DOCSLIB.ORG

Statistical Process Control Lab

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Statistical Process Control Lab Statistical Process Control Lab by Jenna Hartmann Submitted to Dr. C. G. Willson ChE 253M Department of Chemical Engineering The University of Texas at Austin Fall 2007 Statistical Process Control Abstract 2 Contents Introduction 4 Methods 4 Results 6 Conclusions and Recommendations 15 Appendices Appendix A 17 Appendix B 19 Appendix C 21 References 22 List of Tables Table 1: Process Capability Index Values 14 List of Figures Figure 1: Process Flow Diagram 4 Figure 2: Pure Dye Pre-Mixer X-bar Chart, Estimation Method 6 Figure 3: Pure Dye Pre-Mixer R Chart 7 Figure 4: Pure Dye Post-Mixer X-bar Chart, Estimation Method 7 Figure 5: Pure Dye Post-Mixer R Chart 8 Figure 6: Mix Stream Pre-Mixer X-bar Chart, Estimation Method 9 Figure 7: Mix Stream Pre-Mixer R Chart 9 Figure 8: Pre-Mixer Distribution Histogram 10 Figure 9: Mix Stream Pre-Mixer X-bar Chart, Standard 11 Deviation Method Figure 10: Mix Stream Post-Mixer X-bar Chart, Estimation Method 12 Figure 11: Mix Stream Post-Mixer R Chart 12 Figure 12: Post-Mixer Distribution Histogram 13 Figure 13: Modified (plotted without outliers) Post-Mixer Histogram 13 Figure 14: Mix Stream Post-Mixer X-bar Chart, Standard 14 Deviation Method Figure 15: Pre- and Post-Mixer Raw Concentration Data 15 3 Statistical Process Control Introduction The purpose of this lab was to determine whether a continuous mixing process was “in control” by analyzing control charts and quantifying the benefits of an in-line mixer in the process. Statistical process control is a frequently used analytical tool for quality improvement programs. In this lab, the quality of the product stream (green dye and water solution) was analyzed using control charts (X-bar for the operating process level and R for variability) which were created by (1) an estimation method in JMP and (2) a standard deviation method. The resulting control charts were then compared to make quality control suggestions for the operating system. Methods The experimental apparatus for the Statistical Process Control lab is shown below in Figure 1. The apparatus consists of a large 20L container that holds the 0.2 wt% dye solution. Connected to this container is the recycle pump that mixes the solution to a uniform concentration. A peristaltic pump is also connected to the outlet of the container. This pump generates the continuous flow rate of dye solution to the rest of the apparatus. The water used for the dye and water mixture enters from the top left of the apparatus (as shown in Fig 1) at a junction before entering the first spectrometer. The mixed solution travels past the pre-mixer spectrometer before entering the in-line mixer. After the mixer, there is a second (post-mixer) spectrometer. The mixed solution then enters into a large graduated cylinder before exiting the system to the drain (Mullet, 2007). Figure 1: Process Flow Diagram The two detectors in this lab are spectrophotometers, or spectrometers for short, but they have slightly different path lengths (the length that the detecting light passes through the fluid). The first spectrometer has a path length of 0.56 cm, and the second spectrometer has a path length of 0.54 cm. Each of the two detectors uses the same tungsten halogen lamp as a light 4 source. The light is directed from the lamp to the detectors by means of a fiber optic cable. Once the light reaches the spectrometer, it enters the flow cell and is emitted through the flowing fluid to a detector on the other side. The spectrometer measures the intensity of light passing through the fluid. The negative logarithm of the ratio of the intensity of the light that exits the flow cell versus the intensity of light of some reference is the absorbance (shown as Equation 1). I A = − log (1) I O As the weight percent of the solution changes, the absorbance changes. A larger weight percent of dye means a larger absorbent peak due to the presence of more dye, and hence the less light that is able to pass through it. The absorbance is only a function of the amount of dye in the solution because it was “zeroed out” by taking measurements with a stream of pure water. The measurements taken with the pure water stream are then used as the reference intensity for absorbance calculations. The relationship between the amount of absorbance and the concentration is known as Beer’s Law, shown in Equation 2 below: A = εcl (2) where c is the concentration, l is the path length of the flow cell, A is the absorbance, and ε is the