Distance Learning Center

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Distance Learning Center

• Lecture 11
• Problem Solving with MATLAB
• Melissa Lin, IT Consultant
• HenEm, Inc. Parkville, Missouri
• linm_at_ipfw.edu
• http//www.etcs.ipfw.edu/linm

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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

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

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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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11-1 Introduction to Statistics

• Example 1 Student performance comparison
  (continue)
• Statistical Inference Student B Better Than
  Student A????
• Two Possible Answers
• Student Bs average grade 91.80 higher than As
  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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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

if A_avg gt 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
gtgt Student B is better than student A B_avg
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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11-2 Statistical Analysis

• Arithmetic mean or average value
• Where N measurements are designated x1 , x2 , ..
• Or in the closed form as

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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
• gtgt X 0 1 2 3 4 5
• gtgt 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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Variance

• The variance is a measure of how spread out a
  distribution is.
• 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 (123)/3, or 2 ohms, and the
  variance is

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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.

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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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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.
• Find the mean and standard deviation using MATLAB
  mean( ) and std ( ) functions.
• Find the mean and deviation using the formula as
  shown below

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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)
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((Nsum(mpg.2) -
(sum_2)2)/(N(N-1))) 1.18 format short
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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
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Empirical Equation Race Car Speed Prediction

• 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)

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Empirical Equation Race Car Speed Prediction

• Example 5 MATLAB Program
• The linear equation can be described as the
  slope-intercept form V mt b
• where m is the slope and b is the intercept
• Select point A(10,60), point B(40, 185)

185
A
60
10
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 mt b
• to find m and b
• 60 m10 b ----- (1)
• 185 m40 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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Empirical Equation Race Car Speed Prediction

• Example 5 MATLAB Program (continue)

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 0540 V 4.2t1
18.3 hold on plot(t1, V, 'r')
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Summary

• Introduction to Statistics
• Statistical Analysis
• Arithmetic Mean
• Variance
• Standard Deviation
• Empirical Linear Equation

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Question?

• Answers
• Email linm_at_ipfw.edu