Probability and Statistics for Engineers

1
Probability and Statistics for Engineers

• Descriptive Statistics
• Measures of Central Tendency
• Measures of Variability
• Probability Distributions
• Discrete
• Continuous
• Statistical Inference
• Design of Experiments
• Regression

2
Descriptive Statistics

• Numerical values that help to characterize the
  nature of data for the experimenter.
• Example The absolute error in the readings from
  a radar navigation system was measured with the
  following results
• the sample mean, x ?

17 22 39 31 28 52 147
3
Calculation of Mean

• Example The absolute error in the readings from
  a radar navigation system was measured with the
  following results
• _
• the sample mean, X
• (17 22 39 31 28 52 147) / 7
• 48

17 22 39 31 28 52 147
4
Calculation of Median

• Example The absolute error in the readings from
  a radar navigation system was measured with the
  following results
• the sample median, x ?
• Arrange in increasing order
  17 22 28 31 39
  52 147
• n odd median x (n1)/2 , ? 31
• n even median (xn/2 xn/21)/2

17 22 39 31 28 52 147

5
Descriptive Statistics Variability

• A measure of variability
• (Recall) Example The absolute error in the
  readings from a radar navigation system was
  measured with the following results
• sample range Max - Min

17 22 39 31 28 52 147
6
Calculations Variability of the Data

• sample variance,
• sample standard deviation,

7
Other Descriptors

• Discrete vs Continuous
• discrete countable
• continuous measurable
• Distribution of the data
• What does it look like?

8
Graphical Methods

• Dot diagram
• See example in text
• Stem and leaf plot
• example (radar data)
• Stem Leaf Frequency
• 1 7 1
• 2 2 8 2
• 3 1 9 2
• 4
• 5 2 1
• 6
• 7
• 8
• 9
• 10
• 11
• 12
• 13
• 14 7 1

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Graphical Methods (cont.)

• Frequency Distribution (histogram)
• Develop equal-size class intervals bins
• Rules of thumb for number of intervals
• 7-15 intervals per data set
• Square root of n
• Interval width range / of intervals
• Build table
• Identify interval or bin starting at low point
• Determine frequency of occurrence in each bin
• Calculate relative frequency
• Build graph
• Plot frequency vs interval midpoint

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Data for Histogram

• Example stride lengths (in inches) of 25 male
  students were determined, with the following
  results
• What can we learn about the distribution of
  stride lengths for this sample?

Stride Length Stride Length Stride Length Stride Length Stride Length
28.60 26.50 30.00 27.10 27.80
26.10 29.70 27.30 28.50 29.30
28.60 28.60 26.80 27.00 27.30
26.60 29.50 27.00 27.30 28.00
29.00 27.30 25.70 28.80 31.40
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Constructing a Histogram

• Determining frequencies and relative frequencies

Lower Upper Midpoint Frequency Relative Frequency
24.85 26.20 25.525 2 0.08
26.20 27.55 26.875 10 0.40
27.55 28.90 28.225 7 0.28
28.90 30.25 29.575 5 0.20
30.25 31.60 30.925 1 0.04
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Histograms
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Relative Frequency Graph