Sampling

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Sampling is more of a technique than a report. It is the act of collecting a portion of all data and using that portion to draw conclusions. A decision is then made to accept or reject the entire lot based on the results of the sample.

The advantages of using sampling come from reducing the costs you would incur if you had to inspect 100% of the lot. But great care should be taken in order to assure you are collecting the appropriate amount of data. Sampling is recommended in situations where looking at all of the data is either too expensive, too time consuming or destructive.

When To Use Sampling


  • When data is too expensive to gather or too time-consuming
  • When the test is destructive
  • When there is a need to improve inspection time
  • When there is a need for greater accuracy

Some factors that you should identify in order to determine how many samples you need are:

  • What type of data are you handling: discrete or continuous?
  • The acceptance of good lots should be more likely than the acceptance of bad lots.
  • What level of “Producers Risk”, the probability that a good lot will be rejected, do you desire.
  • What level of “Consumers Risk”, the probability that a bad lot will be accepted, do you desire.

The most important aspect of sampling is that it requires randomness in the sample selection. The sample must be representative of the population and not just the product that is easiest to obtain. There are many industry standard sampling plans available. Learn more about acceptance sampling.

Random Sampling


Purpose

Random Sampling is used to ensure that the elements selected for measurement are the result of pure chance.

Selecting samples at random increases the likelihood that the measurements of the samples are representative of the process population.

Anatomy

Random Sampling

Random Sampling From A Production Line

Reference: Juran’s Quality Control Handbook

Terminology

A. Process

B. Process Output for a given period

C. Individual Units - With Random Sampling at any one time each of the remaining units of product has an equal chance of being the next unit selected for sample.

D. List of Random Numbers - To conduct Random Sampling requires that random numbers be generated and that the numbers can be applied to the process output. Random numbers can be generated using calculators, computers, Excel, Minitab, a bowl of numbered chips, random number dice, or random number tables.

E. Unit Numbers corresponding to the list of random numbers.

Sequential Sampling


Purpose

Sequential Sampling plans are used for selecting samples of units produced in consecutive order.

Anatomy

Choosing Samples In Sequence

Sequential Sampling From A Process

Terminology

A. Process

B. Process Output for a given period. Measurements taken and recorded from units produced in sequence at different periods (hourly, weekly, monthly, etc.)

Stratified Sampling


Purpose

Stratified Sampling is used when a lot is the result of multiple operators, machines, shifts, etc.; the product is actually the combination of several smaller lots. Selecting stratified samples increases the likelihood that the measurements of the samples are representative of the combined lots’ process population.

Anatomy

Stratified Sampling From Multiple Processes

Stratified Sampling From Multiple Processes

Reference: Juran’s Quality Control Handbook

Terminology

A. Process - When Stratified Sampling is used the lot is known to come from different machines, shifts, operators, etc. The roman numerals represent these differences.

B. Process Output for a given period from different processes.

C. List of Random Numbers - With Stratified Sampling the samples are selected proportionately from each process.

Within each lot random sampling is used; this means at any one time each of the remaining units of product has an equal chance of being the next unit selected for sample.

To conduct Random Sampling requires that random numbers be generated and that the numbers can be applied to the process output. Random numbers can be generated using calculators, computers, Minitab, Excel, a bowl of numbered chips, or random number dice.

D. Proportionate Samples from each individual process.

E. Sample

Sample Size: Continuous Data -Mean


Purpose

To calculate the sample size required in estimating the population mean of a continuous CT characteristic from a sample of data within a certain margin of error.

Anatomy

Sample Size To Estimate The Mean

Determining The Sample Size to Estimate The Mean

Reference: Juran's Quality Control Handbook

Terminology

A. n - Sample size symbol.

B. Zalpha/2 - Tabulated standard normal (Z) distribution value with probability alpha/2.

C. The known population standard deviation from past experience or the sample standard deviation from a preliminary sample.

D. E - The maximum allowable error or the desired precision.

Major Considerations

The assumption for using this formula is that the data comes from a normal distribution. If the standard deviation of the population is unknown, the sample standard deviation of a preliminary sample of at least 30 can be used.

Application Cookbook

1. Establish alpha level. Usually alpha is 0.05.

2. Obtain the Zalpha/2 value from a software such as Excel, or a table of area under the normal curve (ex. For alpha=0.05, Zalpha/2 = Z 0.025 =1.96).

3. Establish E, which is the maximum allowable error or the desired precision, expressed in the measurement unit of the CT characteristic.

In other words, let's say that we are willing to take a alpha risk that our estimate of the mean of the population will be off by E or more units. As an example, suppose we wish to estimate the true cycle time of a process. We determine that our estimate must be within 2.0 hours of the true mean in order to be useful, then E=2.

4. Obtain standard deviation from past experience or an estimate from a preliminary sample of observations equal to or greater than 30. If only a smaller preliminary sample is available, the following chart can be used.

5. Use the formula presented or the following chart.

To use the chart, first calculate E/s (s is the sample standard deviation) and select the curve line corresponding to the alpha level. The sample size can be read on the Y-axis of the graph.

