Statistical Process Control: Micrometer Readings

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1 Statistical Process Control: Micrometer Readings Timothy M. Baker Wentworth Institute of Technology College of Engineering and Technology MANF 3000: Manufacturing Engineering Spring Semester 2017 Abstract This document provides an example of a statistical process control (SPC) report for a given dataset in a manufacturing setting. By applying statistical and manufacturing methodologies to the recorded set of micrometer measurements, the effectiveness of the process can be determined. To show this, a grouped data analysis, process control charts, and process capability calculations are performed and displayed. I. INTRODUCTION Given twenty-five manufactured components, micrometer readings were taken at a specified location. For each part, a total of four readings were taken using the millimeter micrometer with the datum being 6 millimeters. The lower specification limit (LSL) and upper specification limits (LSL) are 6.3 mm and 6.5 mm respectively. II. BACKGROUND To thoroughly understand the statistical analysis of this manufacturing process, basic statistical concepts should be understood. A. Statistical Process Control Statistical process control (SPC) is a proven method of detecting assignable variability in a process. SPC procedures help monitor the behavior of a process where control charts are perhaps the most successful tool. A control chart facilitates in the differentiation between two different types of process variation: 1. Common cause variation (natural variation) is a caused by unknown factors that result in a steady but random distribution of output around the mean of the data. This variation is caused by categories commonly referred to as Manpower, Material, Method, Measurement, Machine, and Environment. [1] 2. Special cause variation (assignable variation) is caused by known factors that result in a non-random distribution of the output around the mean of the data. It can be directly accounted for and potentially removed or controlled. [2] D. The Normal Distribution The normal distribution (Gaussian distribution) is a function that needs only the mean and dispersion to completely characterize itself. In this case, the data is evenly distributed around a central value with no bias left or right. This means that 50% of values are greater than the mean, and 50% are less than the mean. C. The Mean The mean shows the value that the measurements tend to cluster around (measure of central tendency) and is also known as the average [3]. It is calculated using the equation: x = x 1+x 2 +x 3 + +x n n x = mean, average value x i = measurements n = number of measurements Eq. 1 D. Dispersion Dispersion indicates how much variability is present in a set of measurements. Dispersion also determines the width of the normal curve. Two of the most common measures of dispersion include the range, R, and the standard deviation, s. The range is defined by the following equation: 1

2 R = x max x min Eq. 2 R = range x max = largest measured value x min = smallest measured value The standard deviation can be calculated by the following equation: s = n i=1 (x i x ) 2 n 1 s = sample standard deviation x = mean, average value x i = measurements n = number of measurements Eq. 3 E. Process Capability Process capability determines if a process is capable of meeting specifications. It is the relationship between the specified limits for a value and the limits of the natural variability of the value. [3] The process capability ratio, C p, is a measure that has the natural limits centered within the specified limits. It is calculated by dividing the difference in the specified limits by the total amount of variation expected in the process. For grouped data analysis the process capability is defined as: C p = USL LSL 6σ Eq. 4 For statistical process control (SPC), the process capability is defined as: C p = USL LSL 6σ USL = upper specified limit LSL = lower specified limit σ = sample standard deviation σ = process standard deviation Eq. 5 σ = R d 2 Eq. 6 R = average of subgroup ranges d 2 = function of subgroup size [4] (Table 5) If the C p is less than one, a significant amount of parts will be produced that will be deemed defective, or out of specification. The larger the C p value, the more capable the process is of producing good parts. If C p < 1.33 the process is general considered unacceptable. [3] The process capability index, C pk, numerically describes how close a process is to running to its specification limits relative to the natural variability of the process. It considers both process centering and process capability This determines how close the process is to the specification as well as how consistent the process is around its average performance. The equation below defines the C pk : C pk = nearest specification x 3σ nearest specification = USL or LSL σ = process standard deviation Eq. 7 F. Control Charts A control chart is a plot of a quality characteristic with respect to time. This quality characteristic is compared to the control limits which help determine whether the process is varying within acceptable limits or if it is out of control. These control limits define the range in which the process can be considered acceptable. Two of the most common and useful control charts are the x (x-bar) and R-charts. The x (x-bar) chart shows the quality characteristic of the average of the sample. The R-chart shows the quality characteristic of the range of the sample. The control limits of these charts can be found by using the following equations: 2

