Probability and Statistics for Engineers Descriptive Statistics Measures
Probability and Statistics for Engineers § Descriptive Statistics § Measures of Central Tendency § Measures of Variability § Probability Distributions § Discrete § Continuous § Statistical Inference § Design of Experiments § Regression JMB Chapter 1 EGR 252 Spring 2014 Slide 1
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: 17 22 39 31 28 52 147 § the sample mean, x = ? JMB Chapter 1 EGR 252 Spring 2014 Slide 2
Calculation of Mean § Example: The absolute error in the readings from a radar navigation system was measured with the following results: 17 22 39 31 28 52 147 _ § the sample mean, X = (17+ 22+ 39 + 31+ 28 + 52 + 147) / 7 = 48 JMB Chapter 1 EGR 252 Spring 2014 Slide 3
Calculation of Median § Example: The absolute error in the readings from a radar navigation system was measured with the following results: 17 22 39 31 28 52 147 ~ § the sample median, x = ? § Arrange in increasing order: 17 22 28 31 39 52 147 § n odd median = x (n+1)/2 , → 31 § n even median = (xn/2 + xn/2+1)/2 § If n=8, median is the average of the 4 th and 5 th data values. JMB Chapter 1 EGR 252 Spring 2014 Slide 4
Descriptive Statistics: Variability § A measure of variability § Example: The absolute error in the readings from a radar navigation system was measured with the following results: 17 22 39 31 28 52 147 § sample range = Max – Min = 147 – 17 = 130 JMB Chapter 1 EGR 252 Spring 2014 Slide 5
Calculations: Variability of the Data § sample variance, § sample standard deviation, JMB Chapter 1 EGR 252 Spring 2014 Slide 6
Other Descriptors § Discrete vs Continuous § discrete: countable § continuous: measurable § Distribution of the data § “What does it look like? ” JMB Chapter 1 EGR 252 Spring 2014 Slide 7
Graphical Methods – Stem and Leaf Stem and leaf plot for radar data Stem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 JMB Chapter 1 Leaf 7 2 1 Frequency 1 2 2 8 9 2 1 7 1 EGR 252 Spring 2014 Slide 8
Graphical Methods - Histogram § 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 JMB Chapter 1 EGR 252 Spring 2014 Slide 9
Data for Histogram § Example: stride lengths (in inches) of 25 male students were determined, with the following results: 28. 6 26. 1 28. 6 26. 6 29. 0 Stride Length 26. 5 30. 0 29. 7 27. 3 28. 6 26. 8 29. 5 27. 0 27. 3 25. 7 27. 1 28. 5 27. 0 27. 3 28. 8 27. 8 29. 3 27. 3 28. 0 31. 4 § What can we learn about the distribution (shape) of stride lengths for this sample? JMB Chapter 1 EGR 252 Spring 2014 Slide 10
Constructing a Histogram § Determining frequencies and relative frequencies Lower Upper 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 JMB Chapter 1 Midpoint Frequency Relative Frequency EGR 252 Spring 2014 Slide 11
Computer-Generated Histograms JMB Chapter 1 EGR 252 Spring 2014 Slide 12
Relative Frequency Graph JMB Chapter 1 EGR 252 Spring 2014 Slide 13
Graphical Methods – Dot Diagram § Dot diagram (text) § Dotplot (Minitab) JMB Chapter 1 EGR 252 Spring 2014 Slide 14
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