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The 0. The variables used in an operation are CTQ, or those that are critical to quality. CTQ variables are used to adjust to a known value, say 1. The CTQ variables may be adjusted resulting in an increase or decrease in nail length. In other words, adjustment results in a decrease of error, not a change in the center point. The goal for an operation is to stay within defined limits or control levels around a center point. Other goals are to understand, control, and optimize input variables.

Introduction to Lean Six Sigma Methodology

Calibrated dissolved oxygen DO colored standards were used in the study Cat. The colored standards have a blue dye that absorbs light at nm. The standards varied in the amount of dye, so that an increase in dye concentration results in an increase in absorption. In other words, the more dye present, then the darker blue the standard. The dye solutions were in sealed ampoules that were well within their expiration date, and their color did not change over time. The standards are normally used as a visual comparison reference to determine dissolved oxygen content.

This study used instrumentation to quantitate PPM value by measuring absorption. Typical applications for actual dissolved oxygen measurements include water drinking, sewage, industrial waste and bio-habitats, e. The amount of oxygen has a large effect on a wide-range of measurements, e. Thus, it is imperative that the dissolved oxygen measurements be as accurate and reproducible as possible.

Errors in accuracy and reproducibility can be due to measurement variance as well as sampling procedure. Thus, a major goal is to reduce measurement variance as much as possible. Spectrometers are used to quantify the wavelength and intensity of light waves.

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Visible spectrometers are used to determine the amount of color at one or more specific wavelength. The principle involved is to shine white light, with known intensity, through a material and to determine the Absorbance value. Light transmits through a colored liquid and the amount of absorbed light is determined. The amount of absorbed light Absorbance is proportional to the amount of color. In this way, the color of the DO standards may be measured. The depth of color of a colored liquid in an ampoule was measured and reported as Absorbance.

The acceptable tolerance or variation in measurement is strongly application dependent, but generally the smaller the better. The tolerance or variability of the dissolved oxygen standards was unknown, needed to be determined, and then improved upon. The second step was to determine the variables that affect the measurement values. In general, an ANOVA gauge study determines the contribution of variables to errors in measurements and compares them to errors of the total system.

The specifications and samples were merged together, with known calibration standards being measured. In that manner, the reference and the samples were the same, thus eliminating these variables. Test methods remained consistent throughout the study, eliminating this variable as well.

So the variables that were examined were the within unit variability, unit to unit variability, and operator measurement. Normally, point 1 equates to repeatability, or variation in measurements taken by a single person or instrument on the same item and under the same conditions. Also point 3 equates to reproducibility, or variation in readings contributed by different operators measuring the same sample.

It is very important to note that this study also includes point 2 , which is unit to unit variability. After that, a Pareto analysis was completed to find the largest source of error. Pareto analysis is a statistical technique for decision making. It is used for selecting a limited number of task s to do, which would cause the most significant changes to an overall effect. In this case, focus would be on understanding the largest contributor s to measurement error.

The next step was to determine how to adjust each specific variable to reduce error. To extend the Pareto principle, one needs to limit or remove these critical causes to achieve significant quality improvement. In the current study, the Pareto analysis was used to identify the critical contributors that cause errors in measurement. Next, improvements will focus on decreasing the specific variables. And then a new Pareto assessment will be made verifying the contributors were the areas to focus on, and statistically significant improvements were obtained.

The improvements should be monitored to ensure the variable changes work. As mentioned previously, the main controllable variables were determined to be the individual unit, multiple units, and operator. A plan was made involving the main controllable variables and how they could be quantitated. For every standard, three measurements would be performed. There were nine blue standards of varying PPM levels of dissolved oxygen to be measured. Five units would be used in the study. Two highly-trained operators would perform the measurements. Each standard would be measured on each unit, by each operator, in triplicate.

So, in total measurements would be made based on three variables. Measure Stage The next step was performing the actual measurements. The adapters for the spectrometer were designed to accommodate glass ampoules. The adapter snapped onto the i-LAB unit such that light was focused through the ampoule, to the back side of the adapter, and then reflected through the sample again and to a detector.

The light path was the same for every unit. The glass ampoule standards contained blue-colored aqueous liquids with differing dye concentrations. The degree of color correlated to the amount of dissolved oxygen DO for their ampoules kit K samples. The sample ampoules are meant to measure DO from ppm parts per million using an indigo carmine reaction. Only the sealed colored standard ampoules were used in the study. Two trained operators performed the measurements using the same standard testing procedure and software.

