MDH Chapter 1 EGR 252 Fall 2015 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

MDH Chapter 1 EGR 252 Fall 2015 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 = ?

172239312852

147

MDH Chapter 1 EGR 252 Fall 2015 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

172239312852

147

MDH Chapter 1 EGR 252 Fall 2015 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 If n=8, median is the average of the 4th and 5th data values.

172239312852

147~

MDH Chapter 1 EGR 252 Fall 2015 Slide 5

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:

sample range = Max – Min = 147 – 17 = 130

172239312852

147

MDH Chapter 1 EGR 252 Fall 2015 Slide 6

Calculations: Variability of the Data sample variance,

sample standard deviation,

n

i

i

nxxs

1

22

1

14.452 ss

3.20376

48147...48224817 2222

s

MDH Chapter 1 EGR 252 Fall 2015 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

MDH Chapter 1 EGR 252 Fall 2015 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

MDH Chapter 1 EGR 252 Fall 2015 Slide 9

Graphical Methods - Histogram Frequency Distribution (histogram)

Develop equal-size class intervals – “bins” ‘Rules of thumb’ for number of intervals

Less than 50 observations 5 – 7 intervals 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

MDH Chapter 1 EGR 252 Fall 2015 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 (shape) of stride lengths for this sample?

Stride Length28.6 26.5 30.0 27.1 27.826.1 29.7 27.3 28.5 29.328.6 28.6 26.8 27.0 27.326.6 29.5 27.0 27.3 28.029.0 27.3 25.7 28.8 31.4

MDH Chapter 1 EGR 252 Fall 2015 Slide 11

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

25 1.0

= 2/25

MDH Chapter 1 EGR 252 Fall 2015 Slide 12

Computer-Generated HistogramsExcel 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

Frequ

ency

313029282726

10

8

6

4

2

0

Minitab Histogram of 252dataset2

Bin Size determined using Sturges’ formula= 1+3.3 log (n)= 5.61 round to 6

MDH Chapter 1 EGR 252 Fall 2015 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

Rela

tive

Freq

uenc

y

MDH Chapter 1 EGR 252 Fall 2015 Slide 14

Graphical Methods – Dot Diagram Dot diagram (text) Dotplot (Minitab)

252dataset231.230.429.628.828.027.226.425.6

Dotplot of 252dataset2

Homework and Reading Assignment

ReadingChapter 1:

Introduction to Statistics and Data Analysis pg. 1- 30

Problems1.9 pg. 171.18 pg. 31

MDH Chapter 1 EGR 252 Fall 2015 Slide 15