Data Analysis Review Flashcards
Sample standard deviation
Instead of calculating mean of squared differences by dividing by n, divide by n-1
Standard deviation
- Find mean
- Subtract each data point from mean
- Square each difference
- Average squared differences
- Square root of (nonnegative) average squared difference
Set
Collection of objects of some particular property
Repetitions are not counted as additional elements
Example: {1,2,3,2} and {3,1,2} are the same set of three elements
Three elements in set “S” is denoted |S| = 3
List
Numbers that can all be listed
Two differences form sets:
1. Members are ordered (rearranging created a different list), may not be numerical order (order is “first element”, “second element”)
2. Elements can be repeated and the repetitions matter
Intersection of sets
Intersection is denoted by symbol which looks like upside down U or an arch
It is the set of elements that are in both set S and set T (or whichever two sets are being considered)
Union of sets
Denoted by S U T (upside down arch is shorter and wider)
Includes all elements in both S and T
Disjoint sets
Sets S and T have no elements in common; they’re mutually exclusive
To denote mutually exclusive sets
Intersection of sets B and C is empty
B “intersection” (symbol) C = “empty” (symbol
Empty set symbol is circle with diagonal line from upper right to lower left
Inclusion-exclusion principle to find the number of elements in the union set of sets A and B
Number of elements in union set of A and B = number of elements in A plus the number of elements in B MINUS the number of elements in the intersecting set of A and B (to ensure that no elements are counted twice)
A U B | = |A| + |B| - |A intersection B|
Multiplication principle
If there are k different possibilities for the first choice and m different possibilities for the second choice, then there are km combinations of the pair of choices
Ex. 3 choices of app and 5 choices of main course = 15 choices of meal combinations
For as many variables as exist; ex. 3 different pairs of shoes, 5 different skirts, 7 different shirts, and 4 different sweaters, then there are 3x5x7x4 (420) outfits
Factorial
1! = 1 2! = (2)(1) = 2 3! = (3)(2)(1) = 6 4! = (4)(3)(2)(1) = 24
As a special rule, 0! = 1
Permutations of n objects taken k at a time
= n! / (n-k)!
Example: 7 (n) numbers (objects) taken 5 digits at a time (create how many different five-digit numbers using 7 different digits without repeating any number
7! / (7-5)!
(7)(6)(5)(4)(3)(2!) / 2! = (7)(6)(5)(4)(3) = 2,520 different 5 digit combinations
Combinations of n objects taken k at a time (without counting repeated combination in different order)
n!/ k!(n-k)!
Ex: I want to select three of five letters and know how many combinations of three letters I can make using those five. (n= 5, k=3)
5!/ 3!(5-3)! = 5!/3!(2!) = 120/12= 10 different combinations of 3 letters can be made from a set of 5 letters
Ways to denote “combinations of n objects taken k at a time”
“n choose k”
nCk or
(n
k)
(within same parenthesis set, of course)
n choose 0
n!/0!(n!)
=1, which corresponds to the fact that there is only one subset of S with 0 elements, namely the empty set
