Problem 234 - Applied Statistics and Probability for Engineers

Question

A process is said to be of six-sigma quality if the process mean is at least six standard deviations from the nearest specification. Assume a normally distributed measurement. (a) If a process mean is centered between upper and lower specifications at a distance of six standard deviations from each, what is the probability that a product does not meet specifications? Using the result that 0.000001 equals one part per million, express the answer in parts per million. (b) Because it is difficult to maintain a process mean centered between the specifications, the probability of a product not meeting specifications is often calculated after assuming that the process shifts. If the process mean positioned as in part (a) shifts upward by 1.5 standard deviations, what is the probability that a product does not meet specifications? Express the answer in parts per million. (c) Rework part (a). Assume that the process mean is at a distance of three standard deviations. (d) Rework part (b). Assume that the process mean is at a distance of three standard deviations and then shifts upward by 1.5 standard deviations. (e) Compare the results in parts (b) and (d) and comment.

Step-by-Step Solution

Answer

(a) 0.2 ppm; (b) 30 ppm; (c) 2,700 ppm; (d) 66,800 ppm; (e) More shift impact with lower sigma.

Step 1: Understanding Six-Sigma Quality

A six-sigma process means that the process mean is six standard deviations away from the upper or lower specification limits. Under a normal distribution, this implies that the majority of the process output falls within the specification limits.

Step 2: Calculating Part (a) Probability

For a balanced six-sigma process, the process mean is 6 standard deviations away from the nearest specification limit. This corresponds to the probability in the tail of a standard normal distribution beyond 6 standard deviations. Using Z-tables, the probability is approximately 0.0000002 (or 0.00002%). This equates to about 0.2 parts per million (ppm) as 0.0000002 * 1,000,000 = 0.2 ppm.

Step 3: Calculating Part (b) for Shifted Process

When the process mean shifts by 1.5 standard deviations, it is now 4.5 standard deviations away from the nearest specification limit. Using Z-tables or a standard normal distribution calculator, the probability of falling outside is approximately 0.00003 (or 0.003%), which is 30 parts per million (ppm).

Step 4: Calculating Part (c) with Three Standard Deviations

If the process mean is only 3 standard deviations away from the specification limit (not meeting six-sigma), use Z-tables to find the probability. For 3 standard deviations, the probability of falling outside is about 0.0027 (or 0.27%), resulting in 2,700 parts per million (ppm).

Step 5: Calculating Part (d) with Process Shift

If this same process, initially at 3 standard deviations, shifts by 1.5 standard deviations, it then rests at 1.5 standard deviations from the limit. The probability of a product not meeting specifications is now larger, about 0.0668 (or 6.68%), translating to 66,800 parts per million (ppm).

Step 6: Comparing Parts (b) and (d)

A shift in the process mean greatly affects processes with lower sigma quality. Part (b), starting at a six-sigma quality, resulted in a small increase to 30 ppm. However, part (d) started at three-sigma, and the shift resulted in a dramatic increase to 66,800 ppm. This demonstrates the importance of maintaining higher sigma quality.

Key Concepts

Process Capability

When we talk about process capability, we're describing how well a process can produce outputs within specification limits. It's a measure of the inherent variability of a process in relation to its allowable limits. The higher the process capability, the more reliable it is in consistently meeting specifications.

• Capability Indices: These quantify process performance; the most common are Cp and Cpk. Both compare process spread to specification limits.
• Process Centering: When the process is centered, it's aligned with the midpoint of the specification limits. This reduces the chance of producing defects.
• Impact of Process Capability: A higher capability index means a more robust process, producing fewer defects, and aligning with Six Sigma goals.

In a six-sigma process, the specification limits are far removed from the process average, ensuring most products meet the quality standards expected.

Standard Deviations

Standard deviation is a statistical measurement of variability or dispersion in a data set. It tells us how much individual data points differ from the mean, or average, value. A small standard deviation indicates data points tend to be very close to the mean, while a large standard deviation indicates data are spread out over a large range of values.

• Understanding Sigma Levels: In Six Sigma, the term 'sigma' refers to standard deviations. A six-sigma level means that six standard deviations fit between the process mean and the nearest specification limit.
• Importance in Quality Control: High standard deviations imply greater variability and potential quality issues; thus, reducing variability is key to quality improvement.

By maintaining a low standard deviation, processes remain consistent and predictable, reducing the likelihood of defects.

Normal Distribution

Normal distribution, often called the bell curve, is a fundamental concept in statistics describing data that symmetrically distributes around the mean. In a perfectly normal distribution, the mean, median, and mode are all the same value.

• Characteristics: It's defined by its bell shape. About 68% of data within a normal distribution falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
• Role in Six Sigma: Processes following a normal distribution allow for predictions of defect rates beyond certain sigma levels.
• Relevance in Practice: Assuming normal distribution helps in applying statistical methods for process evaluation and control.

Most process capabilities assume normal distribution, which makes understanding this concept crucial for implementing Six Sigma practices.

Specification Limits

Specification limits define the acceptable range of variation in an output for it to be considered acceptable. They are critical in determining whether a product meets the desired quality standards.

• Types of Limits: The upper specification limit (USL) is the highest acceptable value, while the lower specification limit (LSL) is the lowest.
• Setting the Limits: They are determined based on customer requirements and product design, setting the boundaries for process output.
• Importance in Quality Assurance: Outputs falling outside specification limits indicate defects, making these thresholds vital for monitoring quality.

An effective Six Sigma process aims to operate well within these specification limits, minimizing the chance of defects and enhancing product quality.