Statistical Process Control

What is a process? Inputs PROCESS Outputs A process can be described as a transformation of set of inputs into desired outputs.

Types of Measures Measures where the metric is composed of a classification in one of two (or more) categories is called Attribute data. _ Good/Bad Yes/No Measures where the metric consists of a number which indicates a precise value is called Variable data. Time Miles/Hr Use Minitab to have students record the results and have the students display using Graph..Histogram Note how “rough” the graph looks Redo using Basic Statistics …. Descriptive Statistics and display using the Graphical Summary. Walk through the normal curve transform: Mean (Arithmetic Average) Standard Deviation Skew (How off center the data is skewed -=left) Kurtosis (How flat or peaked the data is -=flat) Show the Box Plot: Quartile (25% of the Data Points) Median (50% of the Data Points on Each Side) Show the 95% Confidence Interval and Explain how it relates to the data. January 16, 2019

A sample is just a subset of all possible values Population Vs. Sample (Certainty Vs. Uncertainty) A sample is just a subset of all possible values population sample Since the sample does not contain all the possible values, there is some uncertainty about the population. Hence any statistics, such as mean and standard deviation, are just estimates of the true population parameters. January 16, 2019

WHY STATISTICS? THE ROLE OF STATISTICS ……… Statistics is the art of collecting, classifying, presenting, interpreting and analyzing numerical data, as well as making conclusions about the system from which the data was obtained. January 16, 2019

Descriptive Statistics Descriptive Statistics is the branch of statistics which most people are familiar. It characterizes and summarizes the most prominent features of a given set of data (means, medians, standard deviations, percentiles, graphs, tables and charts. January 16, 2019

Inferential Statistics Inferential Statistics is the branch of statistics that deals with drawing conclusions about a population based on information obtained from a sample drawn from that population. January 16, 2019

WHAT IS THE MEAN? å The mean is simply the average value of the data. ORDERED DATA SET -5 -3 -1 1 3 -6 -4 -2 2 4 5 6 The mean is simply the average value of the data. n=12 x i = - å mean n 12 17 . Mean January 16, 2019

WHAT IS THE MEDIAN? ORDERED DATA SET -5 -3 -1 1 3 -6 -4 -2 2 4 5 6 If we rank order (descending or ascending) the data set ,we find the value half way (50%) through the data points and is called the median value. Median Value Median 50% of data points January 16, 2019

WHAT IS THE MODE? ORDERED DATA SET -5 -3 -1 1 3 -6 -4 -2 2 4 5 6 If we rank order (descending or ascending) the data set We find that a single value occurs more often than any other. This is called the mode. . Mode January 16, 2019

WHAT IS THE RANGE? x = - 9 ( ) The range is a very common metric . ORDERED DATA SET -5 -3 -1 1 3 -6 -4 -2 2 4 5 6 The range is a very common metric . To calculate the range simply subtract the minimum value in the sample from the maximum value. Range Max Min x MAX MIN = - 9 ( ) January 16, 2019

( ) å WHAT IS THE VARIANCE/STANDARD DEVIATION? s X n 1 61 67 12 5 6 = The variance (s2) is a very robust metric . The standard deviation(s) is the square root of the variance and is the most commonly used measure of dispersion. ( ) s X n i 2 1 61 67 12 5 6 = - å . DATA SET -5 -3 -1 3 -6 -4 -2 4 -.17 -5-(-.17)=-4.83 -3-(-.17)=-2.83 -1-(-.17)=-.83 0-(-.17)=.17 1-(-.17)=1.17 3-(-.17)=3.17 4-(-.17)=4.17 (-4.83)2=23.32 (-2.83)2=8.01 (-.83)2=.69 (.17)2=.03 (1.17)2=1.37 (3.17)2=10.05 (4.17)2=17.39 61.67 January 16, 2019

Statistical Process Control (SPC) Measures performance of a process Uses mathematics (i.e., statistics) Involves collecting, organizing, & interpreting data Objective: Regulate product quality Used to Control the process as products are produced Inspect samples of finished products 4

Functions of a Process Control System are CONTROL CHART Functions of a Process Control System are To signal the presence of assignable causes of variation To give evidence if a process is operating in a state of statistical control

