Six Sigma and Beyond: Statistics and Probability, Volume III
MEAN from RV Range: M discrete values or cells , X k
or MEAN from Sample Space: N discrete individual samples, S i
RANDOM EXPERIMENT
Sum produced by pair of fair six-sided dice.
Random Variable X k defined as sum of the two numbers :
X k = {D 1 + D 2 }
| Cell k | No. in X k | X k | f (X k ) | X k f (X k ) |
|---|---|---|---|---|
| 1 | 1 | 2 | 1/36 | 2/36 |
| 2 | 2 | 3 | 2/36 | 6/36 |
| 3 | 3 | 4 | 3/36 | 12/36 |
| 4 | 4 | 5 | 4/36 | 20/36 |
| 5 | 5 | 6 | 5/36 | 30/36 |
| 6 | 6 | 7 | 6/36 | 42/36 |
| 7 | 5 | 8 | 5/36 | 40/36 |
| 8 | 4 | 9 | 4/36 | 36/36 |
| 9 | 3 | 10 | 3/36 | 30/36 |
| 10 | 2 | 11 | 2/36 | 22/36 |
| 11 | 1 | 12 | 1/36 | 12/36 |
| M = 11 | 36 | Sum 252/36 |
Sample variance and standard deviation
-
Sample variance: Expected value of X i about the mean
or
-
Sample standard deviation: s x ‰
Exercise: Sum of two fair dice: X k = {D 1 + D 2 }
x k
(X k -
) (X k -
) 2 f(X k )
(X k -
) 2 f(X k ) 2
-5
25
1/36
25/36
3
-4
16
2/36
32/36
4
-3
9
3/36
27/36
5
-2
4
4/36
16/36
6
-1
1
5/36
5/36
7
6/36
8
1
1
5/36
5/36
9
2
4
4/36
16/36
10
3
9
3/36
27/36
11
4
16
2/36
32/36
12
5
25
1/36
25/36
Sum 210/36
Variance:
Standard deviation:
s x = 2.415