February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 11

CPET 190 CPET 190

Problem Solving with MATLABProblem Solving with MATLAB

Lecture 11Lecture 11

http://www.ecet.ipfw.edu/~linhttp://www.ecet.ipfw.edu/~lin

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 22

Lecture 11: Solving Basic Lecture 11: Solving Basic Statistics ProblemsStatistics Problems

11-1 Introduction to Statistics 11-1 Introduction to Statistics

11-2 Statistical Analysis11-2 Statistical Analysis• Arithmetic MeanArithmetic Mean• VarianceVariance• Standard DeviationStandard Deviation

11-3 Empirical Linear Equation11-3 Empirical Linear Equation

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 33

11-1 Introduction to Statistics11-1 Introduction to Statistics

Origin of Statistics (18Origin of Statistics (18thth century) century)• Game of chance, and what is now Game of chance, and what is now

called political sciencescalled political sciences• Descriptive statistics: numerical Descriptive statistics: numerical

description of political units (cities, description of political units (cities, provinces, countries, etc); presentation provinces, countries, etc); presentation of data in tables and charts; of data in tables and charts; summarization of data by means of summarization of data by means of numerical descriptionnumerical description

Reference: Chapter 14 Statistics, Engineering Fundamentals and Problem Solving, Arvid Edie, et. al. McGrawHill, 1979

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 44

11-1 Introduction to Statistics11-1 Introduction to Statistics

Statistics InferenceStatistics Inference• Make generalization about collected data using Make generalization about collected data using

carefully controlled variables carefully controlled variables Applications of StatisticsApplications of Statistics

• Decision makingDecision making• Gaming industriesGaming industries• Comparison of the efficiency of production Comparison of the efficiency of production

processesprocesses• Quality ControlQuality Control

Measure of central tendency (mean or average)Measure of central tendency (mean or average) Measure of variation (standard deviation)Measure of variation (standard deviation) Normal curveNormal curve Linear regressionLinear regression

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 55

11-1 Introduction to Statistics11-1 Introduction to Statistics Basic Statistical AnalysisBasic Statistical Analysis

• A large set of measured data or numbersA large set of measured data or numbers• Average value (or arithmetic mean)Average value (or arithmetic mean)• Standard Deviation Standard Deviation • Study and summarize the results of the measured data, and Study and summarize the results of the measured data, and

moremore

Example 1: Student performance comparisonExample 1: Student performance comparison Two ECET students are enrolled in the CPET 190 and each Two ECET students are enrolled in the CPET 190 and each

completed five quizzes: qz1, qz2, qz3, qz4, and qz5. The grades are completed five quizzes: qz1, qz2, qz3, qz4, and qz5. The grades are in the array format:in the array format:• A = [82 61 88 78 80];A = [82 61 88 78 80];• B =[94 98 92 90 85];B =[94 98 92 90 85];

Student A has an average score of (82 + 61 + 88 + 78 + 90)/5 = 71.80Student A has an average score of (82 + 61 + 88 + 78 + 90)/5 = 71.80 Student B has an average score of (94 + 98 +92 + 90 + 85)/5 = 91.80Student B has an average score of (94 + 98 +92 + 90 + 85)/5 = 91.80 Statistical Inference: Student B Better Than Student A????Statistical Inference: Student B Better Than Student A????

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 66

11-1 Introduction to Statistics11-1 Introduction to Statistics

Example 1: Student performance comparison Example 1: Student performance comparison (continue)(continue)

Statistical Inference: Student B Better Than Statistical Inference: Student B Better Than Student A????Student A????

Two Possible Answers:Two Possible Answers:• Student B’s average grade 91.80 higher than Student B’s average grade 91.80 higher than

A’s average grade 71.80, so that student B is a A’s average grade 71.80, so that student B is a betterbetter student? Not quite true. student? Not quite true.

• Student B Student B may be bettermay be better than A. This could be a than A. This could be a more accurate answer.more accurate answer.

