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**Statistical Quality Control**

Quality Control Charts using Excel II

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**Learning Objectives After this class the students should be able to:**

Determine control limits for several types of control charts Use graphics to create statistical control charts with Excel Interpret control charts Create a Pareto chart

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Time management The expected time to deliver this module is 50 minutes. 30 minutes are reserved for team practices and exercises and 20 minutes for lecture.

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Control Charts As long as the points remain between the lower and upper control limits, we assume that the observed variation is controlled variation and that the process is in control

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Control Chart The process is out of control. Both the fourth and the twelfth observations lie outside of the control limits, leading us to believe that their values are the result of uncontrolled variation.

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Control Chart Even control charts in which all points lie between the control limits might suggest that a process is out of control. In particular, the existence of a pattern in eight or more consecutive points indicates a process out of control, because an obvious pattern violates the assumption of random variability.

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Control Chart The first eight observations are below the center line, whereas the second seven observations all lie above the center line. Because of prolonged periods where values are either small or large, this process is out of control.

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Control Chart Other suspicious patterns could appear in control charts. Unfortunately, we cannot discuss them all here. Control chart makes it very easy for you to identify visually points and processes that are out of control without using complicated statistical tests. This makes the control chart an ideal tool for the shop floor, where quick and easy methods are needed.

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**Chart and Hypothesis testing**

The idea underlying control charts is closely related to confidence intervals and hypothesis testing. The associated null hypothesis is that the process is in control; you reject this null hypothesis if any point lies outside the control limits or if any clear pattern appears in the distribution of the process values. Another insight from this analogy is that the possibility of making errors exists, just as errors can occur in standard hypothesis testing. Occasionally a point that lies outside the control limits does not have any special cause but occurs because of normal process variation.

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The Range Chart The x-chart provides information about the variation around the average value for each subgroup. It is also important to know whether the range of values is stable from group to group. If some observations exhibit very large ranges and others very small ranges, you might conclude that the sprayer is not functioning consistently over time. To test this, you can create a control chart of the average subgroup ranges, called a range chart.

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**The Range Chart If a is known, the control limits are:**

See QC Correction Control

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**The Range Chart To create a range chart of the weight values:**

Return to the Coating Data worksheet (in Teaching.XLS). Click StatPlus > QC Charts > Range Chart. Select Weight as your Data Values variable and Time as the Subgroup variable. Verify that the Sigma Known checkbox is unselected. Direct the output to a new chart sheet named Range Chart. Click

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Analysis Each point on the range chart represents the range within each subgroup. The average subgroup range is 3.25, with the control limits going from 0 to According to the range chart shown, only the 27th observation has an out-of-control value. The special cause should be identified if possible. However, in discussing the problem with the operator, sometimes you might not be able to determine a special cause.

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The C-Chart Both the X-chart and the range chart measure the values of a particular variable. Now let's look at an attribute chart that measures an attribute of the process. A C-chart displays control limits for the counts attribute. c is the average number of counts in each subgroup. If the LCL is less than zero, by convention it will set to equal zero, because a negative count is impossible.

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**C-Chart: Factory Accidents**

Team exercise: The Accident data worksheet in Teaching.XLS workbook contains the number of accidents that occurred each month during a period of a few years at a -production site. Using StatPlus, Create control charts of the number of accidents per month to determine whether the process is in control.(15 minutes) To create a C-chart for accidents at this firm: Click StatPlus > QC Charts > C-Chart. Select Accidents as the Data Values variable. Direct the output to a new chart sheet named C-Chart. Click OK.

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P-Chart P-chart is closely related to the C-chart. It depicts the proportion of items with a particular attribute, such as defects. The P-chart is often used to analyze the proportion of defects in each subgroup.

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**P-Chart Let p = average proportion of the sample that is defective.**

The distribution of the proportions can be approximated by the normal distribution, provided that nxp and n(1 - p) are both at least 5. If p is very close to 0 or 1, a very large subgroup size might be required for the approximation to be legitimate.

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**P-Chart: Steel rod defects**

Team exercise: A manufacturer of steel rods regularly tests whether the rods will withstand 50% more pressure than the company claims them to be capable of withstanding. A rod that fails this test is defective. Twenty samples of 200 rods each were obtained over a period of time, and the number and fraction of defects were recorded in the Steel Rod Data worksheet in Teaching .XLS workbook. Using StaPlus, Create control P-charts and analyze it. (15 minutes) Click StatPlus > QC Charts > P-Chart. Click the Proportions button and select Percentage from the list of range names. Click OK. Type 200 in the Sample Size box, because each subgroup has the same sample size. Send the output to a new chart sheet named P-Chart. Click OK.

