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Learning Objectives • To understand the goal of statistical methods of data analysis • To apply multiple techniques (summary statistics, tables, graphics) to describe data, and to understand the benefits of each.

Uncertainty in Engineering • Uncertainty is inherent in all real world problems. • Two types of uncertainty: – Natural randomness of underlying phenomena – Inaccurate estimation of real-world parameters and conditions • Statistics and probability are needed for describing information and forming the bases for design and decision making under uncertainty.

Purpose of Probability/Statistics in Engineering • The purpose of probability and statistics is to deal with uncertainty. • Probability and statistics are widely used in engineering contexts to develop models or information that can be used in engineering decision-making so that the associated risks and costs are effectively managed.

Introduction to Descriptive Statistics Descriptive vs. Inferential Statistics DEFINITIONS – Population – all members of a class or category of interest – Parameter – a summary measure of the population (e.g. average) – Sample – a portion or subset of the population collected as data – Observation – an individual member of the sample (i.e., a data point) – Statistic – a summary measure of the observations in a sample

Populations and Samples Population

Sample

Inference Parameters

Statistics

Summary Statistics • Measures of Central Tendency – Arithmetic mean – Median – Mode • Measures of Dispersion – Variance – Standard deviation – Coefficient of variation (COV)

Measures of Central Tendency 8

6

4

10

3

8

6

8

4

8

5

The sample mean is given by:

The sample median is given by: 3

4

4

5

8

8

10

Measures of Central Tendency The mode of the sample is the value that occurs most frequently.

3

4

4

5

6

8

8

8

10

3

4

4

4

6

8

8

8

10

Bimodal

Measures of Dispersion The most common measure of dispersion is the sample variance:

The sample standard deviation is the square root of sample variance:

Measures of Dispersion Coefficient of variation (COV):

This is a good way to compare measures of dispersion between different samples whose values don’t necessarily have the same magnitude (or, for that matter, the same units!).

Frequency Distribution Vehicle Speeds on Central Avenue Speeds (mph)

Vehicles Counted

40–45

1

45–50

9

50–55

15

55–60

10

60–65

7

65–70

5

70–75

3

TOTAL

50

A frequency distribution is a tabular summary of sample data organized into categories or classes

Histogram

Frequency

15

10

5

0 0 30-35

35-40 4040-45 4545-50 5050-55 5555-60 6060-65 6565-70 7070-75 7575-80 Speeds (mph)

A histogram is a graphical representation of a frequency distribution. Each class includes those observations who’s value is greater than the lower bound and less than or equal to the upper bound of the class.

Histogram Range

10

5

0 0 30-35

35-40

40-45

45-50

50-55

55-60

60-65

65-70

70-75

75-80

Speeds (mph) 15

Class mark

Frequency

Frequency

15

10

5

0 0 32.5

37.5

42.5

47.5

52.5

57.5

Speeds (mph)

62.5

67.5

72.5

77.5

Symmetry and Skewness

Relative Frequency Distribution Vehicle Speeds on Poplar Avenue Speeds (kph)

Vehicles Counted

Pecentage of Sample

40–45

1

2

45–50

9

18

50–55

15

30

55–60

10

20

60–65

7

14

65–70

5

10

70–75

3

6

TOTAL

50

100

Relative Frequency Histogram

Relative Frequency

30%

20%

10%

0% 0 30-35

35-40 4040-45 4545-50 5050-55 5555-60 6060-656565-707070-757575-80 Speeds (mph)

Cumulative Frequency Distributions Vehicle Speeds on Poplar Avenue Speeds (mph)

Vehicles Counted

Percentage of Sample

Cumulative Percentage

40–45

1

2

2

45–50

9

18

20

50–55

15

30

50

55–60

10

20

70

60–65

7

14

84

65–70

5

10

94

70–75

3

6

100

TOTAL

50

100

Cumulative Frequency Diagram Cumulative Frequency

100% 75% 50% 25% 0% 0 30-35 35-404040-454545-505050-555555-606060-656565-707070-757575-80 Speeds (mph)

A good rule of thumb is that the number of classes should be approximately equal to the square root of the number of observations.

Cumulative Frequency Distributions

Boxplots • A boxplot is a graphic that presents the median, the first and third quartiles, and any outliers present in the sample. • The interquartile range (IQR) is the difference between the third and first quartile. This is the distance needed to span the middle half of the data.

Boxplots Steps in the Construction of a Boxplot:

Compute the median and the first and third quartiles of the sample. Indicate these with horizontal lines. Draw vertical lines to complete the box.

Find the largest sample value that is no more than 1.5 IQR above the third quartile, and the smallest sample value that is not more than 1.5 IQR below the first quartile. Extend vertical lines (whiskers) from the quartile lines to these points.

Points more than 1.5 IQR above the third quartile, or more than 1.5 IQR below the first quartile are designated as outliers. Plot each outlier individually.

Boxplots

Boxplots – Example

Boxplots - Example Notice there are no outliers in these data. Looking at the four pieces of the boxplot, we can tell that the sample values are comparatively densely packed between the median and the third quartile. The lower whisker is a bit longer than the upper one, indicating that the data has a slightly longer lower tail than an upper tail. The distance between the first quartile and the median is greater than the distance between the median and the third quartile. This boxplot suggests that the data are skewed to the left.

Excel Tools

Analysis Toolpack in Excel

Other Statistics Software U of M UMWare service: http://tstumware.memphis.edu/

SigmaPlot