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Chapter 1: Problem 21

A Pareto diagram is a variation of a histogram for categorical data resulting from a quality control study. Each category represents a different type of product nonconformity or production problem. The categories are ordered so that the one with the largest frequency appears on the far left, then the category with the second largest frequency, and so on. Suppose the following information on nonconformities in circuit packs is obtained: failed component, 126; incorrect component, 210 ; insufficient solder, 67; excess solder, 54 ; missing component, 131. Construct a Pareto diagram.

Short Answer

Expert verified
Draw bars representing frequencies of categories in decreasing order, add a cumulative line.

Step by step solution

01

Identify Categories and Frequencies

Identify the categories given in the problem and the frequency of each. The categories are: failed component (126), incorrect component (210), insufficient solder (67), excess solder (54), and missing component (131).
02

Sort Categories by Frequency

Arrange the categories in descending order of their frequencies: incorrect component (210), missing component (131), failed component (126), insufficient solder (67), excess solder (54).
03

Create Frequency Bars

Draw a bar for each category. Start with the category on the left having the highest frequency and place the remaining categories in order from highest to lowest frequency. The height of each bar corresponds to its frequency.
04

Add Cumulative Line

Calculate the cumulative frequency after each added category. Incorrect component is 210, then you add missing component to get 341 (210 + 131), followed by failed component to get 467 (341 + 126), then insufficient solder getting 534 (467 + 67), and finally adding excess solder to reach 588 (534 + 54). Plot these cumulative points above each respective bar and connect them with a line.
05

Label Axes and Components

Label the horizontal axis with the category names starting from the highest frequency and the vertical axis with the frequency scale. Clearly indicate frequency values for the bars and the cumulative percentage line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Histogram
A histogram is a graphical representation that organizes a group of data points into user-specified ranges. It is very useful for representing the distribution of numerical data.
  • A histogram displays continuous data. This is different from a bar chart, which is used for categorical data.
  • The bars in a histogram touch each other, indicating that the data is continuous and falls within range intervals or 'bins'.

For our Pareto diagram exercise, consider that a Pareto diagram is a type of specialized histogram. It visualizes categorical data by displaying bars in descending order based on frequency. This helps us quickly identify the most significant factors in a dataset.
Categorical Data
Categorical data is data that can be divided into specific groups or categories. This type of data is often used when information needs to be grouped based on qualitative characteristics.
There are key features of categorical data:
  • It represents characteristics such as a person's gender, marital status, or types of products such as those observed in a Pareto diagram.
  • The data can be nominal, which means the categories do not have an order or hierarchy.

In the context of the exercise, we have various categories of nonconformities in circuit packs, such as 'failed component' and 'incorrect component'. The frequency of each category helps in understanding which nonconformities occur most frequently, making it ideal for effective quality control analysis.
Quality Control
Quality control (QC) involves processes that ensure a product or service meets defined standards of quality. This is crucial in manufacturing to detect and reduce defects.
  • Pareto diagrams are vital tools in QC as they help prioritize the issues that need fixing based on their frequency of occurrence.
  • By tackling the most frequent problems first, efforts in quality improvement become more effective. For example, focusing on the 'incorrect component' category first would likely yield the biggest improvements.

The data on nonconformities within the manufacturing of circuit packs is used to identify areas with major quality issues, thereby guiding corrective actions efficiently.
Cumulative Frequency
Cumulative frequency is a way of summing up data points progressively to show the running total of frequencies from the start up to each point within a dataset. This provides a cumulative representation, which is very helpful for analysis.
  • In a Pareto diagram, this is visualized by a line that starts at the top of each bar and progresses to the next.
  • The cumulative line helps in understanding not just individual frequencies but also the accumulation of those frequencies, making it clear how much each category contributes to the total.

For instance, in the circuit pack example, as categories are plotted by descending frequency, the cumulative line helps pinpoint at what point most issues or nonconformities are being accounted for, which is essential for strategic decision-making.

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Most popular questions from this chapter

Automated electron backscattered diffraction is now being used in the study of fracture phenomena. The following information on misorientation angle (degrees) was extracted from the article "Hbservations on the Faceted Initiation Site in the Dwell-Fatigue Tested Ti-6242 Alloy: Crystallographic Orientation and Size Effects" (Metallurgical and Materials Trans., 2006: 1507-1518). \(\begin{array}{lcccc}\text { Class: } & 0-<5 & 5-<10 & 10-<15 & 15-<20 \\\ \text { Rel freq: } & .177 & .166 & .175 & .136 \\ \text { Class: } & 20-<30 & 30-<40 & 40-<60 & 60-<90 \\ \text { Rel freq: } & .194 & .078 & .044 & .030\end{array}\) a. Is it true that more than \(50 \%\) of the sampled angles are smaller than \(15^{\circ}\), as asserted in the paper? b. What proportion of the sampled angles are at least \(30^{\circ} ?\) c. Roughly what proportion of angles are between \(10^{\circ}\) and \(25^{\circ} ?\) d. Construct a histogram and comment on any interesting features.