Sample Size Chart

Sample Size Chart

Determining The Sample Size Required To Estimate The Mean

Reference: Juran's Quality Control Handbook

Sample Size: Continuous Data - Standard Deviation


Purpose

To calculate the sample size required in estimating the population standard deviation of a continuous CT characteristic from a sample within a certain margin of error.

Anatomy

Estimating Standard Deviation

Estimating Standard Deviation

Determining The Sample Size To Estimate Standard Deviation

Reference: Juran's Quality Control Handbook

Terminology

A. gamma - Confidence level (1-alpha)

B. P% - The maximum allowable error or the desired precision expressed as a percentage

C. Degrees of freedom v = n - 1 , hence sample size = v + 1

Major Considerations

The assumption for using this formula is that the data comes from a normal distribution.

Application Cookbook

1. Establish confidence level (1-alpha). Usually the confidence level is 0.95

2. Establish P%, the allowable percentage deviation of the estimated standard deviation from its true value.

  • As an example, suppose we want to know how large a sample would be required to estimate the standard deviation within 20% of its true value, with a confidence level equal to 0.95, then P=20% (and gamma=0.95).

3. Select the line corresponding to the confidence level. At the intersection between this line and the P% line, the number of degrees of freedom (n-1) are obtained on the Y-axis.

4. Add 1 to obtain the sample size.

Sample Size: Continuous Data – One Sample Mean Test


Purpose

To calculate the sample size required for a one sample Mean Test taking into account the desired alpha and beta risks.

Anatomy

One Sample Mean Test

Determining The Sample Size For A One Sample Mean Test


Reference: Juran's Quality Control Handbook

Terminology

A. Sample size symbol.

B. The known population variance from past experience or the sample variance from a preliminary sample.

C. Tabulated standard normal (Z) distribution.

D. Tabulated standard normal (Z) distribution value.

E. The minimum difference that we want to be able to detect from this test.

Major Considerations

The assumption for using this formula is that the data comes from a normal distribution.

Application Cookbook

1. Establish alpha level and beta level. Usually alpha is 0.05 and beta is 0.1.

2. Obtain the Zalpha/2 and Zbeta values from a software such as Excel, or a table of area under the normal curve (ex. For alpha=0.05, Zalpha/2 = Z 0.025 = 1.96 and for beta=0.10, Zbeta = 1.28).

3. Obtain standard deviation from past experience or from an estimate from a preliminary sample of size equal to or greater than 30.

4. Establish the minimum difference that you want to be able to detect from the test expressed in the measurement units of the CT characteristic. In other words, if the population mean differs by a certain value, we would like to detect this value with a high probability.

For example, suppose we wish to test the true cycle time of a process. We determine that if the mean cycle time differs by as much as 2.0 hours, we would like to be able to detect it, then difference=2.

5. Use the formula presented. The sample size will be the next integer (e.g. n=35.32, sample size will be 36).

Sample Size: Continuous Data – Two Samples Mean Test


Purpose

To calculate the sample size, taking into account desired alpha and beta risks.

Anatomy

Two Sample Means Test

Determining The Sample Size For A Two Samples Means Test

Reference: The Vision of Six Sigma: Tools and Methods for Breakthrough

Terminology

A. Alpha and beta risk.

B. The difference in means that we want to be able to detect with the test expressed in standard deviation units.

C. Corresponding sample size for sample 1 and 2.

D. Formula used in the table where: n1 and n2 represent sample sizes of sample 1 and 2, Zalpha/2 and Zbeta are tabulated standard normal (Z) distribution values with probability alpha/2 and beta, standard deviation is the population standard deviation and g is the difference that we want to be able to detect with the test.

Major Considerations

The assumption for using this formula is that the data comes from a normal population.

Application Cookbook

1. Establish alpha level and beta level. Usually alpha is 0.05 and beta is 0.10.

2. Establish g2, which represents the minimum difference that we want to be able to detect from the test expressed in the measurement units of the CT characteristic. In other words, if the two population means differ by a certain value, we would like to detect this value with a high probability.

For example, suppose we wish to compare the true cycle time of a process before and after a change. We determine that if the mean cycle time differs by as much as 2.0 hours, we would like to be able to detect it, so g2=2.

3. Obtain standard deviation from past experience or an estimate from a preliminary sample.

4. Use the formula or the table presented to obtain the sample size.
If the formula is used, obtain the Zalpha/2 and Zbeta values from a software such as Excel or a table of area under the normal curve (ex. For alpha=0.05, Zalpha/2 = Z0.025 = 1.96 and for beta=0.10, Zbeta = 1.28).

Sample Size: Discrete Data


Purpose

To establish the appropriate sample size for a population of discrete data.

Anatomy

Discrete Data Sample Size

Determining The Sample Size For Discrete Data

Terminology

A. Sample Size - Discrete Data (Proportion)

B. Proportion - The proportion of successes

Major Considerations

Applicable to discrete distributions.

  • To be used only where p is not too close to zero or one.
  • Applies for normal approximation of the binomial distribution.
  • Sample size should be a minimum of 30.
  • Sample size should be sufficiently large such that under practical worst case considerations no data frequency (cell) is greater than or equal to 5.

Application Cookbook

1. Estimate the value of p for the population.

2. Choose a sample size.

3. Verify both equations are satisfied by calculating values for both equations

Discrete Data Sample Size

From Sampling to Six Sigma Tools.

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