Basic Statistics for Engineers.

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Basic Statistics for Engineers.

• Collection, presentation, interpretation and
  decision making.
• Prof. Dudley S. Finch

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Statistics

• Four steps
• Data collection including sampling techniques
• Data presentation
• Data analysis
• Conclusions and decisions based on the analysis

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Data types

• Discrete
• Defined as
• A variable consisting of separate values for
  example the number of bolts in a packet. There
  may be 8 or 9 but there cannot be 8.5
• Continuous
• Defined as
• A variable which may have any value for example
  the diameter of steel bars after machining. Any
  diameter is possible within the allowable
  tolerance to which the machine is set.

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Sampling

• Often not practical to examine every component
  therefore sampling techniques are used.
• Sample should be representative of the complete
  set (the population) of values from which it has
  been chosen.
• Although not guaranteed, we attempt to chose an
  unbiased sample.
• To be unbiased every possible sample must have an
  equal chance of being chosen. Satisfied if sample
  is chosen at random that is, if there is no
  order in the way the sample is chosen. This is
  called a random sample.

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Random samples

• The larger the random sample the more
  representative of the population it is likely to
  be.
• Random sampling can be carried out by allocating
  a number to each member of the population and
  then drawing numbered balls from a bag or using a
  random number generator.
• Sampling techniques involve probability theory
  (will be dealt with later).

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Data presentation
51.4 55.3 56.1 50.5 55.5
52.8 55.6 55.3 50.2 56.1
52.1 54.8 49.6 57.0 52.0
56.5 55.3 54.0 51.6 52.1
57.3 53.9 53.5 56.1 57.2
54.6 55.4 55.9 56.0 52.9
54.1 55.0 54.2 54.2 54.5
53.0 52.7 54.5 54.7 58.4
56.2 55.8 54.1 56.0 55.1
55.1 54.4 57.2 53.2 55.4
53.9 50.9 54.5 56.9 54.0
56.4 53.1 51.8 52.8 50.5
53.7 52.8 54.0 56.4 55.0
53.8
Measured weights of a casting (lbs).
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Frequency distribution
The class interval should be one that emphasizes
any pattern in the data. Typically between 8 and
15 class intervals should be chosen. In the
example used, a class interval of 1lb is chosen.
50lbs therefore includes 49.5 to 50.4lbs. We can
therefore compile a frequency distribution table.
Mass of casting 50 51 52 53 54 55 56 57 58
Number of castings (frequency)f 2 4 5 8 13 15 12 6 1
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Bar chart
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Histogram
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Frequency polygon
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Frequency curve
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Pie chart showing relative frequency
Relative frequency class frequency / total
frequency of the sample e.g. the relative
frequency of the 53lb class is 8/66 or 0.121
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Numerical methods of a distribution

• A frequency distribution can be represented by
  two numerical quantities
• Central tendency or average value of the
  distribution
• Dispersion or scatter of variables about the
  average value

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Numerical measures of central tendency

• Mid point of range
• Difference between the largest and smallest
  values of the variable
• Generally poor measure of central tendency since
  it depends only on the extreme values of the
  variable and is not influenced by the form of the
  distribution.
• Mode
• The most frequently occurring value of the
  variable
• Easily obtained from frequency table. For the
  casting the mode 55lbs.

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• Arithmetic mean
• Determined by adding all the values of the
  variable and dividing this by the total number of
  values. If x1, x2, x3, .xn are the N values then

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For frequency distribution tables
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To calculate standard deviation
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Estimation

• Applies to the difficulty of obtaining data about
  the population from which the sample was drawn
  and in setting up a mathematical model to
  describe this population.
• Two components estimation and testing of
  hypotheses about the chosen model.

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Two types of estimates

• Point estimate
• Estimate of a population parameter expressed as a
  single number
• This method gives no indication as to the
  accuracy of the estimate
• Interval estimate
• Estimate of a population parameter expressed as
  two numbers
• This method is preferable as it gives an
  indication as to where the population parameter
  is expected to lie

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Confidence intervals

• In practice, the true standard deviation, ?, is
  unknown and that the sample standard deviation,
  s, is used to estimate ?.
• If a random sample size n is drawn, an estimate
  of the standard error of the sample mean is
  given by
• Need to determine the confidence interval for the
  true mean, ?.
• For ngt30 a good approximation can be obtained.
  For small samples a wider interval is used.

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Use of Student t-distribution tables

• Look up value for (n-1) and use desired
  confidence limits (0.01 98, 0.005 99, 0.001
  99.8, etc.).
• Find
• The true mean ? sample mean
• ? t½?,n-1

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For castings example
Sample mean 54.3lbs Standard deviation, s
1.83lbs n 66 Using t0.005, 65 the true mean ?
is given by 54.3 ? 2.66 x 0.225 0.599 Thus we
can be 99 confident that the true mean lies
between 53.7 and 54.9