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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engr/Math/Physics 25 Chp7 Statistics-1
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 2 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Learning Goals Use MATLAB to solve Problems in Statistics Probability Use Monte Carlo (random) Methods to Simulate Random processes Properly Apply Interpolation or Extrapolation to Estimate values between or outside of know data points
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Histogram Histograms are COLUMN Plots that show the Distribution of Data Height Represents Data Frequency Some General Characteristics Used to represent continuous grouped, or BINNED, data –BIN SubRange within the Data Usually Does not have any gaps between bars Areas represent %-of-Total Data
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 4 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods HistoGram ≡ Frequency Chart A HistoGram shows how OFTEN some event Occurs Histograms are often constructed using Frequency Tables
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 5 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Histograms In MATLAB MATLAB has 6 Forms of the Histogram Cmd The Simplest Hist(y) The Plot Statement Generates a Histogram with 10 bins Example: Max Temp at Oakland AirPort in Jul-Aug08 TmaxOAK = [70, 75, 63, 64, 65, 66, 65, 65, 67, 78, 75, 73, 79, 71, 72, 67, 69, 69, 70, 74, 71, 72, 71, 74, 77, 77, 86, 90, 90, 70, 71, 66, 66, 72, 68, 73, 72, 82, 91, 82, 76, 75, 72, 72, 69, 70, 68, 65, 67, 65, 63, 64, 72, 70, 68, 71, 77, 65, 63, 69, 69, 67] hist(TmaxOAK), ylabel('No. Days'), xlabel('Max. Temp (°F)'), title('Oakland Airport - Jul-Aug08')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Result for Oakland It was COLD in Summer 08 Bin Width = (91-63)/10 = 2.8 °F
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Histograms In MATLAB Next Example: Max Temp at Stockton AirPort in Jul-Aug08 Hist(y) The Plot Statement Generates a Histogram with 10 bins TmaxSTK = [94, 98, 93, 94, 91, 96, 93, 87, 89, 94, 100, 99, 103, 103, 103, 97, 91, 83, 84, 90, 89, 95, 94, 99, 97, 94, 102, 103, 107, 98, 86, 89, 95, 91, 84, 93, 98, 104, 105, 107, 103, 91, 90, 96, 93, 86, 92, 93, 95, 95, 86, 81, 93, 97, 96, 97, 101, 92, 89, 92, 93, 94] hist(TmaxSTK), ylabel('No. Days'), xlabel('Max. Temp (°F)'), title(‘Stockton Airport - Jul-Aug08')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 8 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Result for Stockton It was HOT in Summer 08 Bin Width = (107-81)/10 = 2.6 °F
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 9 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Command Refinements Adjust The number and width of the bins using hist(y,N) hist(y,x) Where –N an integer specifying the NUMBER of Bins –x A vector that Specs CENTERs of the Bins Consider Summer 08 Max-Temp Data from Oakland and Stockton Make 2 Histograms 17 bins 60F→110F by 2.5’s
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 10 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Plots 17 Bins >> hist(TmaxSTK,17), ylabel('No. Days'), xlabel('Max. Temp (°F)'), title('Stockton, CA - Jul- Aug08')>> hist(TmaxOAK,17), ylabel('No. Days'), xlabel('Max. Temp (°F)'), title('Oakland, CA - Jul- Aug08')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Plots Same Scale >> x = [60:2.5:110]; >> hist(TmaxSTK,x), ylabel('No. Days'), xlabel('Max. Temp (°F)'), title('Stockton, CA - Jul- Aug08') >> x = [60:2.5:110]; hist(TmaxOAK,x), ylabel('No. Days'), xlabel('Max. Temp (°F)'), title('Oakland, CA - Jul- Aug08')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 12 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Numerical Output Hist can also provide numerical Data about the Histogram n = hist(y) Gives the number of values in each of the (default) 10 Bins For the Stockton data k = 2 5 1 10 16 7 9 2 7 3 We can also spec the number and/or Width of Bins >> k13 = hist(TmaxSTK,13) k13 = 2 2 4 4 6 10 10 7 5 2 6 2 2 >> k2_5s = hist(TmaxOAK,x)
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 13 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods hist Numerical Output Bin-Count and Bin-Locations (Frequency Table) for the Oakland Data >> [u, v] = hist(TmaxOAK,x) u = 0 3 11 7 1 5 9 6 4 1 2 1 0 30 0 0 0 0 0 0 0 v = 60.0000 62.5000 65.0000 67.5000 7 0.0000 72.5000 75.0000 77.5000 80.0000 82.5000 85.0000 87.5000 90.0000 92.5000 95.0000 97.5000 100.0000 102.5000 105.0000 107.5000 110.0000
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 14 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Histogram Commands - 1 CommandDescription bar(x,y) Creates a bar chart of y versus x. hist(y) Aggregates the data in the vector y into 10 bins evenly spaced between the minimum and maximum values in y. hist(y,n) Aggregates the data in the vector y into n bins evenly spaced between the minimum and maximum values in y. hist(y,x) Aggregates the data in the vector y into bins whose center locations are specified by the vector x. The bin widths are the distances between the centers.
