page.

32

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Are two standard deviations significantly different? F-Test Two-Sample for Variances Time Interval Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail

1 to 3 4 to 6 150.6667 168.3333 320.3333 4.333333 3 3 2 2 73.92308 0.013347 19.00003

33

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Are two averages significantly different? ƒ Earlier calculations from Exercise 4 showed that the averages for

the C3S values for the 4 time intervals were as follows: Interval Avg

A 151

B 168

C 158

D 147

ƒ Are these averages equal? In particular are the averages different

for A and B? How about for B and C? ƒ In Excel® open the spreadsheet with the C3S values for the 4 time

intervals. Then click on “Tools”, “Data Analysis”, “t-Test: Two sample assuming unequal variances”. This routine is used because of the results from the F-test.

ƒ Put the cursor in the first window, and highlight the data for interval

A. Put the cursor in the second window, and highlight the data for interval B. Accept the default risk of deciding the two averages are not equal when in fact they are equal. This will give a 95% confidence for the decision you are making. 34

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Are two averages significantly different? ƒ In Excel® the input screen is as follows for comparing time intervals

A and B:

ƒ The output from this program is shown on the next page. ƒ Exercise 11. Are the averages for time intervals B and C unequal?

How confident are you of your decision? Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

35 12.12.2005 PECP02 - 2002

Are two averages significantly different? ƒ In Excel® the output is as follows for comparing time intervals A

and B: t-Test: Two-Sample Assuming Unequal Variances Variable 1 Variable 2 Mean 150.6667 168.3333 Variance 320.3333 4.333333 Observations 3 3 Hypothesized 0 df 2 Avg are equal t Stat -1.69823 P(T<=t) one-ta 0.115781 t Critical one-ta 2.919987 P(T<=t) two-ta 0.231561 t Critical two-ta 4.302656 36

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Review of Session 1 ƒ There are several ways to summarize the central tendency of data.

Always start by using the AVERAGE function in Excel®. But if outliers are possible in small data sets, use the MEDIAN function. ƒ There are several ways to summarize the variation. The range [use

the functions MAX()-MIN()] is useful if the number of observations is small, say 2 to 5. Otherwise use the STDEV function in Excel®. ƒ When data are time ordered, use the Spcl=MEDIAN(MR)/.954 This

is often a better estimate of how the process is capable of performing and is better for determining when the process shifts.

ƒ Outliers can be detected using:

Range Ratio method for small samples sizes, 3 to 5 Least Significant Difference MR’s if the data are time ordered

37

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

Review of Session 1 ƒ The F-test can be used to determine with specified confidence

whether or not two standard deviations are different. The usual confidence level is 5%. This means that when one decides two standard deviations are not equal, one could still be wrong. But this type of error will only occur 5% of the time. ƒ There are several t-tests that are used to determine whether or not

two averages are significantly different with specified confidence. One t-test is appropriate if the standard deviations are not equal while the other is appropriate if the standard deviations are equal. If you are in doubt which test to use, use the case for unequal standard deviations (or variances). If the variances are actually equal, the test will be essentially equivalent.

38

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002

End of Session 1

39

Holcim (US) Inc - CTS

Geology, Raw Materials and Raw Mix

12.12.2005 PECP02 - 2002