Essential features of a control chart Upper Control Limit Central Line Variable Values Lower Control Limit Time

Control Chart Purposes Show changes in data pattern e.g., trends Make corrections before process is out of control Show causes of changes in data Assignable causes Data outside control limits or trend in data Natural causes Random variations around average 9

Quality Characteristics Variables Attributes 1. Characteristics that you measure, e.g., weight, length 2. May be in whole or in fractional numbers 3. Continuous random variables 1. Characteristics for which you focus on defects 2. Classify products as either ‘good’ or ‘bad’, or count # defects e.g., radio works or not 3. Categorical or discrete random variables 6

Types of Control Charts for Attribute Data Description Type Sample Size Control Chart for proportion non conforming units p Chart May change Control Chart for no. of non conforming units in a sample np Chart Must be constant Control Chart for no. of non conformities in a sample c Chart Control Chart for no. of non conformities per unit u Chart May Change

Control Chart Types Control Charts Variables Attributes Charts Charts ` X P C Chart Chart Chart Chart 13

X Chart Type of variables control chart Interval or ratio scaled numerical data Shows sample means over time Monitors process average and tells whether changes have occurred. These changes may due to 1. Tool wear 2. Increase in temperature 3. Different method used in the second shift 4. New stronger material Example: Weigh samples of coffee & compute means of samples; Plot 15

R Chart Type of variables control chart Interval or ratio scaled numerical data Shows sample ranges over time Difference between smallest & largest values in inspection sample Monitors variability in process, it tells us the loss or gain in dispersion. This change may be due to: 1. Worn bearing 2. A loose tool 3. An erratic flow of lubricant to machine 4. Sloppiness of machine operator Example: Weigh samples of coffee & compute ranges of samples; Plot 17

Construction of X and R Charts Step 1: Select the Characteristics for applying a control chart. Step 2: Select the appropriate type of control chart. Step 3: Collect the data. Step 4: Choose the rational sub-group i.e Sample Step 5: Calculate the average ( X) and range R for each sample. Step 6: Cal Average of averages of X and average of range(R)

Construction of X and R Charts Steps 7:Cal the limits for X and R Charts. Steps 8: Plot Centre line (CL) UCL and LCL on the chart Steps 9: Plot individual X and R values on the chart. Steps 10: Check whether the process is in control (or) not. Steps 11: Revise the control limits if the points are outside.

X Chart Control Limits From Tables Sub group average X = x1 + x2 +x3 +x4 +x5 / 5 Sub group range R = Max Value – Min value 16

R Chart Control Limits From Tables 18

Problem8.1 from TQM by V.Jayakumar Page No 8.5

p Chart for Attributes Type of attributes control chart Nominally scaled categorical data e.g., good-bad Shows % of nonconforming items Example: Count # defective chairs & divide by total chairs inspected; Plot Chair is either defective or not defective 19

p Chart p = np / n where p = Fraction of Defective np = no of Defectives n = No of items inspected in sub group p= Avg Fraction Defective = ∑np/ ∑n = CL

p Chart Control Limits z = 3 for 99.7% limits 20

Purpose of the p Chart Identify and correct causes of bad quality The average proportion of defective articles submitted for inspection,over a period. To suggest where X and R charts to be used. Determine average Quality Level.

Problem Problem 9.1 Page no 9.3 TQM by V.Jayakumar

np CHART P and np are quiet same Whenever subgroup size is variable,p chart is used. If sub group size is constant, then np is used. FORMULA: Central Line CLnp = n p Upper Control Limit, UCLnp = n p +3√ n p (1- p ) Lower Control Limit, LCLnp = n p -3 √ n p (1- p ) Where p = ∑ np/∑n =Average Fraction Defective n = Number of items inspected in subgroup.

Problem Problem No 9.11 page No 9.11 in TQM by V.Jayakumar

c Chart Type of attributes control chart Discrete quantitative data Shows number of nonconformities (defects) in a unit Unit may be chair, steel sheet, car etc. Size of unit must be constant Example: Count no of defects (scratches, chips etc.) in each chair of a sample of 100 chairs; Plot 21

c Chart Control Limits Use 3 for 99.7% limits 22