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 77

Example 1: The MATLAB SolutionExample 1: The MATLAB Solution% ex10_1.m% ex10_1.m% By M. Lin% By M. Lin% Student Performance % Student Performance

ComparisonComparisonformat bank % 2 digitsformat bank % 2 digitsA = [82 61 88 78 80];A = [82 61 88 78 80];B =[94 98 92 90 85];B =[94 98 92 90 85];A_total = 0;A_total = 0;B_total = 0;B_total = 0;for n = 1: length(A)for n = 1: length(A) A_total = A_total + A(n);A_total = A_total + A(n);endendA_avgA_avg = A_total/length(A) = A_total/length(A)%71.80%71.80for n = 1: length(B)for n = 1: length(B) B_total = B_total + B(n);B_total = B_total + B(n);endendB_avg B_avg = B_total/length(B)= B_total/length(B)% 91.80% 91.80

if A_avg > B_avg;if A_avg > B_avg; disp('Student A is better than disp('Student A is better than student B')student B') A_avgA_avgelseelse disp('Student B is better than disp('Student B is better than student A')student A') B_avgB_avgendendformat short % 4 digitsformat short % 4 digits

>> Student B is better than student A

B_avg = 91.80

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 88

11-2 Statistical Analysis11-2 Statistical Analysis

Statistical AnalysisStatistical Analysis• Data grouping and classifying dataData grouping and classifying data• Measures of tendency Measures of tendency

Arithmetic mean or average value.Arithmetic mean or average value.• Measures of variationMeasures of variation

VarianceVariance Standard Deviation Standard Deviation

• Predict or forecast the outcome of certain Predict or forecast the outcome of certain eventsevents

Linear regression (the simplest one)Linear regression (the simplest one)

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11-2 Statistical Analysis11-2 Statistical Analysis

Arithmetic mean or average valueArithmetic mean or average value

Where N measurements are designated xWhere N measurements are designated x11 , x, x22 , .., ..

Or in the closed form asOr in the closed form as

N

xxxxMean n

...321

N

iixN

Mean1

1

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MEAN() - MATLAB Function for MEAN() - MATLAB Function for Calculating Average or Mean ValuesCalculating Average or Mean Values

MEAN Average or mean value.MEAN Average or mean value. For vectors, MEAN(X) is the mean value of the elements in X. For vectors, MEAN(X) is the mean value of the elements in X.

For matrices, MEAN(X) is a row vector containing the mean For matrices, MEAN(X) is a row vector containing the mean value of each column. For N-D arrays, MEAN(X) is the mean value of each column. For N-D arrays, MEAN(X) is the mean value of the elements along the first non-singleton dimension value of the elements along the first non-singleton dimension of X.of X.

Example 2:Example 2: If X = [0 1 2 3 4 5], then mean(X) = 2.5000 If X = [0 1 2 3 4 5], then mean(X) = 2.5000

>> X = [0 1 2 3 4 5]; >> X = [0 1 2 3 4 5]; >> mean(X)>> mean(X)ans = 2.5000ans = 2.5000 Verify the answer by hand:Verify the answer by hand:(0 + 1 + 2 + 3 + 4 + 5)/6 = 15/6 = 2.5.(0 + 1 + 2 + 3 + 4 + 5)/6 = 15/6 = 2.5.

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VarianceVariance The variance is a measure of how spread out a distribution is.The variance is a measure of how spread out a distribution is.

Where x is each measurement, Where x is each measurement, μμ is the mean, and N is the is the mean, and N is the number of measurementnumber of measurement

It is computed as the average squared deviation of each It is computed as the average squared deviation of each number from its mean.number from its mean.

Example 3:Example 3: we measure three resistors in a bin and read the we measure three resistors in a bin and read the resistances 1 ohm, 2 ohms, and 3 ohms, the mean is resistances 1 ohm, 2 ohms, and 3 ohms, the mean is (1+2+3)/3, or 2 ohms, and the variance is(1+2+3)/3, or 2 ohms, and the variance is

667.0

3

)23()22(21 2222

N

x

22 )(

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Standard DeviationStandard Deviation

A measure of the dispersion (or spread) of a set of data A measure of the dispersion (or spread) of a set of data from its mean. from its mean.

