Probability and Statistics in Engineering

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Probability and Statistics in Engineering

• Philip Bedient, Ph.D.

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Probability Basic Ideas

• Terminology
• Trial each time you repeat an experiment
• Outcome result of an experiment
• Random experiment one with random outcomes
  (cannot be predicted exactly)
• Relative frequency how many times a specific
  outcome occurs within the entire experiment.

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Statistics Basic Ideas

• Statistics is the area of science that deals with
  collection, organization, analysis, and
  interpretation of data.
• It also deals with methods and techniques that
  can be used to draw conclusions about the
  characteristics of a large number of data
  points--commonly called a population--
• By using a smaller subset of the entire data.

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For Example

• You work in a cell phone factory and are asked to
  remove cell phones at random off of the assembly
  line and turn it on and off.
• Each time you remove a cell phone and turn it on
  and off, you are conducting a random experiment.
• Each time you pick up a phone is a trial and the
  result is called an outcome.
• If you check 200 phones, and you find 5 bad
  phones, then
• relative frequency of failure 5/200 0.025

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Statistics in Engineering

• Engineers apply physical and chemical laws and
  mathematics to design, develop, test, and
  supervise various products and services.
• Engineers perform tests to learn how things
  behave under stress, and at what point they might
  fail.

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Statistics in Engineering

• As engineers perform experiments, they collect
  data that can be used to explain relationships
  better and to reveal information about the
  quality of products and services they provide.

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Frequency Distribution

• Scores for an engineering class are as follows
  58, 95, 80, 75, 68, 97, 60, 85, 75, 88, 90, 78,
  62, 83, 73, 70, 70, 85, 65, 75, 53, 62, 56, 72,
  79
• To better assess the success of the class, we
  make a frequency chart

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• Now the information can be better analyzed.
• For example, 3 students did poorly, and 3 did
  exceptionally well. We know that 9 students were
  in the average range of 70-79. We can also show
  this data in a freq. histogram (PDF).

Divide each no. by 26
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Cumulative Frequency

• The data can be further organized by calculating
  the cumulative frequency (CDF).
• The cumulative frequency shows the cumulative
  number of students with scores up to and
  including those in the given range. Usually we
  normalize the data - divide 26.

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Measures of Central Tendency Variation

• Systematic errors, also called fixed errors, are
  errors associated with using an inaccurate
  instrument.
• These errors can be detected and avoided by
  properly calibrating instruments
• Random errors are generated by a number of
  unpredictable variations in a given measurement
  situation.
• Mechanical vibrations of instruments or
  variations in line voltage friction or humidity
  could lead to random fluctuations in observations.

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• When analyzing data, the mean alone cannot signal
  possible mistakes. There are a number of ways to
  define the dispersion or spread of data.
• You can compute how much each number deviates
  from the mean, add up all the deviations, and
  then take their average as shown in the table
  below.

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• As exemplified in Table 19.4, the sum of
  deviations from the mean for any given sample is
  always zero. This can be verified by considering
  the following
• Where xi represents data points, x is the
  average, n is the number of data points, and d,
  represents the deviation from the average.

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• Therefore the average of the deviations from the
  mean of the data set cannot be used to measure
  the spread of a given data set.
• Instead we calculate the average of the absolute
  values of deviations. (This is shown in the third
  column of table 19.4 in your textbook)
• For group A the mean deviation is 290, and Group
  B is 820. We can conclude that Group B is more
  scattered than A.

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Variance

• Another way of measuring the data is by
  calculating the variance.
• Instead of taking the absolute values of each
  deviation, you can just square the deviation and
  find the means.
• (n-1) makes estimate unbiased

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• Taking the square root of the variance which
  results in the standard deviation.
• The standard deviation can also provide
  information about the relative spread of a data
  set.

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• The mean for a grouped distribution is calculated
  from
• Where
• x midpoints of a given range
• f frequency of occurrence of data in the
  range
• n ?f total number of data points

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• The standard deviation for a grouped distribution
  is calculated from

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Normal Distribution

• We could use the probability distribution from
  the figures below to predict what might happen in
  the future. (i.e. next years students
  performance)

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Normal Distribution

• Any probability distribution with a bell-shaped
  curve is called a normal distribution.
• The detailed shape of a normal distribution curve
  is determined by its mean and standard deviation
  values.

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• THE NORMAL CURVE
• Using Table 19.11, approx. 68 of the data will
  fall in the interval of -s to s, one std
  deviation
• 95 of the data falls between -2s to 2s, and
  approx all of the data points lie between -3s to
  3s
• For a standard normal distribution, 68 of the
  data fall in the interval of z -1 to z 1.

zi (xi - x) / s
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• AREAS UNDER THE NORMAL CURVE
• z -2 and z 2 (two standard deviations below
  and above the mean) each represent 0.4772 of the
  total area under the curve.
• 99.7 or almost all of the data points lie
  between -3s and 3s.

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• Analysis of Two Histograms
• Graph A is class distribution of numbers 1-10
• Graph B is class distribution of semester credits
• Data for A 5.64 /- 2.6 (much greater spread
  than B)
• Data for B 15.7 /- 1.96 (smaller spread)
• Skew of A -0.16 and Skew B 0.146
• CV of A 0.461 and CV of B 0.125 (CV
  SD/Mean)