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 15 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Histogram Commands - 2 CommandDescription [z,x] = hist(y) Same as hist(y) but returns two vectors z and x that contain the frequency count and the 10 bin locations. [z,x] = hist(y,n) Same as hist(y,n) but returns two vectors z and x that contain the frequency cnt and the n bin locations. [z,x] = hist(y,x) Same as hist(y,x) but returns two vectors z and x that contain the frequency count and the bin locations. The returned vector x is the same as the user-supplied vector x.
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 16 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Data Statistics Tool - 1 Make Line- Plot of Temp Data for Stockton, CA Use the Tools Menu to find the Data Statistics Tool
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Data Statistics Tool - 2 Use the Tool to Add Plot Lines for The Mean ±StdDev
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 18 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Data Statistics Tool - 3 Quite a Nice Tool, Actually The Result The Avg Max Temp Was 96.97 °F
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Probability Probability The LIKELYHOOD that a Specified OutCome Will be Realized The “Odds” Run from 0% to 100% What are the Odds of winning the California MEGA-MILLIONS Lottery? Class Question: What are the Odds of winning the California MEGA-MILLIONS Lottery?
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 20 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 175 711 536... EXACTLY???!!! To Win the MegaMillions Lottery Pick five numbers from 1 to 56 Pick a MEGA number from 1 to 46 The Odds for the 1 st ping-pong Ball = 5 out of 56 The Odds for the 2 nd ping-pong Ball = 4 out of 55, and so On The Odds for the MEGA are 1 out of 46
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 175 711 536... Calculated Calc the OverAll Odds as the PRODUCT of each of the Individual OutComes This is Technically a COMBINATION
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 175 711 536... is a DEAL! The ORDER in Which the Ping-Pong Balls are Drawn Does NOT affect the Winning Odds If we Had to Match the Pull-Order: This is a PERMUTATION
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 23 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 1 Consider Data on the Height of a sample group of 20 year old Men We can Plot this Frequency Data using bar >> y_abs=[1,0,0,0,2,4,5, 4,8,11,12,10,9,8,7,5, 4,4,3,1,1,0,1]; >> xbins = [64:0.5:75]; >> bar(xbins, y_abs), ylabel('No.'), xlabel('Height (Inches'), title('Height of 20 Yr-Old Men')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 24 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 2 We can also SCALE the Bar/Hist such that the AREA UNDER the CURVE equals 1.00, exactly The Game Plan for Scaling Calc the Height of Each Bar To Get the Total Area = [Bin Width] x [Σ(individual counts)] The individual Bar Area = [Bin Width] x [individual count] %-Area any one bar → [Bar Areas]/[Total Area]
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 25 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 3 We can Use bar to Plot the Scaled-Area Hist. >>y_abs=[1,0,0,0,2,4,5,4,8,11,12,10,9,8,7,5,4,4,3,1,1,0,1] ; >> xbins = [64:0.5:75]; >> TotalArea = sum(0.5*y_abs) >> y_scale = 100*y_abs/TotalArea; >> bar(xbins, y_scale), ylabel('Fraction (%/inch)'), xlabel('Height (inches)'), title('Height of 20 Yr-Old Men')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 26 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 4 This is a Good Time for a UNITS Check Remember, our GOAL → the Area Under the Curve = 1 Recall From the Plot the UNITS for the y-axis → %/inch (?) The Units come from these MATLAB Statements So TotalArea is in inchesNo. Now y_scale TotalArea = sum(0.5*y_abs) Bin Width in INCHES y_scale = 100*y_abs/TotalArea; Cont. on Next Slide
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 27 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 5 The Units Analysis for y-scale Recall From MTH1 that for y = f(x) displayed in BAR Form the Area Under the Curve y_scale = 100*y_abs/TotalArea;
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 28 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 6 In this Case y(x) → y_scale in %/inch Δx → Bin Width = 0.5 in inches Then The Units Analysis for Our “integration” Check the integration Example
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 29 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution - 7 Example 71” The 71” Bar Area = HgtWidth: Alternatively from the Absolute values The Total Abs Area = 50 No.inch
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 30 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Probability Distribution Fcn (PDF) Because the Area Under the Scaled Plot is 1.00, exactly, The FRACTIONAL Area under any bar, or set-of-bars gives the probability that any randomly Selected 20 yr-old man will be that height e.g., from the Plot we Find 67.5 in → 8 %/in 68 in → 16 %/in 68.5 in → 22%/in Summing → 46 %/in Multiply the Uniform BinWidth of 0.5 in → 23% of 20 yr-old men are 67.25- 68.75 inches tall
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 31 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Random Variable A random variable x takes on a defined set of values with different probabilities; e.g.. If you roll a die, the outcome is random (not fixed) and there are 6 possible outcomes, each of which occur with equal probability of one-sixth. If you poll people about their voting preferences, the percentage of the sample that responds “Yes on Proposition 101” is a also a random variable –the %-age will be slightly differently every time you poll. Roughly, probability is how frequently we expect different outcomes to occur if we repeat the experiment over and over (“frequentist” view)
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 32 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Random variables can be Discrete or Continuous Discrete random variables have a countable number of outcomes Examples: Dead/Alive, Red/Black, Heads/Tales, dice, counts, etc. Continuous random variables have an infinite continuum of possible values. Examples: blood pressure, weight, Air Temperature, the speed of a car, the real numbers from 1 to 6.
