POPULATION VERSUS SAMPLE Definition A population consists of all elements – individuals, items, or objects – whose characteristics are being studied. The population that is being studied is also called the target population. A portion of the population selected for study is referred to as a sample.

Figure: Population and sample.

POPULATION VERSUS SAMPLE cont. Definition: A survey that includes every number of the population is called a census. The technique of collecting information from a portion of the population is called a sample survey.

Normally Distributed Curve

Skewed Distributions

Characteristics of the Normal Distribution It is symmetrical -- Half the cases are to one side of the center; the other half is on the other side. The distribution is single peaked, not bimodal or multi-modal Most of the cases will fall in the center portion of the curve and as values of the variable become more extreme they become less frequent, with “outliers” at each of the “tails” of the distribution few in number. It is only one of many frequency distributions but the one we will focus on for most of this course. The Mean, Median, and Mode are the same. Percentage of cases in any range of the curve can be calculated.

Family of Normal Curves

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 Interval or ratio scaled numerical data 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

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.

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.

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