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. 001 Spring 2009 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. 001 Spring 2009 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. 001 Spring 2009 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 JMB Chapter 1 EGR 252. 001 Spring 2009 4

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: 17 22 39 31 28 52 147 § sample range: Max - Min JMB Chapter 1 EGR 252. 001 Spring 2009 5

Calculations: Variability of the Data § sample variance, § sample standard deviation, JMB Chapter 1 EGR 252. 001 Spring 2009 6

Other Descriptors § Discrete vs Continuous § discrete: countable § continuous: measurable § Distribution of the data § “What does it look like? ” JMB Chapter 1 EGR 252. 001 Spring 2009 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 8 9 Frequency 1 2 2 2 1 7 1 EGR 252. 001 Spring 2009 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. 001 Spring 2009 9

Data for Histogram § Example: stride lengths (in inches) of 25 male students were determined, with the following results: Stride Length 28. 60 26. 50 30. 00 27. 10 27. 80 26. 10 29. 70 27. 30 28. 50 29. 30 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 § What can we learn about the distribution of stride lengths for this sample? JMB Chapter 1 EGR 252. 001 Spring 2009 10

Constructing a Histogram § Determining frequencies and relative frequencies Relative Midpoint Frequency 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 EGR 252. 001 Spring 2009 11