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 33 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Probability Distribution Functions A Probability Distribution Function (PDF) maps the possible values of x against their respective probabilities of occurrence, p(x) p(x) is a number from 0 to 1.0, or alternatively, from 0% to 100%. The area under a probability distribution function curve is always 1 (or 100%).
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 34 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Discrete Example: Roll The Die 1/6 145623 xp(x) 1p(x=1)=1/6 2p(x=2)=1/6 3p(x=3)=1/6 4p(x=4)=1/6 5p(x=5)=1/6 6p(x=6)=1/6
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 35 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Continuous Case The probability function that accompanies a continuous random variable is a continuous mathematical function that integrates to 1. The Probabilities associated with continuous functions are just areas under a Region of the curve (→ Definite Integrals) Probabilities are given for a range of values, rather than a particular value e.g., the probability of getting a math SAT score between 700 and 800 is 2%).
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 36 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Continuous Case PDF Example Recall the negative exponential function (in probability, this is called an “exponential distribution”): This Function Integrates to 1 zero to infinity as required for all PDF’s
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 37 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Continuous Case PDF Example x p(x)=e -x 1 For example, the probability of x falling within 1 to 2: The probability that x is any exact value (e.g.: 1.9976) is 0 we can ONLY assign Probabilities to possible RANGES of x x 1 12 p(x)=e -x NO Area Under a LINE
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 38 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Gaussian Curve The Man-Height HistroGram had some Limited, and thus DISCRETE, Data If we were to Measure 10,000 (or more) young men we would obtain a HistoGram like this As We increase the number and fineness of the measurements The PDF approaches a CONTINUOUS Curve
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 39 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Gaussian Distribution A Distribution that Describes Many Physical Processes is called the GAUSSIAN or NORMAL Distribution Gaussian (Normal) distribution Gaussian → famous “bell-shaped curve” –Describes IQ scores, how fast horses can run, the no. of Bees in a hive, wear profile on old stone stairs... All these are cases where: –deviation from mean is equally probable in either direction –Variable is continuous (or large enough integer to look continuous)
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 40 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution Real-valued PDF: f(x) → −∞ < x < +∞ 2 independent fitting parameters: µ, σ (central location and width) Properties: Symmetrical about Mode at µ, Median = Mean = Mode, Inflection points at ±σ Area (probability of observing event) within: ± 1σ = 0.683 ± 2σ = 0.955 For larger σ, bell shaped curve becomes wider and lower (since area =1 for any σ)
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 41 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Distribution Mathematically Where –σ 2 = Variance –µ = Mean The Area Under the Curve
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 42 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 68-95-99.7 Rule for Normal Dist 68% of the data 95% of the data 99.7% of the data σσ 2σ2σ2σ2σ 3σ3σ3σ3σ
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 43 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 68-95-99.7 Rule in Math terms… Using Definite-Integral Calculus
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 44 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods How Good is the Rule for Real? Check some example data: The mean, µ, of the weight of a large group of women Cross Country Runners = 127.8 lbs The standard deviation (σ) for this Group = 15.5 lbs
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 45 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 127.8143.3112.3 68% of 120 =.68x120 = ~ 82 runners In fact, 79 runners fall within 1σ (15.5 lbs) of the mean
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 46 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 127.896.8 95% of 120 =.95 x 120 = ~ 114 runners In fact, 115 runners fall within 2σ of the mean 158.8
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 47 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 127.881.3 99.7% of 120 =.997 x 120 = 119.6 runners In fact, all 120 runners fall within 3σ of the mean 174.3
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 48 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Estimating µ & σ (1) The Location & Width Parameters, µ & σ, are Calculated from the ENTIRE POPULATION Mean, µ Variance, σ 2 Standard Deviation, σ For LARGE Populations it is usually impractical to measure all the x k In this case we take a Finite SAMPLE to ESTIMATE µ & σ
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 49 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Estimating µ & σ (2) Say we want to characterize Miles/Yr driven by Every Licensed Driver in the USA We assume that this is Normally Distributed, so we take a Sample of N = 1013 Drivers We Take the Mean of the SAMPLE Use the SAMPLE- Mean to Estimate the POPULATION-Mean