extinction coefficient. Beer’s Law shows that a larger concentration is directly proportional to a larger absorbance (Mullet, 2007). Before the dye solution could be mixed with the tap water, a calibration curve needed to be created using a series of settings for the rotameter. The rotameter flow rate was measured at settings of 26%, 36%, and 51% of the maximum flow rate. The flow rate of the peristaltic pump was also measured by use of a bucket and stopwatch method. Dye solution was sent through the system without mixing with tap water, and the absorbance was taken before and after the in-line mixer. Then, dye solution and tap water were mixed, and the absorbance was measured before and after the in-line mixer. All the intensity data collected by the computer were then used to calculate the absorbance, and hence concentration, by Beer’s Law. Finally, control charts were created using the estimation and standard deviation methods. Random fluctuations in a process are always present, so processes are studied to determine if non-random variations exist. Non-random variations mean that the process can be modified to obtain statistically better data because the problems are from “assignable causes.” Control charts are an excellent way to interpret the data to determine what types of fluctuations are present. When looking at a control chart, the process is said to be “in control” if the data is varying only by random fluctuations. The term “in control” only corresponds to the statistical data, not the specifications needed on the product. A system is said to be out of control when there are both random and non-random variations. The capability index incorporates specified quality limits and predicts the product quality that can be expected from a process (Mullet, 2007). 5 Results The experimental results confirmed that the in-line mixer provided benefits for the process, but not enough to be considered for further use (e.g. in a factory). The mixed stream’s pre-mixer concentration data followed a random but non-normal distribution, and the post-mixer data followed a normal but non-random distribution. For a system to truly be in control, the data must be both normal and random (Mullet, 2007). Since neither the pre- nor post-mixer data met both of these requirements, the system was determined to be out of control. The mixer reduced the variability of the process (R bar = 0.017 pre-mixer and R bar = 0.00069 post-mixer), but the post-mixer x-bar chart shows trends in the data and an out of control system. The mixer proved capable of producing a ± 0.002% product in the pure dye stream (Cp pre-mixer = 2.28 and Cp post-mixer = 1.82) but not in the mix stream (water and dye) experimental runs (Cp = 0.092 and 1.13 pre- and post-mixer, respectively). First, the experimenters calibrated the spectrometer by taking a dark reference measurement using a stream of pure dye. Figures 2 and 3 display the pre-mixer X-bar and R charts for the pure dye stream. Figure 2 exhibits the non-random behavior of the system prior to entering the mixer. The control chart shows that the dye concentration varies around the lower control limit at first, then follows an upward trend, and finally maintains steady fluctuations around the upper control limit. This trend may have resulted because the experimenters did not allow the process to reach steady state before taking data readings. Figure 3 shows that the concentration of the dye before the mixer maintained a wide range of variability. The process did not vary to a state of being out of control, but the R chart shows that the pre-mixer conditions result in a wide range of resultant product, in this case, a wide range of dye concentration. 0.1705 0.1703 0.1701 UCL=0.170113 0.1699 Avg=0.16987 0.1697 LCL=0.169627 0.1695 0.1693 DYE Pre-MixerDYE Concentration (wt%) 16 32 48 64 80 96 112 Sample Figure 2: Pure Dye Pre-Mixer X-bar Chart, Estimation Method 6 0.0011 0.0009 UCL=0.000889 0.0007 0.0005 Avg=0.000421 0.0003 0.0001 LCL=0 -0.0001 DYE Pre-MixerDYE Concentration (wt%) 16 32 48 64 80 96 112 Sample Figure 3: Pure Dye Pre-Mixer R Chart Figures 4 and 5 present the control charts for the post-mixer results of the pure dye stream. Again, Figure 4 shows the same upward trend in the X-bar chart as in the pre-mixer X- bar chart but to a lesser degree. In the case of the post-mixer, the process appears to operate under more control, with the exception of one outlying point. However, although the post-mixer results show the benefits of the mixer, the concentration still varies over a wide range of values, as shown by the R chart in Figure 5.
