Statistical Quality Control

Contents

• Concept of variation

• Reasons for variation

• Statistical quality control

• Objectives of SQC

• Benefits of SQC

• Concept of probability

• Fundamentals of statistics

• Basic definitions

• Frequency Distribution

• Measures of central tendency

• Mean

• Median

• Mode

• Measures of dispersion

• Range

• Variance

• Standard Deviation

• Probability distribution

• Discrete distributions

• Hypergeometric

• Binomial

• Poisson

• Continuous distributions

• Normal

• Exponential

• Weibull

• Gamma

• Lognormal

Concept of variation

• No two items will be perfectly identical even if extreme care is taken to make them identical in some respect.

• Variation is a fact of nature and manufacturing processes are not exceptions to this.

• Types of variation (for the purpose of analysis) • Within the part

• Among the parts produced during the same period of time

• Among the parts produced at different period of time

Reasons for Variation

• Tool wear

• Machine vibration

• Loose bearings

• Faulty jigs and fixtures

• Poor quality of raw material

• Poor maintenance

• Carelessness of operator

• Untrained operator

• Fatigue caused to the operator

Statistical quality control (SQC) • Statistics - is science that deals with the collection,

classification, analysis, and making of inferences from data or information.

• Quality - is a relative term and is generally explained with reference to the end use of the product.

• Control - is a system for measuring and checking or inspecting a phenomenon. It suggests when to inspect, how often to inspect and how much to inspect.

• SQC refers to the use of statistical methods in the monitoring and maintaining of the quality of products and services.

• The tools of SQC are helpful in evaluating the quality of products or services.

• Different tools to analyze quality problem.

• Descriptive Statistics - involves describing quality characteristics and relationships.

• Statistical Process Control (SPC) - involves inspect random sample of output from process for characteristic. Also known as Inferential statistics.

• Acceptance Sampling - involve batch sampling by inspection. It does not control or improve the quality level of the process.

Objectives of SQC • Quality control includes quality of product or service

given to customer, leadership, commitment of management, continuous improvement, fast response, actions based on facts, employee participation and a quality driven culture.

• Objective of SQC is to control of material reception, internal rejections, clients, claims, providers and evaluations of the same corrective actions are related to their follow-up.

• These systems and methods guide all quality activities. The development and use of performance indicators is linked, directly or indirectly, to customer requirements and satisfaction, and to management.

Benefits of SQC

• It provides a means of detecting error at inspection. • It leads to more uniform quality of production. • It improves the relationship with the customer. • It reduces inspection costs. • It reduces the number of rejects and saves the cost of

material. • It provides a basis for attainable specifications. • It points out the bottlenecks and trouble spots. • It provides a means of determining the capability of

the manufacturing process. • It promotes the understanding and appreciation of

quality control.

• The probability of an event describes the chance of occurrence of that event.

• Probability of an event A is given by

• Complement of A,

• Additive law

• Multiplicative law

• Independent events

• Mutually exclusive events

Concept of Probability

In the production of metal plates for an assembly, it is known from past experience that 5% of the plates do not meet the length requirement. Also, from historical records, 3% of the plates do not meet the width requirement. Assume that there are no dependencies between the processes that make the length and those that trim the width. a) What is the probability of producing a plate that meets both

the length and width requirements? Let A be the outcome that the plate meets the length requirement B be the outcome that the plate meets the width requirement. From the problem statement, P(Aᶜ) = 0.05 and P(Bᶜ) = 0.03. Then P(A) = 1 - P(Aᶜ) = 1 - 0.05 = 0.95 P(B) = 1 - P{Bᶜ) = 1 - 0.03 = 0.97 Using the special case of the multiplicative law for independent events, we have P(meeting both length and width requirements) = P(A∩B) = P(A)P(B) (since A and B are independent events) = (0.95) (0.97) =0.9215

b) What proportion of the parts will not meet at least one of the requirements?

The required probability = P(Aᶜor Bᶜ or both).

Using the additive law,

P(Aᶜ or Bᶜ or both) = P{Aᶜ) + P(Bᶜ) - P(Aᶜ ∩Bᶜ)

= 0.05 + 0.03 - (0.03)(0.05) = 0.0785

Therefore, 7.85% of the parts will have at least one characteristic (length, width, or both) not meeting the requirements.

c) What proportion of parts will meet neither length nor width requirements?

We need to find P(Aᶜ∩Bᶜ)

P(Aᶜ∩Bᶜ) = P(Aᶜ)P(Bᶜ) = (0.05)(0.03) = 0.0015

0.15% of the parts will meet neither length nor width requirements

d) Suppose the operations that produce the length and the width are not independent. If the length does not satisfy the requirement, it causes an improper positioning of the part during the width trimming and thereby increases the chances of nonconforming width. From experience, it is estimated that if the length does not conform to the requirement, the chance of producing nonconforming widths is 60%. Find the proportion of parts that will neither conform to the length nor the width requirements.