The operators measured the standards on the same day, with the same instruments to minimize environmental variations. The measuring procedure consisted of initially calibrating the spectrometer with an ampoule filled with clear, distilled water and also with an ampoule filled with an opaque, black ink. The calibration procedure was performed to ensure that the spectrometer LEDs and sensor contained the correct optical specifications. After calibration, the operator measured the ampoule filled with clear, distilled water to obtain the background. Next, the standard colored ampoules samples were cleaned, placed, and measured.

Three absorbance measurements at nm were taken for each standard by each operator on each unit. Two trained operators performed the tests. In this manner, the 5x3x2 study yielded 30 reference points for each part of the study. So, together with the nine blue dissolved oxygen standards, the amount of measurements was two points.

Analysis Stage The data was analyzed in multiple ways. After the measurements were done in triplicate by each operator on five spectrometers, the average Absorbance ABS values were calculated and are shown in Table 1.

### Training in statistics

Variation amongst measurements with differing units, variation within units and operators are also apparent, and will be explored. The graphs show linearity between Absorbance and the standard PPM values. The connected lines also show that each unit has variances, and that each unit performed differently for each operator. Error bars are at 2 standard deviations 2 sigma levels. Figure 3: ABS vs. The second step is to quantify their contribution to variations. Further classification by variables can also ease in seeing their contribution and magnitude.

So, in detail, for each colored PPM standard, the overall average absorbance values were determined. The difference delta between the measured absorbance value for each point and the average absorbance value was then calculated and plotted in Figure 5. Figure 5 highlights that there is variation in all three areas; unit to unit, between operators, and within unit.

The variables were further segmented and analyzed statistically for the degree of error and its significance.

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4. The f-test is used to determine whether the expected values of a variable differ significantly from each other. In the case of comparing multiple units, the ANOVA analysis indicates if the instruments get similar results of if they differ substantially. Inherent in the test is a Fisher or F number that is calculated and is defined as the between group sum of squares divided by the within group sum of squares, Equation 1.

## Statistical Methods for Six Sigma: In R&D and Manufacturing - Semantic Scholar

The probability is that of observing a result as extreme or unlikely than a result using a null hypothesis that the sample measurements are not alike. The Fisher method is often used to determine the statistical likelihood, actually unlikelihood, for different measurements coming from the same variable Fisher, ; Miller, Jr. In Equation 1, Y is the measurement average for a specific unit or measurement; i is the number of data values per group; j is the within group number; and a is the number of groups.

After calculating the F number, a probability factor p was determined. Table 2 shows the results of the Fisher calculations for each of the PPM levels tested. This shows that the variation between units is statistically significant reproducibility is compromised. Thus the five units differ substantially in yielding expected results.

Drastic reduction in variation was achieved after much brainstorming and testing. These alterations are outlined in the improvement section. Table 3 shows the probability, p value, was significantly greater than 0. These values indicate there is a very high probability that each unit will produce values that are alike. These are again outlined in the improvement section. This probability indicates that the measurements made by one operator are the same as those made by the other.

So, the variation in measurements between operators is not statistically significant.

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In Figure 6, the center green bar shows the average difference between operators for all measurements with respect to the PPM numbers. The red and blue bars show the average delta and the standard deviations for operator 1 and operator 2, respectively. From this analysis and a Fischer analysis, it was determined that operator error did not play a significant role, relative to the other two variables, in contributing to the overall error. For this reason, improvement strategies were not explored for this variable.

The graph shows the majority of error in measurements can be attributed to differences between unit to unit variations. These results are not surprising and were shown before in Figure 5. In other words, determine what can be done to decrease variation. Typically, improvements for one variable should not affect others. Understanding the concepts and specific steps involved in each statistical method is critical for achieving consistent and on-target performance.

Emphasizing practical learning, applications, and performance improvement, Dr. It covers a large number of useful statistical methods compactly, in a language and depth necessary to make successful applications. Statistical methods in this book include: variance components analysis, variance transmission analysis, risk-based control charts, capability and performance indices, quality planning, regression analysis, comparative experiments, descriptive statistics, sample size determination, confidence intervals, tolerance intervals, and measurement systems analysis.

The book also contains a wealth of case studies and examples, and features a unique test to evaluate the reader?