Load more

Recommended publications
  • 2. Graphing Distributions
    2. Graphing Distributions A. Qualitative Variables B. Quantitative Variables 1. Stem and Leaf Displays 2. Histograms 3. Frequency Polygons 4. Box Plots 5. Bar Charts 6. Line Graphs 7. Dot Plots C. Exercises Graphing data is the first and often most important step in data analysis. In this day of computers, researchers all too often see only the results of complex computer analyses without ever taking a close look at the data themselves. This is all the more unfortunate because computers can create many types of graphs quickly and easily. This chapter covers some classic types of graphs such bar charts that were invented by William Playfair in the 18th century as well as graphs such as box plots invented by John Tukey in the 20th century. 65 by David M. Lane Prerequisites • Chapter 1: Variables Learning Objectives 1. Create a frequency table 2. Determine when pie charts are valuable and when they are not 3. Create and interpret bar charts 4. Identify common graphical mistakes When Apple Computer introduced the iMac computer in August 1998, the company wanted to learn whether the iMac was expanding Apple’s market share. Was the iMac just attracting previous Macintosh owners? Or was it purchased by newcomers to the computer market and by previous Windows users who were switching over? To find out, 500 iMac customers were interviewed. Each customer was categorized as a previous Macintosh owner, a previous Windows owner, or a new computer purchaser. This section examines graphical methods for displaying the results of the interviews. We’ll learn some general lessons about how to graph data that fall into a small number of categories.
    [Show full text]
  • Chapter 16.Tst
    Chapter 16 Statistical Quality Control- Prof. Dr. Samir Safi TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. 1) Statistical process control uses regression and other forecasting tools to help control processes. 1) 2) It is impossible to develop a process that has zero variability. 2) 3) If all of the control points on a control chart lie between the UCL and the LCL, the process is always 3) in control. 4) An x-bar chart would be appropriate to monitor the number of defects in a production lot. 4) 5) The central limit theorem provides the statistical foundation for control charts. 5) 6) If we are tracking quality of performance for a class of students, we should plot the individual 6) grades on an x-bar chart, and the pass/fail result on a p-chart. 7) A p-chart could be used to monitor the average weight of cereal boxes. 7) 8) If we are attempting to control the diameter of bowling bowls, we will find a p-chart to be quite 8) helpful. 9) A c-chart would be appropriate to monitor the number of weld defects on the steel plates of a 9) ship's hull. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Which of the following is not a popular definition of quality? 1) A) Quality is fitness for use. B) Quality is the totality of features and characteristics of a product or service that bears on its ability to satisfy stated or implied needs.
    [Show full text]
  • Section 2.2: Bar Charts and Pie Charts
    Section 2.2: Bar Charts and Pie Charts 1 Raw Data is Ugly Graphical representations of data always look pretty in newspapers, magazines and books. What you haven’tseen is the blood, sweat and tears that it some- times takes to get those results. http://seatgeek.com/blog/mlb/5-useful-charts-for-baseball-fans-mlb-ticket-prices- by-day-time 1 Google listing for Ru San’sKennesaw location 2 The previous examples are informative graphical displays of data. They all started life as bland data sets. 2 Bar Charts Bar charts are a visual way to organize data. The height or length of a bar rep- resents the number of points of data (frequency distribution) in a particular category. One can also let the bar represent the percentage of data (relative frequency distribution) in a category. Example 1 From (http://vgsales.wikia.com/wiki/Halo) consider the sales …g- ures for Microsoft’s videogame franchise, Halo. Here is the raw data. Year Game Units Sold (in millions) 2001 Halo: Combat Evolved 5.5 2004 Halo 2 8 2007 Halo 3 12.06 2009 Halo Wars 2.54 2009 Halo 3: ODST 6.32 2010 Halo: Reach 9.76 2011 Halo: Combat Evolved Anniversary 2.37 2012 Halo 4 9.52 2014 Halo: The Master Chief Collection 2.61 2015 Halo 5 5 Let’s construct a bar chart based on units sold. Translating frequencies into relative frequencies is an easy process. A rela- tive frequency is the frequency divided by total number of data points. Example 2 Create a relative frequency chart for Halo games.