The following data on distilled alcohol content (\%) for a sample of 35 port wines was extracted from the article "A Method for the Estimation of Alcohol in Fortified Wines Using Hydrometer Baum茅 and Refractometer Brix" (Amer. J. Enol. Vitic., 2006: \(486-490\) ). Each value is an average of two duplicate measurements. $$ \begin{array}{lllllllll} 16.35 & 18.85 & 16.20 & 17.75 & 19.58 & 17.73 & 22.75 & 23.78 & 23.25 \\ 19.08 & 19.62 & 19.20 & 20.05 & 17.85 & 19.17 & 19.48 & 20.00 & 19.97 \\ 17.48 & 17.15 & 19.07 & 19.90 & 18.68 & 18.82 & 19.03 & 19.45 & 19.37 \\ 19.20 & 18.00 & 19.60 & 19.33 & 21.22 & 19.50 & 15.30 & 22.25 & \end{array} $$ Use methods from this chapter, including a boxplot that shows outliers, to describe and summarize the data.

For each of the following hypothetical populations, give a plausible sample of size 4 : a. All distances that might result when you throw a football b. Page lengths of books published 5 years from now c. All possible earthquake-strength measurements (Richter scale) that might be recorded in California during the next year d. All possible yields (in grams) from a certain chemical reaction carried out in a laboratory

Allowable mechanical properties for structural design of metallic aerospace vehicles requires an approved method for statistically analyzing empirical test data. The article "Establishing Mechanical Property Allowables for Metals" (J. of Testing and Evaluation, 1998: 293-299) used the accompanying data on tensile ultimate strength (ksi) as a basis for addressing the difficulties in developing such a method. \(\begin{array}{lllllllll}122.2 & 124.2 & 124.3 & 125.6 & 126.3 & 126.5 & 126.5 & 127.2 & 127.3 \\ 127.5 & 127.9 & 128.6 & 128.8 & 129.0 & 129.2 & 129.4 & 129.6 & 130.2 \\ 130.4 & 130.8 & 131.3 & 131.4 & 131.4 & 131.5 & 131.6 & 131.6 & 131.8 \\ 131.8 & 132.3 & 132.4 & 132.4 & 132.5 & 132.5 & 132.5 & 132.5 & 132.6 \\ 132.7 & 132.9 & 133.0 & 133.1 & 133.1 & 133.1 & 133.1 & 133.2 & 133.2 \\ 133.2 & 133.3 & 133.3 & 133.5 & 133.5 & 133.5 & 133.8 & 133.9 & 134.0 \\ 134.0 & 134.0 & 134.0 & 134.1 & 134.2 & 134.3 & 134.4 & 134.4 & 134.6 \\ 134.7 & 134.7 & 134.7 & 134.8 & 134.8 & 134.8 & 134.9 & 134.9 & 135.2 \\ 135.2 & 135.2 & 135.3 & 135.3 & 135.4 & 135.5 & 135.5 & 135.6 & 135.6 \\ 135.7 & 135.8 & 135.8 & 135.8 & 135.8 & 135.8 & 135.9 & 135.9 & 135.9 \\ 135.9 & 136.0 & 136.0 & 136.1 & 136.2 & 136.2 & 136.3 & 136.4 & 136.4 \\ 136.6 & 136.8 & 136.9 & 136.9 & 137.0 & 137.1 & 137.2 & 137.6 & 137.6 \\ 137.8 & 137.8 & 137.8 & 137.9 & 137.9 & 138.2 & 138.2 & 138.3 & 138.3 \\ 138.4 & 138.4 & 138.4 & 138.5 & 138.5 & 138.6 & 138.7 & 138.7 & 139.0 \\ 139.1 & 139.5 & 139.6 & 139.8 & 139.8 & 140.0 & 140.0 & 140.7 & 140.7 \\ 140.9 & 140.9 & 141.2 & 141.4 & 141.5 & 141.6 & 142.9 & 143.4 & 143.5 \\ 143.6 & 143.8 & 143.8 & 143.9 & 144.1 & 144.5 & 144.5 & 147.7 & 147.7\end{array}\) a. Construct a stem-and-leaf display of the data by first deleting (truncating) the tenths digit and then repeating each stem value five times (once for leaves 1 and 2 , a second time for leaves 3 and 4 , etc.). Why is it relatively easy to identify a representative strength value? b. Construct a histogram using equal-width classes with the first class having a lower limit of 122 and an upper limit of 124 . Then comment on any interesting features of the histogram.

Mercury is a persistent and dispersive environmental contaminant found in many ecosystems around the world. When released as an industrial by-product, it often finds its way into aquatic systems where it can have deleterious effects on various avian and aquatic species. The accompanying data on blood mercury concentration \((\mu \mathrm{g} / \mathrm{g})\) for adult females near contaminated rivers in Virginia was read from a graph in the article "Mercury Exposure Effects the Reproductive Success of a Free-Living Terrestrial Songbird, the Carolina Wren" (The Auk, 2011: 759_769; this is a publication of the American Ornithologists' Union). \(\begin{array}{rrrrrrrr}.20 & .22 & .25 & .30 & .34 & .41 & .55 & .56 \\ 1.42 & 1.70 & 1.83 & 2.20 & 2.25 & 3.07 & 3.25 & \end{array}\) a. Determine the values of the sample mean and sample median and explain why they are different. [Hint: \(\left.\Sigma x_{1}=18.55 .\right]\) b. Determine the value of the \(10 \%\) trimmed mean and compare to the mean and median. c. By how much could the observation \(.20\) be increased without impacting the value of the sample median?
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