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 50 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Estimating µ & σ (3) Now Calc the SAMPLE Variance & StdDev Estimate Number decreased from N to (N – 1) To Account for case where N = 1 –In this case x-bar = x 1, and the S 2 result is meaningless standard deviation: positive square root of the variance –small std dev: observations are clustered tightly around a central value –large std dev: observations are scattered widely about the mean
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 51 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Sample Mean and StdDev Sample Mean Calculate the Population Variance, σ 2, from: Sample Variance But we cannot know the true population mean µ so the practical estimate for the sample variance and standard deviation would be:
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 52 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Error Function (erf) & Probability Guass’s Defining Eqn This looks a lot Like the normal dist Now Let Consider the Gaussian integral Or
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 53 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Error Function (erf) & Probability Subbing for x & dx As ReArranging
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 54 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Error Function (erf) & Probability Now the Limits Plotting This Fcn is Symmetrical about y = 0 Recall And the erf properties erf(0) = 0 erf(h) = 1
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 55 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Error Function (erf) & Probability By Symmetry about y = 0 for Thus So Finally integrating − h to B
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 56 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Error Function (erf) & Probability Note That for a Continuous PDF Probability that x is Less or Equal to b Probability that x is between a & b The probability for the Normal Dist But
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 57 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Error Function (erf) & Probability If We Scale this Properly we can Cast these Eqns into the ½erf Form MATLAB has the erf built-in, so if we have the sample Mean & StdDev We can Calc Probabilities for Normally Distributed Quantities
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 58 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods All Done for Today Gaussian? Or Normal? Normal distribution was introduced by French mathematician A. De Moivre in 1733. Used to approximate probabilities of coin tossing Called it the exponential bell-shaped curve 1809, K.F. Gauss, a German mathematician, applied it to predict astronomical entities… it became known as the Gaussian distribution. Late 1800s, most believe majority of physical data would follow the distribution called normal distribution Recall De Moivre’s Theorem
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 59 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engr/Math/Physics 25 Appendix
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 60 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Basic Fitting Demo File % Bruce Mayer, PE % ENGR25 * 11Apr10 % file = Demo_Basic_Fitting_Stockton_Temps_1004.m % TmaxSTK = [94, 98, 93, 94, 91, 96, 93, 87, 89, 94, 100, 99, 103, 103, 103, 97, 91, 83, 84, 90, 89, 95, 94, 99, 97, 94, 102, 103, 107, 98, 86, 89, 95, 91, 84, 93, 98, 104, 105, 107, 103, 91, 90, 96, 93, 86, 92, 93, 95, 95, 86, 81, 93, 97, 96, 97, 101, 92, 89, 92, 93, 94] Ntot = length(TmaxSTK) nday = [1:Ntot]; plot(nday, TmaxSTK, '-dk'), xlabel('No. Days after 31Jun08'), ylabel('Max. Temp (°F)'), title('Stockton, CA - Jul-Aug08')
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 61 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal or Gaussian? Normal distribution was introduced by French mathematician A. De Moivre in 1733. Used to approximate probabilities of coin tossing Called it exponential bell-shaped curve 1809, K.F. Gauss, a German mathematician, applied it to predict astronomical entities… it became known as Gaussian distribution. Late 1800s, most believe majority data would follow the distribution called normal distribution
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 62 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Carl Friedrich Gauss
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 63 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Normal Dist Data Ht (in)No.Area (BW*No.)No./TotAreaBW*(No./TotArea) 64 10.5 0.02001.00% 64.5 000.00000.00% 65 000.00000.00% 65.5 000.00000.00% 66 210.04002.00% 66.5 420.08004.00% 67 52.50.10005.00% 67.5 420.08004.00% 68 840.16008.00% 68.5 115.50.220011.00% 69 1260.240012.00% 69.5 1050.200010.00% 70 94.50.18009.00% 70.5 840.16008.00% 71 73.50.14007.00% 71.5 52.50.10005.00% 72 420.08004.00% 72.5 420.08004.00% 73 31.50.06003.00% 73.5 10.50.02001.00% 74 10.50.02001.00% 74.5 000.00000.00% 75 10.50.02001.00% 50.0 100.00%
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BMayer@ChabotCollege.edu ENGR-25_Lec-19_Statistics-1.ppt 64 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods SPICE Circuit
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