3 UCL x = x + 3σ x = x + A 2 R Eq. 8 LCL x = x 3σ x = x A 2 R Eq. 9 UCL R = R + 3σ R = x + D 4 R Eq. 10 UCL R = R + 3σ R = x + D 4 R Eq. 11 x = avaerage of the subgroup of averages σ x = standard deviation of the subgroup averages R = average of the subgroup ranges σ = standard deviation of the subgroup ranges A 2,D 4 = functions of the subgroup size [4] (Table 5) By using Eq. 12 we can reconfigure the data into its final measurement value. Table 2 below displays the translated micrometer measurements in mm interpreted from the raw data. Table 2. III. GROUPED DATA ANALYSIS The raw data displayed in Table 1 is a set of micrometer measurements in millimeters where the values are based off of a 6 mm datum. The sample calculation below displays how the raw data can be properly translated: n 1,x 1 = 6.35 n 1,x 2 = 6.40 n 1,x 3 = 6.32 n y,x x = (x x ) Eq. 12 n y = subgroup x x = sample measurement Table 1 From these micrometer measurements, a grouped data analysis can be performed to help evaluate the process. By using Eq. 1-4 and 7, the mean, dispersion, and process capabilities can be calculated. Table 3 displays their values for this process. Table 3. Based on these values we can see that the average micrometer measurement is 6.41, which approximately in the middle of our USL 3

4 and LSL. However, it is important to note that the process capability, C p, is well below 1 which makes the process unacceptable. Also from the translated data, a frequency array can be constructed. Table 4 displays the frequency of a micrometer measurement occurs within a specific range using a 0.05 mm step. Table 4. several variables that must be calculated. Listed below are the variables to be calculated and the equations to be used to do so: 1. Average of each sample (Eq. 1) 2. Range of each sample (Eq. 2) 3. Average of all of the sample averages (Eq. 1) 4. x upper control limit (Eq. 7) 5. x lower control limit (Eq. 8) 6. Average of all of the sample ranges (Eq. 1) 7. R upper control limit (Eq. 9) 8. R lower control limit (Eq. 10) When calculating the control limits, constants based on the size of the subgroup are needed. In this case, the subgroup size is four. Refer to Table 5 for constants A 2 and D 4 when calculating the x-bar and R control limits. Table 5. This frequency array allows us to create a histogram (Fig. 1) in which the average and specification limits are also displayed. From this histogram we can see that the micrometer measurements create a mostly normal distribution with the most measurements clustered around the average. There are, however, several measurements that lie outside of the upper and lower tolerances for this process. These values directly affect the dispersion of the measurements which result in a lower process capability. Table 6 below displays the calculated values of variables 1-8 described above. Table 6. Fig. 1. IV. STATISTICAL PROCESS CONTROL In order to properly construct the control charts for this manufacturing process, there are 4

5 Using the values from Table 6, an x-bar chart (Fig. 2a) and R chart (Fig. 2b) can be plotted. Notice that each chart shows the upper and lower control limits as well as the value in which the sample measurements should be centered around. In the x-bar chart that would be the x value and in the R chart it would be the R value. Fig. 2 (a) V. CONCLUSION By performing a grouped data analysis on the given micrometer measurements, it is easy to see that the process is not performing up to specifications. Looking at the histogram shows us that there are several parts produced outside of the specification limits. When analyzing the process from a SPC standpoint, the control charts show that the process is out of control. The random dispersion of sample measurements on both the x-bar and R charts show that there is outlying data No trend can be seen from these charts, thus resulting in an undetermined root cause. From both quality analyses, the process capability was calculated with the grouped data C p being and the SPC C p being Both of these capabilities show that the process is no good as it does not have the minimum C p standard value of 1.3. References Fig. 2 (b) Lastly, the process capability based on the statistical process control can be calculated and evaluated. By using Eq. 5 and Eq. 6 [1] "Special Cause Variation, isixsigma", Isixsigma.com, [Online]. [2] Common Cause Variation, isixsigma, Isixsigma.com, [Online]. [3] P. D. Rufe, Fundamentals of manufacturing [4] Quality America Inc., Control Chart Constants. [Online]. Center/statisticalprocesscontrol/control_chart_constants.ph p. σ = R d 2 = = C p = USL LSL 6σ = (0.043) = = This process capability is well below 1, thus making the process unacceptable. 5