The probability of interest is P(Aᶜ∩Bᶜ). The problem states that P(Bᶜ|Ac) = 0.60. Using the general from of the multiplicative law P(Aᶜ∩Bᶜ)= P(Aᶜ)P(Bᶜ|Aᶜ) = (0.05) (0.06) = 0.03 So 3% of the parts will meet neither the length nor the width requirements. Notice that this value is different from the answer to part (c), where the events were assumed to be independent.

Fundamentals of statistics

• Data can be collected in several ways.

• direct observation.

• indirect observations

• Data on quality characteristics are categorized as

• Continuous – can assume any number on continuous scale within a range. (Variable)

For eg. - Viscosity of fluid, length of fabric

• Discrete – they are the count of an event. They assume a finite or countably infinite number of values. (Attributes)

• For eg. – No. of defective bolts, satisfied customers

Basic definitions

• Population is the set of all items that possess a certain characteristic of interest.

• Sample is a subset of a population.

• Parameter is a characteristic of a population, something that describes it.

• Statistic is a characteristic of a sample, used to make inferences on the population parameters that are typically unknown.

• Accuracy - degree of uniformity of the observations around a desired value.

• Precision - the degree of variability of the observations.

• Frequency distribution – is a tabulation of data obtained from measurement, arranged in ascending or descending order of the required measurement.

• It is a practical method of analyzing the quality of production process in terms of product specification limits.

• It is useful in

• New design control

• Product control

• Incoming material control

• Special process studies

The other uses of frequency distribution in SQC

• For predicting the characteristic of entire lot.

• For determining the process capability.

• For comparison of inspection results between two units or sections.

• For examining the difference between the dimensional characteristics of similar parts produced from two different processes.

• For examining accuracy of fit between mating parts.

• For analyzing the effect of tool wear during a long production run or a machine tool.

Mean - simple average of the observations in a data set.

Measures of Central Tendency

Sample mean Population mean

The population mean is sometimes denoted as E(X), the expected value of the random variable X. It is also called the mean of the probability distribution of X.

Median is the value in the middle, when the observations are ranked. Mode is the value that occurs most frequently in the data set.

Measures of Dispersion • Gives information about the extent to which data

is scattered about the zone of central tendency

• Range – simplest of all. It is the difference between the largest and smallest value of data set.

• Variance - measures the fluctuation of the observations around the mean. The larger the value, the greater the fluctuation.

Variance of population Variance of sample

• Standard deviation - measures the variability of the observations around the mean. It is equal to the positive square root of the variance.

Std. deviation of sample

Std. deviation of population

• Data set with the largest standard deviation has the most variability

A random sample of 10 observations of the output voltage of transformers is taken. The values (in volts, V) are as follows:

9.2, 8.9, 8.7, 9.5, 9.0, 9.3, 9.4, 9.5, 9.0, 9.1

• Sample mean is 9.16 V.

X

• Sample variance is given by

2V0.0716110

0.644

• Sample standard deviation

Probability distributions • Data values in a population are described by a

probability distribution.

• For discrete data, a probability distribution shows the values that the random variable can assume and their corresponding probabilities.

• A discrete random variable X, which takes on the values x₁, x₂, and so on, a probability distribution function p(x) has the following properties:

• Continuous data can take on an infinite number of values, so the probability distribution is usually expressed as a mathematical function of the random variable.

• When X is a continuous random variable, the probability density function is represented by f(x), which has the following properties:

Discrete distributions

• The discrete class of probability distributions deals with those random variables that can take on a finite or countably infinite number of values.

• Hypergeometric distribution

Hypergeometric distribution • It is useful in sampling from a finite population without

replacement when the outcomes can be categorized either as success or failure.

• If we consider finding a nonconforming item a success, the probability distribution of the number of nonconforming items (x) in the sample is given by

The mean (or expected value) of a hypergeometric distribution is given by

The variance of a hypergeometric random variable is given by

A lot of 20 chips contains 5 nonconforming ones. If an inspector randomly samples 4 items, find the probability of 3 nonconforming chips. In this problem, N = 20, D = 5, n = 4, and x = 3.

Binomial distribution

Probability

Mean

Variance

It is also used for situations in which items are selected from an ongoing process (i.e., the population size is very large).

It is applicable to sampling without replacement from a population (or lot) that is large compared to the sample, or to sampling with replacement from a finite population.

A manufacturing process is estimated to produce 5% nonconforming items. If a random sample of five items is chosen, find the probability of getting two nonconforming items.

Here, n = 5,p = 0.05 (if success is defined as getting a nonconforming item), and x = 2.

