 working as an associate professor in the Department of Mechanical Engineering at Vaalchen Institute of Technology, Sholapur. In this video, we will discuss on control chart for variables. Learning outcome At the end of this session, students will be able to apply and analyze the control chart for variables which are associated with quality. Contents Variability Control chart for variables First is X bar chart Second is R chart Example on control chart Reference Introduction Variability In nature, two extremely similar things are difficult to obtain. If at all we come across exactly similar things, it must be only by chance. It is said that even the twins cannot be alike. So, if we take any operation, there is a variation which is bound to happen. The variation may be due to different aspects. For example, if suppose we wanted to drill a hole on a metal, the variation could be based on the material, that is the hardness of the material, may be due to the machine tool, may be due to the cutting tool, may be due to the environment, may be due to the man who is doing that operation. So, variation is bound to happen. Because of that variation, we used to give the tolerances for the products. Now, there are two types of variations due to assignable cause. So, this assignable cause is high in magnitude and it is easily traced or detected. So, this assignable causes are due to machines, worker, material, equipment and based on these machines, worker and material over the time. Second is variations due to chance cause. So, this is inevitable in any process and it is very difficult to trace out. Say for example, there is a play between the nut and the bolt. That leads to the variation. So, it is very difficult to trace or find out. Now, what is the meaning of the control chart or the definition of the control chart? Control chart is a graphical representation of the collected information. The information may pertain to measuring quality characteristics of samples. It detects the variation in processing and warns if there is any departure from the specified tolerance limit. Now, there are different types of quality characteristics. For example, technological is a quality characteristic. Under technological, we can take length, diameter, thickness, viscosity, frequency, pressure, temperature strength of the material. Now, this quality characteristics can be plotted by means of a control chart. Where the data is collected from the machines and that data is plotted. And we wanted to find out based on the data whether the process is in control or not. So, in order to have a control chart, we require the data of the variable as I told in the technological characteristics. Now, there are two types of charts which comes under the variable control chart. One is the X bar chart and second is the R chart. X bar is the centering of the process. That is, how much dimension is varying from the centre that we can calculate from X bar chart. And R chart is uniformity and consistency of the process. Whether the process is behaving at the centre or not, that can be determined by means of R chart. Now, control charts for variables, just now as we have discussed, there are two types. One is X bar chart and another is R chart. Now, what is the calculation procedure for this control chart for variable? Now, we collected data. That data is to be plotted in the form of a graph or the control chart. First is, calculate the average X bar and range for each subgroup. Now, we have taken a sample for a particular product. For that sample, we have to take the technological dimensions or the characteristics as far as the quality is concerned. So, take the average of that. That average could be related with a shift or for a particular day or for a particular week. And the range for each product. So, that is the minimum dimension to the maximum dimension in terms of variation. Then calculate the grand average that is X bar and range R bar. So, that is called as a grand average. Average of average is nothing but a grand average. Then calculation of three sigma limits on control chart for X chart. And calculation of three sigma limits on control chart for R chart. So, these are the steps which are involved in order to find out the control limits and to plot a particular control chart for variable. Now, let us think for a moment, what is the obligation of this variable control chart? In my opinion, if a worker is working on a particular machine tool, and if we measure the dimensions of the product which are coming out of that particular machine tool, we can plot a control chart for that particular variable. Could it be a length, could it be a diameter or could it be a thickness? Now, one man, one machine or one man who is operating three or more machines for that purpose, we can plot a control chart in order to see how the process is behaving. Now, let us take an example on control chart for variable. In a capability study of a length, in turning a shaft to a diameter of 23.75 plus minus 0.1 mm, a sample of six consecutive pieces, that is nothing but a subgroup. Six is a subgroup, was taken each day for five days for subgroup of six pieces. The diameter of these shafts are as given below. Compete control limits for X bar and R chart. So, here the data for five days is given and each day is having six diameters. So, we will take this problem in order to compute the control chart for X bar and R chart. Now, in a capability study of a length, in a turning shaft to a diameter of 23.75 plus minus 0.1 mm, a sample of six consecutive pieces was taken each day for five days. So, these are the five days, one, two, three, four, five. Each day there are certain diameters 23.77 to 23.75, these are the six dimensions. Now, for one day six dimensions, for the second day these are the samples drawn. This is the third day, this is the fourth day, this is the fifth day. What is the first step? The first step is we must have to calculate X1 bar. Now, how do we calculate X1 bar? So, for the first day let us say that X1 plus X2 plus X3 plus X4 plus X5 plus X6 divided by six. So, this is for the first day six samples are there. We have taken the dimensions of these six components that is X1 to X6 divided by six. So, we are getting X1 bar is 23.77. So, similarly for the second day it is taken. So, here X2 bar is 23.77. For the third day X3 bar is equal to 23.77. For the fourth day same, for the fifth day same. So, how do you calculate X double bar? That is the average of a average or grand average. That is X1 bar plus X2 bar that is plus X6 bar divided by six. So, we will calculate this X double bar 23.77. Now, for range, range is nothing but the minimum dimensions on that particular day. That is for example, for the first day 23.73 is the minimum dimension and the maximum dimension is 23.80. So, it is a range from the range of 23.73 to 23.80. So, it is 23.80 minus 23.73 is nothing but 0.07. So, similarly we can calculate the ranges for the different days. And we can calculate R bar. R bar is equal to R1 plus R2 plus dash dash plus R6 divided by six. So, here we can calculate R bar that is average of ranges 0.0675. Now, in order to calculate the upper control limit, upper control limit is equal to X double bar plus A2 R bar. Where A2 is a constant, it is dependent on the subgroup. And lower control limit is equal to X double bar minus A2 R bar. Here, we have already calculated this X double bar. We have already calculated R bar. We know A2. So, here A2 is equal to 1.48. So, we can calculate for X bar chart that is upper control limit is 23.80 and lower control limit is 23.70. And for R chart, upper control limit is equal to D4 R bar. Where D4 is given 2.28, lower control limit is equal to D3 R bar. Here D3 is given that is 0. So, it is 0. This already we have calculated 0.1350. So, in this way we can calculate the control limits for the X bar and R chart that is the variable. So, these are the references which are used for this particular video.