Transcript Lecture 11 - College of Engineering, Technology, and

CPET 190
Problem Solving with MATLAB
Lecture 11
http://www.ecet.ipfw.edu/~lin
February 28, 2005
Lecture 11 - By Paul Lin
1
Lecture 11: Solving Basic
Statistics Problems
11-1 Introduction to Statistics
11-2 Statistical Analysis
• Arithmetic Mean
• Variance
• Standard Deviation
11-3 Empirical Linear Equation
February 28, 2005
Lecture 11 - By Paul Lin
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11-1 Introduction to Statistics
Origin of Statistics (18th century)
• Game of chance, and what is now
called political sciences
• Descriptive statistics: numerical
description of political units (cities,
provinces, countries, etc); presentation
of data in tables and charts;
summarization of data by means of
numerical description
Reference: Chapter 14 Statistics, Engineering Fundamentals and
Problem Solving, Arvid Edie, et. al. McGrawHill, 1979
February 28, 2005
Lecture 11 - By Paul Lin
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11-1 Introduction to Statistics
Statistics Inference
• Make generalization about collected data using
carefully controlled variables
Applications of Statistics
• Decision making
• Gaming industries
• Comparison of the efficiency of production
processes
• Quality Control




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Measure of central tendency (mean or average)
Measure of variation (standard deviation)
Normal curve
Linear regression
Lecture 11 - By Paul Lin
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11-1 Introduction to Statistics

Basic Statistical Analysis
• A large set of measured data or numbers
• Average value (or arithmetic mean)
• Standard Deviation
• Study and summarize the results of the measured data, and
more
Example 1: Student performance comparison

Two ECET students are enrolled in the CPET 190 and each
completed five quizzes: qz1, qz2, qz3, qz4, and qz5. The grades are
in the array format:
• A = [82 61 88 78 80];
• B =[94 98 92 90 85];
 Student 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 = 91.80

Statistical Inference: Student B Better Than Student A????
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Lecture 11 - By Paul Lin
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11-1 Introduction to Statistics
Example 1: Student performance comparison
(continue)
 Statistical Inference: Student B Better Than
Student A????
 Two Possible Answers:
• Student B’s average grade 91.80 higher than
A’s average grade 71.80, so that student B is a
better student? Not quite true.
• Student B may be better than A. This could be a
more accurate answer.
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Lecture 11 - By Paul Lin
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Example 1: The MATLAB Solution
% ex10_1.m
% By M. Lin
% Student Performance
Comparison
format bank % 2 digits
A = [82 61 88 78 80];
B =[94 98 92 90 85];
A_total = 0;
B_total = 0;
for n = 1: length(A)
A_total = A_total + A(n);
end
A_avg = A_total/length(A)
%71.80
for n = 1: length(B)
B_total = B_total + B(n);
end
B_avg = B_total/length(B)
% 91.80
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if A_avg > B_avg;
disp('Student A is better than
student B')
A_avg
else
disp('Student B is better than
student A')
B_avg
end
format short % 4 digits
>> Student B is better
than student A
B_avg =
Lecture 11 - By Paul Lin
91.80
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11-2 Statistical Analysis

Statistical Analysis
• Data grouping and classifying data
• Measures of tendency
 Arithmetic mean or average value.
• Measures of variation
 Variance
 Standard Deviation
• Predict or forecast the outcome of certain
events
 Linear regression (the simplest one)
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Lecture 11 - By Paul Lin
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11-2 Statistical Analysis

Arithmetic mean or average value
x1  x2  x3  ...  xn
Mean 
N
Where N measurements are designated x1 , x2 , ..
Or in the closed form as
1
Mean   
N
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N
x
i 1
Lecture 11 - By Paul Lin
i
9
MEAN() - MATLAB Function for
Calculating Average or Mean Values
MEAN Average or mean value.
For vectors, MEAN(X) is the mean value of the elements in X.
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 the elements along the first non-singleton dimension
of X.
Example 2: If X = [0 1 2 3 4 5], then mean(X) = 2.5000
>> X = [0 1 2 3 4 5];
>> mean(X)
ans = 2.5000
Verify the answer by hand:
(0 + 1 + 2 + 3 + 4 + 5)/6 = 15/6 = 2.5.
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Lecture 11 - By Paul Lin
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Variance

The variance is a measure of how spread out a distribution is.
2 
2
(
x


)

N
Where x is each measurement, μ is the mean, and N is the
number of measurement

It is computed as the average squared deviation of each
number from its mean.
Example 3: we measure three resistors in a bin and read the
resistances 1 ohm, 2 ohms, and 3 ohms, the mean is
(1+2+3)/3, or 2 ohms, and the variance is
2
2
2


1

2

(
2

2
)

(
3

2
)
2 
 0.667
3
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Lecture 11 - By Paul Lin
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Standard Deviation





