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 Spring 2009

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
( )/7 = 48
order: ↑
median = x (n+1)/2 , n odd
median = (xn/2 + xn/2+1)/2
JMB Chapter 1
EGR Spring 2009

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
= ( ) / 7
= 48 17 22 39 31 28 52
147
( )/7 = 48
order: ↑
median = x (n+1)/2 , n odd
median = (xn/2 + xn/2+1)/2
JMB Chapter 1
EGR Spring 2009

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:
n odd median = x (n+1)/2 , → 31
n even median = (xn/2 + xn/2+1)/2 17 22 39 31 28 52
147 ~
( )/7 = 48
order: ↑
median = x (n+1)/2 , n odd
median = (xn/2 + xn/2+1)/2
JMB Chapter 1
EGR Spring 2009

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
range = max – min (useful measure, but very susceptible to extreme values and doesn’t say much about what happens in between)
= 147 – 17 = 130
variance – measures the spread of the data around the mean (more on next page …)
JMB Chapter 1
EGR Spring 2009

6. Calculations: Variability of the Data
sample variance,
sample standard deviation, 17 22 39 31 28 52
147
mean
median
variance
std dev
JMB Chapter 1
EGR Spring 2009

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

8. Graphical Methods – Stem and Leaf
Stem and leaf plot for radar data
Stem Leaf Frequency 4 6 7 8 9 10 11 12 13
Stem Leaf Frequency 4 6 7 8 9 10 11 12 13
JMB Chapter 1
EGR Spring 2009

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 Spring 2009

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
26.80
27.00
26.60
29.50
28.00
29.00
25.70
28.80
31.40
JMB Chapter 1
EGR Spring 2009

11. Constructing a Histogram
Determining frequencies and relative frequencies
Lower
Upper
Midpoint
Frequency
Relative Frequency
24.85
26.20
25.525 2
0.08
27.55
26.875 10
0.40
28.90
28.225 7
0.28
30.25
29.575 5
0.20
31.60
30.925 1
0.04
Class intervals – divide max-min by 5 (sqrt of 25), then either add that number to successive intervals OR let Excel find the bins (caution – Excel bins are determined differently … you may want to play a little to get a good picture of the data…)
JMB Chapter 1
EGR Spring 2009

12. Computer-Generated Histograms
There is no single convention for the x-axis.
JMB Chapter 1
EGR Spring 2009

13. Relative Frequency Graph
Preferred method is to use cell midpoints on x-axis
JMB Chapter 1
EGR Spring 2009

14. Graphical Methods – Dot Diagram
Dot diagram (text)
Dotplot (Minitab)
General rule: one dot for each data point
JMB Chapter 1
EGR Spring 2009