    [Show full text]
  • Bar Charts and Box Plots Normal Poisson Exponential Creating a Simple Yet Effective Plot Requires an 02468 C 1.0 Exponential Uniform Understanding of Data and Tasks
    THIS MONTH a Uniform POINTS OF VIEW Normal Poisson Exponential 02468 b Uniform Bar charts and box plots Normal Poisson Exponential Creating a simple yet effective plot requires an 02468 c 1.0 Exponential Uniform understanding of data and tasks. λ = 1 Min = 5.5 Normal Max = 6.5 0.5 Poisson σμ= 1, = 5 μ = 2 Bar charts and box plots are omnipresent in the scientific literature. 0 02468 They are typically used to visualize quantities associated with a set of Figure 2 | Representation of four distributions with bar charts and box plots. items. Representing the data accurately, however, requires choosing (a) Bar chart showing sample means (n = 1,000) with standard-deviation the appropriate plot according to the nature of the data and the task at error bars. (b) Box plot (n = 1,000) with whiskers extending to ±1.5 × IQR. hand. Bar charts are appropriate for counts, whereas box plots should (c) Probability density functions of the distributions in a and b. λ, rate; be used to represent the characteristics of a distribution. µ, mean; σ, standard deviation. Bar charts encode quantities by length, which is a highly accurate visual encoding and preferred over the angle-based strategy used in to represent such uncertainty. However, because the bars always start pie charts (Fig. 1a). Often the counts that we want to represent are at zero, they can be misleading: for example, part of the range covered sums over multiple categories. There are several options to visualize by the bar might have never been observed in the sample.
    [Show full text]
  • Measurement & Transformations
    Chapter 3 Measurement & Transformations 3.1 Measurement Scales: Traditional Classifica- tion Statisticians call an attribute on which observations differ a variable. The type of unit on which a variable is measured is called a scale.Traditionally,statisticians talk of four types of measurement scales: (1) nominal,(2)ordinal,(3)interval, and (4) ratio. 3.1.1 Nominal Scales The word nominal is derived from nomen,theLatinwordforname.Nominal scales merely name differences and are used most often for qualitative variables in which observations are classified into discrete groups. The key attribute for anominalscaleisthatthereisnoinherentquantitativedifferenceamongthe categories. Sex, religion, and race are three classic nominal scales used in the behavioral sciences. Taxonomic categories (rodent, primate, canine) are nomi- nal scales in biology. Variables on a nominal scale are often called categorical variables. For the neuroscientist, the best criterion for determining whether a variable is on a nominal scale is the “plotting” criteria. If you plot a bar chart of, say, means for the groups and order the groups in any way possible without making the graph “stupid,” then the variable is nominal or categorical. For example, were the variable strains of mice, then the order of the means is not material. Hence, “strain” is nominal or categorical. On the other hand, if the groups were 0 mgs, 10mgs, and 15 mgs of active drug, then having a bar chart with 15 mgs first, then 0 mgs, and finally 10 mgs is stupid. Here “milligrams of drug” is not anominalorcategoricalvariable. 1 CHAPTER 3. MEASUREMENT & TRANSFORMATIONS 2 3.1.2 Ordinal Scales Ordinal scales rank-order observations.
    [Show full text]
  • Chapter 6 Presenting Your Data in Graphic Form Political Orientations
    UNIVARIATE ANALYSIS Chapter 6 Presenting Your Data in Graphic Form Political Orientations Now let’s turn our attention from religion to politics. Some people feel so strongly about politics that they joke about it being a religion. The General Social Survey distribute(GSS) data set has several items that reflect political issues. Two are key political items: POLVIEWS and PARTYID. These items will be the primary focus in this chapter. In the process of examining these variables, we are going to learn not only more about the politicalor orientations of respon- dents to the 2012 GSS but also how to use SPSS Statistics to produce and interpret data in graphic form. You will recall that in the previous chapter, we focused on a variety of ways of displaying univariate distributions (frequency tables) and summarizing them (measures of central ten- dency and dispersion). In this chapter, we are going to build on that discussion by focusing on several ways of presenting your data graphically. We will begin by focusing on two charts thatpost, are useful for variables at the nominal and ordinal levels: bar charts and pie charts. We will then consider two graphs appropriate for interval/ratio variables: histograms and line charts. Graphing Data With Direct “Legacy” Dialogs SPSS Statistics gives us acopy, variety of ways to present data graphically. Clicking the Graphs option on the menu bar will give you three options for building charts: Chart Builder, Interactive, and Legacy Dialogs. These take you to two chart-building methods developed by SPSS over the past 20 years. Chart Builder is the most recent.