A measure of the dispersion (or spread) of a set of data
from its mean.
The more spread apart the data is, the higher the deviation.
A statistic about how tightly all the various measurement are
clustered around the mean in a set of data.
When the examples are pretty tightly bunched together and
the bell-shaped curve is steep, the standard deviation is
small.
When the examples are spread apart and the bell curve is
relatively flat, that tells you have a relatively large standard
deviation.
N
std 
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N
N  x  ( xi ) 2
i 1
2
i
i 1
N ( N  1)
Lecture 11 - By Paul Lin
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MATLAB Function for Standard
Deviation
STD Standard deviation.
For vectors, STD(X) returns the standard deviation. For
matrices, STD(X) is a row vector containing the
standard deviation of each column. For N-D arrays,
STD(X) is the standard deviation of the elements along
the first non-singleton dimension of X.
STD(X) normalizes by (N-1) where N is the sequence
length. This makes STD(X).^2 the best unbiased
estimate of the variance if X is a sample from a normal
distribution.
Example: If X = [4 -2 1 9 5 7]
then std(X) = 4 is standard deviation. This is a large
number which means that the data are spread out.
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Lecture 11 - By Paul Lin
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Mean and Standard Deviation
Example 4: Mr. A purchased a new car and want
to find the MEAN and the Standard Deviation
of gas consumption (miles per gallon)
obtained in 10 test-runs.
1) Find the mean and standard deviation using
MATLAB mean( ) and std ( ) functions.
2) Find the mean and deviation using the
formula as shown below:
1
Mean   
N
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N
 xi
i 1
std 
Lecture 11 - By Paul Lin
N
N
i 1
i 1
N  xi2  ( xi ) 2
N ( N  1)
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Example 4: Continue
Miles per gallon obtained in 10 test-runs:
%Miles Per Gallon
mpg = [20 22 23 22 23 22 21 20 20 22];
% ex10_4.m
% By M. Lin
% Student Performance
% Comparison
format bank
% Miles Per Gallon
mpg = [20 22 23 22 23
22 21 20 20 22];
N = length(mpg);
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% calculation method 1
avg_1 = mean(mpg) % 21.50
std_1 = std(mpg) % 1.18
% calculation method 2
sum_2 = sum(mpg);
avg_2 = sum(mpg)/N % 21.50
std_2 = sqrt((N*sum(mpg.^2) (sum_2)^2)/(N*(N-1)))
%1.18
format short
Lecture 11 - By Paul Lin
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10-3 Empirical Equation – Race
Car Speed Prediction
Example 5: A racing car is clocked at various times t
and velocities V
t = [0 5 10 15 20 25 30 35 40]; % Second
velocity = [24 33 62 77 105 123 151 170 188]; % m/sec


Determine the equation of a straight line
constructed through the points plotted using
MATLAB
Once the equation is determined, velocities at
intermediate values can be computed or
estimated from this equation
Reference: Engineering Fundamentals and Problem
Solving, Arvid Edie, et. al., pp. 67-68, McGrawHill, 1979
February 28, 2005
Lecture 11 - By Paul Lin
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Empirical Equation – Race Car
Speed Prediction
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Velocity vs Time
200
180
160
Velocity - meter/sec
Example 5: MATLAB Program
% ex10_5.m
% By M. Lin
t = [0 5 10 15 20 25 30 35 40];
velocity = [24 33 62 77 105
123 151 170 188];
plot(t, velocity,'o'), grid on
title(' Velocity vs Time');
xlabel('Time - second');
ylabel('Velocity - meter/sec')
hold on
plot(t, velocity)
140
120
100
80
60
40
20
0
Lecture 11 - By Paul Lin
5
10
15
20
25
Time - second
30
35
40
17
Empirical Equation – Race Car
Speed Prediction
Velocity vs Time
200
180
160
Velocity - meter/sec
Example 5: MATLAB
185
Program

The linear equation
can be described as
the slope-intercept
form V = m*t +b;
where m is the slope
and b is the
intercept
60

Select point
A(10,60), point B(40,
185)
140
120
100
80
A
60
40
20
0
5
10
10
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Lecture 11 - By Paul Lin
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20
25
Time - second
30
35
40
40
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Empirical Equation – Race Car
Speed Prediction
Example 5: MATLAB Program (continue)

We substitute A(10,60), and B(40, 185) into the
equation V = m*t +b;
to find m and b
60 = m*10 + b ----- (1)
185 = m*40 + b ---- (2)
We then solve the two equations for the two
unknowns m and b:
m = 4.2
b = 18.3

Now we have the equation
V = 4.2 t + 18.3
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Lecture 11 - By Paul Lin
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Empirical Equation – Race Car
Speed Prediction
Example 5: MATLAB Program (continue)
February 28, 2005
Velocity vs Time
200
150
Velocity - meter/sec
t = [0 5 10 15 20 25 30 35 40];
velocity = [24 33 62 77 105
123 151 170 188];
plot(t, velocity,'o'), grid on
title(' Velocity vs Time');
xlabel('Time - second');
ylabel('Velocity - meter/sec')
hold on
plot(t, velocity)
m = 4.2;
b = 18.3;
t1 = 0:5:40;
V = 4.2*t1 + 18.3;
hold on
plot(t1, V, 'r')
100
50
0
0
5
Lecture 11 - By Paul Lin
10
15
20
25
Time - second
30
35
40
20
Summary



February 28, 2005
Introduction to Statistics
Statistical Analysis
• Arithmetic Mean
• Variance
• Standard Deviation
Empirical Linear Equation
Lecture 11 - By Paul Lin
21