The more spread apart the data is, the higher the deviation. The more spread apart the data is, the higher the deviation. A statistic about how tightly all the various measurement are A statistic about how tightly all the various measurement are

clustered around the mean in a set of data. clustered around the mean in a set of data. When the examples are pretty tightly bunched together and When the examples are pretty tightly bunched together and

the bell-shaped curve is steep, the standard deviation is the bell-shaped curve is steep, the standard deviation is small. small.

When the examples are spread apart and the bell curve is When the examples are spread apart and the bell curve is relatively flat, that tells you have a relatively large standard relatively flat, that tells you have a relatively large standard deviation.deviation.

)1(

)(1 1

22

NN

xxNstd

N

i

N

iii

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MATLAB Function for Standard MATLAB Function for Standard DeviationDeviation

STD Standard deviation.STD Standard deviation. For vectors, STD(X) returns the standard deviation. For For vectors, STD(X) returns the standard deviation. For

matrices, STD(X) is a row vector containing the matrices, STD(X) is a row vector containing the standard deviation of each column. For N-D arrays, standard deviation of each column. For N-D arrays, STD(X) is the standard deviation of the elements along STD(X) is the standard deviation of the elements along the first non-singleton dimension of X.the first non-singleton dimension of X.

STD(X) normalizes by (N-1) where N is the sequence STD(X) normalizes by (N-1) where N is the sequence length. This makes STD(X).^2 the best unbiased length. This makes STD(X).^2 the best unbiased estimate of the variance if X is a sample from a normal estimate of the variance if X is a sample from a normal distribution.distribution.

Example: If X = [4 -2 1 9 5 7]Example: If X = [4 -2 1 9 5 7] then std(X) = 4 is standard deviation. This is a large then std(X) = 4 is standard deviation. This is a large

number which means that the data are spread out.number which means that the data are spread out.

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 1414

Mean and Standard DeviationMean and Standard Deviation

Example 4:Example 4: Mr. A purchased a new car and want Mr. A purchased a new car and want to find the MEAN and the Standard Deviation to find the MEAN and the Standard Deviation of gas consumption (miles per gallon) of gas consumption (miles per gallon) obtained in 10 test-runs.obtained in 10 test-runs.

1)1) Find the mean and standard deviation using Find the mean and standard deviation using MATLAB mean( ) and std ( ) functions.MATLAB mean( ) and std ( ) functions.

2)2) Find the mean and deviation using the Find the mean and deviation using the formula as shown below:formula as shown below:

N

iixN

Mean1

1

)1(

)(1 1

22

NN

xxNstd

N

i

N

iii

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 1515

Example 4: ContinueExample 4: Continue

Miles per gallon obtained in 10 test-runs:Miles per gallon obtained in 10 test-runs:%Miles Per Gallon%Miles Per Gallon

mpg = [20 22 23 22 23 22 21 20 20 22];mpg = [20 22 23 22 23 22 21 20 20 22];

% ex10_4.m% By M. Lin% Student Performance % Comparisonformat bank% Miles Per Gallonmpg = [20 22 23 22 23 22 21 20 20 22];N = length(mpg);

% calculation method 1avg_1 = mean(mpg) % 21.50std_1 = std(mpg) % 1.18% calculation method 2sum_2 = sum(mpg);avg_2 = sum(mpg)/N % 21.50std_2 = sqrt((N*sum(mpg.^2) - (sum_2)^2)/(N*(N-1)))%1.18format short

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 1616

10-3 Empirical Equation10-3 Empirical Equation – Race – Race Car Speed PredictionCar Speed Prediction

Example 5: A racing car is clocked at various times t Example 5: A racing car is clocked at various times t and velocities Vand velocities V

t = [0 5 10 15 20 25 30 35 40]; % Secondt = [0 5 10 15 20 25 30 35 40]; % Second

velocity = [24 33 62 77 105 123 151 170 188]; % m/secvelocity = [24 33 62 77 105 123 151 170 188]; % m/sec Determine the equation of a straight line Determine the equation of a straight line

constructed through the points plotted using constructed through the points plotted using MATLABMATLAB

Once the equation is determined, velocities at Once the equation is determined, velocities at intermediate values can be computed or intermediate values can be computed or estimated from this equationestimated from this equation

