JMB Chapter 1 EGR 252.001 Spring 2010 Slide 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

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 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

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 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

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 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 (n+1)/2 , → 31

n even median = (xn/2 + xn/2+1)/2

17

22

39

31

28

52

147

~

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 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

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 6

Calculations: Variability of the Data

sample variance,

sample standard deviation,

n

i

i

n

xxs

1

22

1

14.452 ss

3.2037

6

48147...48224817 2222

s

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 7

Other Descriptors Discrete vs Continuous

discrete: countable continuous: measurable

Distribution of the data “What does it look like?”

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 2 4 6 8

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 2 4 6 8

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 8

Graphical Methods – Stem and Leaf

Stem and leaf plot for radar dataStem Leaf Frequency1 7 12 2 8 23 1 9 245 2 167891011121314 7 1

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 9

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 2010 Slide 10

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

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

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 11

Constructing a Histogram Determining frequencies and relative frequencies

Lower Upper Midpoint FrequencyRelative 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

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 12

Computer-Generated Histograms

Excel Chart Using Bar Graph Function

0

5

10

15

25.525 26.875 28.225 29.575 30.925Cell Midpoint

Freq

uenc

y

Excel-Generated Histogram

0

5

10

15

26.20 27.55 28.90 30.25 31.60Bin Upper Bound

Freq

uenc

y

252dataset2

Frequency

313029282726

10

8

6

4

2

0

Minitab Histogram of 252dataset2

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 13

Relative Frequency Graph

Relative Frequency Histogram

0.00

0.20

0.40

0.60

25.53 26.88 28.23 29.58 30.93

Cell Midpoint

Rel

ativ

e

Fre

qu

ency

JMB Chapter 1 EGR 252.001 Spring 2010 Slide 14

Graphical Methods – Dot Diagram

Dot diagram (text) Dotplot (Minitab)

252dataset231.230.429.628.828.027.226.425.6

Dotplot of 252dataset2