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**P-Chart: Steel rod defects**

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**P-Chart: Steel rod defects**

The lower control limit is , or a defect percentage of about 1%. The upper control limit is , or about 11%. The average defect percentage is , about 6%. The control chart clearly demonstrates that no point is anywhere near the three-s limits. The lower control limit is , or a defect percentage of about 1%. The upper control limit is , or about 11%. The average defect percentage is , about 6%. The control chart clearly demonstrates that no point is anywhere near the three-s limits. Not all out-of-control points indicate the existence of a problem. For example, suppose that another sample of 200 rods was taken and that only one rod failed the stress test. In other words, only one-half of 1% of the sample was defective. In this case, the proportion is 0.005, which falls below the lower control limit, so technically it is out of control. Yet you would not be concerned about the process being out of control in this case, because the proportion of defects is so low. Still, you might be inclined to investigate, just to see whether you could locate the source of your good fortune and then duplicate it!

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**Control chart for individual observations**

Sometimes it's not possible to group your data into subgroups. This could occur when each measurement represents a single batch in a process or when the measurements are widely spaced in time. With a subgroup size of 1, it's not possible to calculate subgroup ranges. This makes many of the regular formulas impractical to apply.

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**Control chart for individual observations**

The method is to create a "subgroup" consisting of each consecutive observation and then calculate the moving average of the data. The subgroup variation is determined by the variation from one observation to another, and that variation will be used to determine the control limits for the variation between subgroups. The limits are: Here x is the sample average of all of the observations, R is the average range of consecutive values in the data set, and d2 is the control limit factor shown earlier in QC Correction Factor. We are using a moving average of size 2. Here x is the sample average of all of the observations, R is the average range of consecutive values in the data set, and d2 is the control limit factor from the QC Correction factors,

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**Range chart for individual observations**

We can also create a moving range chart of the moving range values; that is, the range between consecutive values. The limits are:

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The tensile strength

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Analysis The chart shown gives the values of the individual observations (not the moving averages) plotted alongside the upper and lower control limits. No values fall outside the control limits, which would lead us to conclude that the process is in control. However, the last eight observations were all either above or near the center line, which might indicate a process going out of control toward the end of the process. This is something that should be investigated further.

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Moving Range I-chart

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Analysis The chart shows additional indications of a process that is not in control. The last seven values all fall below the center line, and there appears to be a generally downward trend to the ranges from the sixth observation on. We would conclude that there is sufficient evidence to warrant further investigation and analysis.

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Pareto Chart Pareto chart create a bar chart of the causes of the problem in order from most to least frequent so that you can focus attention on the most important elements or combination of elements .

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Baby powder example Part of the process of company that manufactures baby powder involves a machine called a filler, which pours the powder into bottles to a specified limit. The quantity of powder placed in the bottle varies because of uncontrolled variation, but the final weight of the bottle filled with powder cannot be less than grams. Any bottle weighing less than this amount is rejected and must be refilled manually (at a considerable cost in terms of time and labor). Bottles are filled from a filler that has 24 valve heads so that 24 bottles can be filled at one time.

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Exercise Data Sometimes a head is clogged with powder, and this causes the bottles being filled on that head to receive less than the minimum amount of powder. To gauge whether the machine is operating within limits, random samples of 24 bottles (one from each head) are selected at about 1-minute intervals over the nighttime shift at the factory. The teams are examine the data and determine which part of the filler is most responsible for defective fills (20 minutes). The worksheet Powder in workbook Teaching.XLS contains the data. Use Pareto Chart.

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**Baby Powder example To create the Pareto chart:**

Click StatPlus > QC Charts > Pareto Chart. Click the Values in separate columns option button. Click the Data Values button and then select the range names from Head 01 to Head 24 in the range names list (do not select the Time variable). Click OK. Click the Defects occur when the data value is drop-down list box and select Less than. Type in the text box below the drop-down list box. Click the Output button and direct the output to a new chart sheet named Pareto Chart. Click OK. Figure shows the completed dialog box.

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Analysis The Pareto chart shows that a majority of the rejects come from a few heads. Filler head 18 accounts for 87 of the defects, and the first three heads in the chart (18, 14, and 23) account for almost 40% of all of the defects. There might be something physically wrong with the heads that made them more liable to clogging up with powder. If rejects were being produced randomly from the filler heads, you would expect that each filler head would produce 24 , or about 4%, of the total rejects. Using the Pareto chart you might want to repair or replace those three heads in order to reduce clogging.

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QC Correction Factors Source: Adapted from "1950 ASTM Manual on Quality Control of Materials," American Society for Testing and Materials, in J. M. Juran, ed., Quality Control Handbook (New York: McGraw-Hill, 1974), Appendix II, p. 39.

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Reference “Data Analysis with Excel”. Berk & Carey, Duxbury, 2000, chapter 12, p

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