    [Show full text]
  • Stacked Bar Charts……………………………………………12
    Effective Use of Bar Charts Purpose This tool provides guidelines and tips on how to effectively use bar charts to communicate research findings. Format This tool provides guidance on bar charts and their purposes, shows examples of preferred practices and practical tips for bar charts, and provides cautions and examples of misuse and poor use of bar charts and how to make corrections. Audience This tool is designed primarily for researchers from the Model Systems that are funded by the National Institute on Disability, Independent Living, and Rehabilitation Research (NIDILRR). The tool can be adapted by other NIDILRR-funded grantees and the general public. The contents of this tool were developed under a grant from the National Institute on Disability, Independent Living, and Rehabilitation Research (NIDILRR grant number 90DP0012-01-00). The contents of this fact sheet do not necessarily represent the policy of Department of Health and Human Services, and you should not assume endorsement by the Federal Government. 1 Overview and Organization Simple Bar Charts……………………………………..……....3 Clustered Bar Charts…………………………………………10 Stacked Bar Charts……………………………………………12 Paired Bar Charts…………………………………………..…14 Simple Bar Chart – Categorical Comparisons A bar chart is basically a vertical column chart oriented horizontally instead. Data values are displayed as horizontal bars. The magnitude of each data element is represented by the length of the bar. Can be used to display values for categorical items (diabetes prevalence by state, hospital performance rankings on a preferred clinical practice measure etc). Shows comparisons among the categorical groups on the measure. Categories displayed on the vertical axis. Simple Bar Chart – Categorical Comparisons Bar charts are preferred over column charts: When the categorical axis labels are lengthy; When you have 12 or more categories; When the metric to be displayed is duration (such as clinic lobby wait time per health center).
    [Show full text]
  • Which Chart Or Graph Is Right for You?
    Which chart or graph is right for you? Tracy Rodgers, Product Marketing Manager Robert Bloom, Tableau Engineer You’ve got data and you’ve got questions. You know there’s a chart or graph out there that will find the answer you’re looking for, but it’s not always easy knowing which one is best without some trial and error. This paper pairs appropriate charts with the type of data you’re analyzing and questions you want to answer. But it won’t stop there. Stranding your data in isolated, static graphs limits the number and depth of questions you can answer. Let your analysis become your organization’s centerpiece by using it to fuel exploration. Combine related charts. Add a map. Provide filters to dig deeper. The impact? Immediate business insight and answers to questions that actually drive decisions. So, which chart is right for you? Transforming data into an effective visualization (any kind of chart or graph) or dashboard is the first step towards making your data make an impact. Here are some best practices for conducting meaningful visual analysis: 2 Bar chart Bar charts are one of the most common data visualizations. With them, you can quickly highlight differences between categories, clearly show trends and outliers, and reveal historical highs and lows at a glance. Bar charts are especially effective when you have data that can be split into multiple categories. Use bar charts to: • Compare data across categories. Bar charts are best suited for data that can be split into several groups. For example, volume of shirts in different sizes, website traffic by referrer, and percent of spending by department.
    [Show full text]
  • Visualiation of Quantitative Information
    Visualisation of quantitative information Overview 1. Visualisation 2. Approaching data 3. Levels of measurement 4. Principals of graphing 5. Univariate graphs 6. Graphical integrity James Neill, 2011 2 Is Pivot a turning point for web exploration? (Gary Flake) (TED talk - 6 min. ) 4 Approaching Entering & data screening Approaching Exploring, describing, & data graphing Hypothesis testing 5 6 Describing & graphing data THE CHALLENGE: to find a meaningful, accurate way to depict the ‘true story’ of the data 7 Clearly report the data's main features 10 Levels of measurement • Nominal / Categorical • Ordinal • Interval • Ratio 12 Discrete vs. continuous Discrete - - - - - - - - - - Continuous ___________ Each level has the properties of the preceding 14 13 levels, plus something more! Categorical / nominal Ordinal / ranked scale • Conveys a category label • Conveys order , but not distance • (Arbitrary) assignment of #s to e.g. in a race, 1st, 2nd, 3rd, etc. or categories ranking of favourites or preferences e.g. Gender • No useful information, except as labels 15 16 Ordinal / ranked example: Ranked importance Interval scale Rank the following aspects of the university according to what is most • Conveys order & distance important to you (1 = most important • 0 is arbitrary through to 5 = least important) e.g., temperature (degrees C) __ Quality of the teaching and education • Usually treat as continuous for > 5 __ Quality of the social life intervals __ Quality of the campus __ Quality of the administration __ Quality of the university's reputation 17 18 Interval example: 8 point Likert scale Ratio scale • Conveys order & distance • Continuous, with a meaningful 0 point e.g. height, age, weight, time, number of times an event has occurred • Ratio statements can be made e.g.