Reference: Engineering Fundamentals and Problem Solving, Arvid Edie, et. al., pp. 67-68, McGrawHill, 1979

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 1717

Empirical EquationEmpirical Equation – Race Car – Race Car Speed PredictionSpeed Prediction

Example 5: MATLAB ProgramExample 5: MATLAB Program% ex10_5.m% ex10_5.m

% By M. Lin% By M. Lin

t = [0 5 10 15 20 25 30 35 40];t = [0 5 10 15 20 25 30 35 40];

velocity = [24 33 62 77 105 velocity = [24 33 62 77 105 123 151 170 188];123 151 170 188];

plot(t, velocity,'o'), grid onplot(t, velocity,'o'), grid on

title(' Velocity vs Time');title(' Velocity vs Time');

xlabel('Time - second');xlabel('Time - second');

ylabel('Velocity - meter/sec')ylabel('Velocity - meter/sec')

hold onhold on

plot(t, velocity)plot(t, velocity)0 5 10 15 20 25 30 35 40

20

40

60

80

100

120

140

160

180

200 Velocity vs Time

Time - second

Velo

city

- m

eter

/sec

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 1818

Empirical EquationEmpirical Equation – Race Car – Race Car Speed PredictionSpeed Prediction

Example 5: MATLAB Example 5: MATLAB ProgramProgram

The linear equation The linear equation can be described as can be described as the slope-intercept the slope-intercept form form V = m*t +b;V = m*t +b;where m is the slope where m is the slope and b is the and b is the intercept intercept

Select point Select point A(10,60), point B(40, A(10,60), point B(40, 185)185)

0 5 10 15 20 25 30 35 4020

40

60

80

100

120

140

160

180

200 Velocity vs Time

Time - second

Velo

city

- m

eter

/sec

10

60 A

40

185

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 1919

Empirical EquationEmpirical Equation – Race Car – Race Car Speed PredictionSpeed Prediction

Example 5: MATLAB Program (continue)Example 5: MATLAB Program (continue) We substitute A(10,60), and B(40, 185) into the We substitute A(10,60), and B(40, 185) into the

equation V = m*t +b;equation V = m*t +b;to find m and b to find m and b 60 = m*10 + b ----- (1)60 = m*10 + b ----- (1)

185 = m*40 + b ---- (2)185 = m*40 + b ---- (2) We then solve the two equations for the two We then solve the two equations for the two

unknowns m and b:unknowns m and b: m = 4.2m = 4.2

b = 18.3b = 18.3 Now we have the equation Now we have the equation

V = 4.2 t + 18.3V = 4.2 t + 18.3

February 28, 2005February 28, 2005 Lecture 11 - By Paul LinLecture 11 - By Paul Lin 2020

Empirical EquationEmpirical Equation – Race Car – Race Car Speed PredictionSpeed Prediction

Example 5: MATLAB Program (continue)Example 5: MATLAB Program (continue)

t = [0 5 10 15 20 25 30 35 40];t = [0 5 10 15 20 25 30 35 40];velocity = [24 33 62 77 105 velocity = [24 33 62 77 105 123 151 170 188];123 151 170 188];plot(t, velocity,'o'), grid onplot(t, velocity,'o'), grid ontitle(' Velocity vs Time');title(' Velocity vs Time');xlabel('Time - second');xlabel('Time - second');ylabel('Velocity - meter/sec')ylabel('Velocity - meter/sec')hold onhold onplot(t, velocity)plot(t, velocity)m = 4.2;m = 4.2;b = 18.3;b = 18.3;t1 = 0:5:40;t1 = 0:5:40;V = 4.2*t1 + 18.3;V = 4.2*t1 + 18.3;hold onhold onplot(t1, V, 'r')plot(t1, V, 'r')

0 5 10 15 20 25 30 35 400

50

100

150

200 Velocity vs Time

Time - second

Vel

ocity

- m

eter

/sec

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SummarySummary

Introduction to Statistics Introduction to Statistics Statistical AnalysisStatistical Analysis

• Arithmetic MeanArithmetic Mean• VarianceVariance• Standard DeviationStandard Deviation

Empirical Linear EquationEmpirical Linear Equation