1. South Dakota School of Mines & Technology Data Analysis Industrial Engineering

2. Data Analysis Statistical Measures Industrial Engineering

3. Aside: Mean, Variance   Mean: Variance:   xp x discrete ( ) ,  2    ( ) x p

4. Example Consider the discrete uniform die example: x 1 2 3 4 5 6
p(x) /6 1/6 1/6 1/6 1/6 1/6
 = E[X] = 1(1/6) + 2(1/6) + 3(1/6)
+ 4(1/6) + 5(1/6) + 6(1/6)
= 3.5

5. Example Consider the discrete uniform die example: x 1 2 3 4 5 6
p(x) /6 1/6 1/6 1/6 1/6 1/6
2 = E[(X-)2] = (1-3.5)2(1/6) + (2-3.5)2(1/6) + (3-3.5)2(1/6)
+ (4-3.5)2(1/6) + (5-3.5)2(1/6) + (6-3.5)2(1/6)
= 2.92

6. å Binomial Mean  p ÷ ø ö ç è æ ) 1 ( )! !   xp x ( )
= 1p(1) + 2p(2) + 3p(3) np(n) x n p - = ÷ ø ö ç è æ å ) 1 ( )! !

7. å Binomial Mean  p ÷ ø ö ç è æ ) 1 ( )! !   xp x ( )
= 1p(1) + 2p(2) + 3p(3) np(n) x n p - = ÷ ø ö ç è æ å ) 1 ( )! !
Miracle 1 occurs
= np

8. Binomial Measures   Mean: Variance:   xp ( x ) = np     ( ) x2    ( ) x p
= np(1-p)

9. Binomial Distribution
0.0
0.1
0.2
0.3
0.4
0.5 1 2 3 4 5 x
P(x)
0.0
0.1
0.2
0.3
0.4
0.5 1 2 3 4 5 x
P(x)
n=5, p=.3
n=8, p=.5
n=20, p=.5
n=4, p=.8
0.0
0.1
0.2
0.3
0.4
0.5 2 4 x
P(x)
0.0
0.1
0.2
0.3
0.4
0.5 1 2 3 4 5 6 7 8
P(x) x

10. Measures of Centrality
Mean
Median
Mode

11. Measures of Centrality
Mean    xp x
discrete ( ) ,      xf x dx
continuous ( ) ,
Sample Mean å = n i x X 1

12. Measures of Centrality
Exercise: Compute the sample mean for the student Gpa data
2.4, 2.7, 2.8, 2.9, 3.0, 3.0, 3.1, 3.3, 3.5, 3.9

13. Measures of Centrality
Failure Data X 1 . 19 =

14. Measures of Centrality
Median
Compute the median for the student Gpa data
2.4, 2.7, 2.8, 2.9, 3.0, 3.0, 3.1, 3.3, 3.5, 3.9 . 3 2 = + X )

15. Measures of Centrality
Mode
Class mark of most frequently occurring interval
For Failure data, mode = class mark first interval ( . 5 = X

16. Measures of Centrality
Measure Student Gpa Failure Data
Mean
Median
Mode
Sample mean X is a blue estimator of true mean m.
X m E[ X ] = m
u.b.

17. Measures of Dispersion
Range
Sample Variance

18. Measures of Dispersion
Range
Compute the range for the student Gpa data
2.4, 2.7, 2.8, 2.9, 3.0, 3.0, 3.1, 3.3, 3.5, 3.9
Min = 2.4 Max = 3.9
Range = = 1.5

19. Measures of Dispersion
Variance   2    ( ) x p   2      ( ) x f dx
Sample variance x 1 2 - = å n s i

20. Measures of Dispersion
Exercise: Compute the sample variance for the student Gpa data
2.4, 2.7, 2.8, 2.9, 3.0, 3.0, 3.1, 3.3, 3.5, 3.9 x 1 2 - = å n s i

21. Measures of Dispersion
Exercise: Compute the variance for failure time data
s2 =

22. An Aside
For Failure Time data, we now have three measures for the data
Expontial ??
s2 = X 1 . 19 =

23. An Aside
Recall that for the exponential distribution
m = 1/l
s2 = 1/l2
If E[ X ] = m and E [s2 ] = s2, then
1/l = 19.1 or
1/l2 = X 1 . 19 =
s2 = l
0524 . ˆ = l
0575 . ˆ =