    [Show full text]
  • Which Chart Or Graph Is Right for You? Tell Impactful Stories with Data
    Which chart or graph is right for you? Tell impactful stories with data Authors: Maila Hardin, Daniel Hom, Ross Perez, & Lori Williams January 2012 p2 You’ve got data and you’ve got questions. Creating a Interact with your data chart or graph links the two, but sometimes you’re not sure which type of chart will get the answer you seek. Once you see your data in a visualization, it inherently leads to more questions. Your bar graph reveals that This paper answers questions about how to select the sales tanked in the second quarter in the Southeast. best charts for the type of data you’re analyzing and the A scatter plot shows an unexpected concentration of questions you want to answer. But it won’t stop there. product defects in one category. Donations from older Stranding your data in isolated, static graphs limits the alumni are significantly down according to a heat map. number of questions you can answer. Let your data In each example, your reaction is the same: why? become the centerpiece of decision making by using it Equip yourself to answer these questions by making to tell a story. Combine related charts. Add a map. your visualization interactive. Doing so creates the Provide filters to dig deeper. The impact? Business opportunity for you and others to analyze your data insight and answers to questions at the speed of visually and in real-time, letting you answer questions Which chart or graph is right for you? Tell impactful stories with data Tell Which chart or graph is right for you? thought.
    [Show full text]
  • Bar Charts This Set of Notes Shows How to Construct a Bar Chart Using
    Revised July 2018 Bar Charts This set of notes shows how to construct a bar chart using Stata. It assumes that you have set Stata up on your computer (see the “Getting Started with Stata” handout), and that you have read in the set of data that you want to analyze (see the “Reading in Stata Format (.dta) Data Files” handout). In Stata, most tasks can be performed either by issuing commands within the “Stata command” window, or by using the menus. These notes illustrate both approaches, using the data file “GSS2016.DTA” (this data file is posted here: https://canvas.harvard.edu/courses/53958). Bar charts display differences in means for a quantitative variable across two or more groups, as opposed to histogram comparisons which display percentage distributions of a categorical variable within groups. For example, we can construct a bar chart of the mean hours per day of television viewing by self-attributed class: 1 Simple bar charts can be constructed easily via the Stata command line. The general command structure is graph bar <varname1>, over(<varname2>) where you enter a quantitative variable in place of “varname1" and a categorical variable in place of “varname2". Using the variables in the above graph, you would issue the command graph bar tvhours, over(class) This command produces a bare-bones graph with no enhancements. To enhance and improve the display, it is probably easier to use the Stata menus. For a bar chart, Click on “Graphics” Click on “Bar chart” When you do this, the “graph bar – Bar chart” window shown below opens up: 2 Under “Type of data” select “Graph of summary statistics.” Enter the name of your quantitative variable in the “Variable(s)” box in the middle of the window.
    [Show full text]
  • ١ Topics: Sources of Variation
    Islamic University, Gaza - Palestine Topics: • Control Charts for Variables • Some Special Difficulties of Estimating Indirect Costs in Economy Studies • Did it Pay to Use Statistical Quality Control? • Costs and Product Liability • SQC Techniques in Relation to certain Economic Aspects of Product liability • Control Charts for Variables Islamic University, Gaza - Palestine Sources of Variation • Variation exists in all processes. • Variation can be categorized as either; – Common or Random causes of variation, or • Random causes that we cannot identify • Unavoidable • e.g. slight differences in process variables like diameter, weight, service time, temperature Assignable causes of variation • Causes can be identified and eliminated • e.g. poor employee training, worn tool, machine needing repair ١ Islamic University, Gaza - Palestine Control charts identify variation • Chance causes - “common cause” – inherent to the process or random and not controllable – if only common cause present, the process is considered stable or “in control” • Assignable causes - “special cause” – variation due to outside influences – if present, the process is “out of control” Islamic University, Gaza - Palestine Control Charts for x and R Notation and values • Ri = range of the values in the ith sample Ri = xmax - xmin • R = average range for all m samples • is the true process mean • is the true process standard deviation ٢ Islamic University, Gaza - Palestine Control Charts for Xbar and R Statistical Basis of the Charts • Assume the quality characteristic
    [Show full text]