Engineering Fundamentals and Problem Solving, 6e Chapter 10 Statistics

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Objectives Analyze a wide variety of data sets using descriptive techniques (mean, mode, variance, standard deviation, and correlation) Learn to apply the appropriate descriptive statistical techniques in a variety of situations Create graphical representations of individual and grouped data points with graphs and histograms 2

The Need for Statistical Methods “Quality is job one” “…..The basic concept of using statistical signals to improve performance can be applied to any area where output exhibits variation, such as component dimensions, bookkeeping error rates, performance characteristics of a computer information system, or transit times for incoming materials…..” Continuous Process Control Manual Ford Motor Company

The Need for Statistical Methods “…..As world competition intensifies, understanding and applying statistical concepts and tools is becoming a requirement for all employees. Those individuals who get these skills in school will have a real advantage when they apply for their first job.” Paul H. O’Neill CEO, Aluminum company of America

The Need for Statistical Methods “ The competitive position of industry in the US demands that we greatly increase the knowledge of statistics among our engineering graduates…….. The economic survival in today’s world cannot be ensured without access to modern productivity tools, notably applications of statistical methods.” Arno Penzias VP at AT&T Bell laboratories

A Model for Problem Solving State the problem or question Collect and analyze data Interpret the data and make decisions Implement and verify the decisions Plan next actions

From Data Tables to Probability Goal: Improving the quality of any process Solution: Using tools of statistics to make decisions from data in an organized way. How do we obtain good data on which to base these decisions? Most good plans for collecting data make use of randomization which is tied to probability

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Descriptive Statistics Used to summarize or describe important features of a data set Parameters are calculated from available observations Engineers generally contend with samples rather than entire populations of data 8

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Numerical Summaries of Data Data are the numeric observations of a phenomenon of interest The totality of all observations is called a population (finite or infinite) A portion used for analysis is a random sample from the population The collection described in terms of shape, outliers, center, and spread (SOCS) Center  mean; Spread  variance

Population vs. Sample Population described by its parameters ( ,  ) Sample described by its statistics (, s) The statistics are used to estimate the parameters.

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Measures of Central Tendency MEDIAN – “Middle” value in a sample MODE – The most common value in the sample (there may be more than one mode) MEAN – Arithmetic average – Geometric average – Harmonic average 11

Mean If the n observations in a random sample are denoted by x 1, x 2,..., x n, the sample mean is For a finite population with N equally likely values, the population mean is

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Measures of Variation Represent the amount of disparity (dispersal, scatter) between the data points and the mean Variance Standard Deviation Note: n-1 is the number of degrees of freedom left after calculating n 13

Variance If the n observations in a random sample are denoted by x 1, x 2,..., x n, the sample variance is For a finite population with N equally likely values, the population variance is

Frequency Distributions A frequency distribution is a compact summary of data, expressed as a table, graph, or function. The data is gathered into bins or cells, defined by class intervals. The number of classes, multiplied by the class interval, should exceed the range of the data. Number of bins approximately equal to square root of the sample size The boundaries of the class intervals should be convenient values, as should the class width.

Frequency Distribution Table

Histograms A histogram is a visual display of a frequency distribution, similar to a bar chart or a stem-and- leaf diagram. Steps to build one with equal bin widths: 1. Label the bin boundaries on the horizontal scale. 2. Mark & label the vertical scale with the frequencies or relative frequencies. 3. Above each bin, draw a rectangle whose height = the frequency or relative frequency.

Shape of Frequency Distribution

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 20 Interstate Safety Corridors are established on certain roadways with a propensity for strong cross winds, blowing dust, and frequent fatal accidents. A driver is expected to turn on the headlights and pay special attention to the posted speed limit in these corridors. In one such Safety Corridor in northern New Mexico, the posted speed limit is 75 miles per hour. The Department of Public Safety set up a radar checkpoint and the actual speed of 36 vehicles that passed the checkpoint is shown in the Table below. Example 10.2

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Actual speeds of cars in Safety Corridor Example – cont’d

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. a.) Make a frequency distribution table using 5 as a class width (e.g – 64.9) b.) Construct a histogram IntervalFrequency Example – cont’d

Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. c) The mean d) The standard deviation e) The variance 23 Example – cont’d