Poisson distribution • It is used to model the number of events that happen within

• a product unit (number of defective rivets in an airplane wing),

• space or volume (blemishes per 200 square meters of fabric), or

• time period (machine breakdowns per month). It is assumed that the events happen randomly and independently.

• The Poisson random variable is denoted by X. An observed value of X is represented by x.The probability distribution (or mass) function of the number of events (x) is given by

The mean and the variance of Poisson distribution are equal and are given by

It is estimated that the average number of surface defects in 20 m2 of paper produced by a process is 3. What is the probability of finding no more than 2 defects in 40 m2 of paper through random selection?

λ = 6,

Continuous distributions

• Continuous random variables may assume an infinite number of values over a finite or

• infinite range. The probability distribution of a continuous random variable X is often called the probability density function f(x).

• The total area under the probability density function is 1.

Normal Distribution • The probability density function of a normal

random variable is given by

where μ is the population mean, and σ is the population standard deviation.

Effects of the parameters μ and σ on the normal distribution.

• Area within certain limits for any normal distribution can be found by looking up tabulated areas for a standard normal distribution.

• The standardized normal random variable Z is given by

• The distribution of the standardized normal random variable has a mean of 0 and a variance of 1. It is represented as an N(0,1) variable, where the first parameter represents the mean and the second the variance

xZ

Graphical representation of General and Standardized form

The length of a machined part is known to have a normal distribution with a mean of 100 mm and a standard deviation of 2 mm.

a) What proportion of the parts will be above 103.3 mm?

Let X = length of the part. μ = 100 and σ = 2.

The standardized value of 103.3 corresponds to

65.12

1003.103xZ

Thus,P(X> 103.3) = P(Z> 1.65). From Appendix A-3, P(Z< 1.65) = 0.9505, which also equals P(X< 103.3). So P(Z> 1.65) = l-P(Z< 1.65) = 1 - 0.9505 =0.0495

b) What proportion of the output will be between 98.5 and 102.0 mm?

The standardized values are computed as

12

100102Z1

75.0

2

1005.98Z2

From Appendix A-3, we have P(Z< 1.00) = 0.8413 and P(Z< -0.75) = 0.2266. The required probability equals 0.8413 - 0.2266 = 0.6147. Thus, 61.47% of the output is expected to be between 98.5 and 102.0 mm

c) What proportion of the parts will be less than 96.5 mm? We want P(X < 96.5), which is equal to )5.96P(X

75.12

1005.96Z1

Using Appendix A-3, P(Z<=1.75)= 0.0401. Thus, 4.01% of the parts will have a length less than 96.5 mm. d) It is important that not many of the parts exceed the desired length. If a manager stipulates that no more than 5% of the parts should be oversized, what specification limit should be recommended? Let the specification limit be A. From the problem data, P(X >= A) =0.05 and P(X<A) = 1-0.05 = 0.95.

2

100x645.1 i

Thus, A should be set at 103.29 mm to achieve the desired stipulation.

X=103.29

Exponential Distribution Used in reliability analysis to describe the time to the failure of a component or system.

It is known that a battery for a video game has an average life of 500 hours. The failures of batteries are known to be random and independent and may be described by an exponential distribution.

(a) Find the probability that a battery will last at least 600 hours.

Since the average life, or mean life, of a battery is given to be 500 hours, the failure rate is λ = 1/500.

If the life of a battery is denoted by X, we wish to find P(X > 600).

b) Find the probability of a battery failing within 200 hours.

c) Find the probability of a battery lasting between 300 and 600 hours.

(d) Find the standard deviation of the life of a battery. σ=1/λ= 500 hr

(e) If it is known that a battery has lasted 300 hours, what is the probability that it will last at least 500 hr?

Weibull Distribution

• It is typically used in reliability analysis to describe the time to failure of mechanical and electrical components.

• It is a three-parameter distribution. • A Weibull probability density function is given by

The mean and the variance of the distribution are

The time to failure for a cathode ray tube can be modeled by a Weibull distribution with parameters γ = 0, β =1/300 and α = 200 hours.

a) Find the mean time to failure and its standard deviation.

Gamma Distribution

Applications in reliability analysis. Its probability density function is given by

where k is a shape parameter, k > 0; and λ is a scale parameter, λ > 0. The mean and the variance of the gamma distribution are μ = k/ λ variance = k/ λ

Lognormal Distribution # Quality characteristics such as tensile strength or compressive strength are modeled. (Failure distributions due to accumulated damage, such as crack propagation or wear) # A random variable X has a lognormal distribution if ln(x) has a normal distribution with mean, μ and variance, where In represents the natural logarithm. Its probability density function is given by

The mean and variance of the lognormal distribution are