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Engineering Fundamentals and Problem Solving, 6e Chapter 10 Statistics 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 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 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 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 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 8 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 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Population vs. Sample Population described by its parameters (, ) Sample described by its statistics ( x , s) The statistics are used to estimate the parameters. 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 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 11 Mean If the n observations in a random sample are denoted by x1, x2, . . ., xn, the sample mean is x1 x 2 xn 1 n x xi n n i 1 For a finite population with N equally likely values, the population mean is 1 x i fX (x i ) N i 1 n N x i 1 i Measures of Variation • Represent the amount of disparity (dispersal, scatter) between the data points and the mean • Variance 2 ( x x ) i s2 n 1 • Standard Deviation s s 2 Note: n-1 is the number of degrees of freedom left after calculating n Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 13 Variance If the n observations in a random sample are denoted by x1, x2, . . ., xn, the sample variance is n n 1 1 2 2 2 2 s (x i x ) x i n x n 1 i 1 n 1 i 1 For a finite population with N equally likely values, the population variance is 2 1 N N 2 ( x ) i i 1 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-andleaf 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 Example 10.2 • 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. Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 20 Example – cont’d Actual speeds of cars in Safety Corridor 70 85 71 75 65 66 78 69 82 76 90 78 70 68 69 85 77 91 80 61 71 72 89 69 86 81 62 63 76 80 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 76 80 71 72 70 92 21 Example – cont’d a.) Make a frequency distribution table using 5 as a class width (e.g. 60.0 – 64.9) b.) Construct a histogram Highway Speed 9 Interval Frequency 60-64.9 3 65-69.9 6 70-74.9 8 75-79.9 7 80-84.9 5 85-89.9 4 2 90-94.5 3 1 8 7 Frequency 6 5 4 3 0 60-64.9 65-69.9 70-74.9 75-79.9 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 80-84.9 85-89.9 90-94.5 22 Example – cont’d c) The mean x x i n 2716 75.44 36 d) The standard deviation n( xi ) ( xi ) n(n 1) 2 2 1/ 2 36(207360) (2716) 36 ( 35 ) 2 1/ 2 8.3715 e) The variance 2 70.0825 Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. 23