Statistics, Sets, Counting, Probability, Estimation, and Series

Question 1

what are mean and median?

Answer 1

measures that locate centre of data !!

Question 2

other names for mean

Answer 2

• average

- ARITHMETIC mean

Question 3

what is a property of the median?

when is this property “violated”?

Answer 3

usually, half of numbers are below median and half are above.
Violated when the median is repeated lots of times and list of numbers is small, because many number are not smaller / larger than median but are EQUAL to it

Question 4

what are most frequent measures of spread?

Answer 4

• range (simplest)
• standard deviation
• variance
• • coefficient of variation (= st dev / | mean | )

Question 5

what are properties of standard deviation? (2)

Answer 5

• usually if greater spread from mean, larger standard deviation, but not always. sometimes larger spread from mean but smaller st dev
• always non-negative

Question 6

what is frequency distribution and what can be measured from it?

Answer 6

• one of the MANY ways to show visually how data are spread

- can calculate mean, median, mode, st dev

Question 7

what to watch out for when calculating statistical measures from graphs?

Answer 7

• watch out for axis, different lines might refer to different axis (to the left and to the right of the graph)
• approximate points’ value well (even when close to zero!)
• don’t make calculation errors (calculate numbers correctly)!!!!

Question 8

definition of set

Answer 8

a collection of numbers or objects

Question 9

how to list elements in set listed and with what conditions, and what are its properties?

Answer 9

• listed inside brackets with commas IF FINITE!! (otherwise no)
• ORDER does NOT matter

Question 10

important definitions in a set (4) and definitions (4)

Answer 10

• union - set made of all elements that are in A or in B
• intersection - set of all elements that are BOTH in A and in B
• subset - a set that is entirely contained in another set (e.g. A is a subset of B if all elements of A are present in B too)
• mutually exclusive / disjoint set (two sets that have NO elements in common!!)

Question 11

number of elements in a set, two they are calculated in a union and how is this method called

Answer 11

• denoted by |A|
• number of elements in an union = number of elements in A + B - number of elements in intersection of A and B! (only first two terms if disjoint sets)
• general addition rule for two sets

Question 12

1. venn Diagram definition
2. shapers of Venn Diagram
3. what can they be used for?

Answer 12

1. regions on a plane showing two ore more sets
2. classic Venn diagram only if not mutually exclusive/disjoint or subsets
3. word problems !!!

Question 13

what does counting mean in sets?

Answer 13

refers to the number of:

• pairs/combination of 1 choice from each set (multiplication principle)
• permutations in a possible set
• combinations = k choices in a set

Question 14

1. what is the multiplication principle?

2. what can it be applied to?

Answer 14

1.
rule that says that to know number of possible sets that can arise from 1 object apiece from each set is n*m where n is number of objects in first set from where to chose from and m is number of objects in second set from where to choose from

2.
rule can be used to find:
- combinations of 1 object apiece in many sets
- how many options you have if an experiment is repeated n times
- how many options you have if you chose one object from a decreasing set (ie decreasing by one each time) = permutation = ordering a list

Question 15

1. formula for combination in sets
2. property of it
3. formula of a factorial
4. properties of factorials

Answer 15

1.
(nk) = (n!)/(k!(n-k)!

2.
(n k) = (n (n-k)

3.
n(n-1)(n-2)….(3)(2)(1)

4.
n! = n(n-1)!
(n+1)! = (n+1)n! = (n+1)(n)(n-1)!

Question 16

flow of what counting (in sets / probability) talks about

Answer 16

talks about counting between different sets:

• counting 1 object apiece = multiplication rule
• counting 1 object apiece when sets have same dimension
• counting 1 object apiece when sets have decreasing dimension

the last one talks about factorisation: multiplying every set’s number of objects (decreasing by one) by each other is like taking the factorial of the largest set / of the set’s initial size, so then we transition into what factorisation can be used for, or counting among 1 set:

• permutation (how many different ways of ordering objects there are)
• combination (how many different combinations of choices there are)

Question 17

combination principle of equality

Answer 17

combination of k choices and n-k choices (e.g. of 2 out of 7 choices and 5 out of 7 choices are the same number, because the possible combinations of 5 out of 7 are the ones you didn’t pick when choosing the 2 out of 7.

Question 18

1. definition of DISCRETE (rather than continuous)

2. main terminology of DISCRETE probability

Answer 18

1.
probability concerning a SET of FINITE number of events

2.

• experiment = an occurrence that can take on various values, all with uncertainty
• event = a set of possible outcomes (or one possible outcome) that arises from an experiment
• outcome = what the experiment with uncertainty ends up yielding

Question 19

probability range and meaning (3)

Answer 19

• range = 0 ≤ P ≤ 1
• P = 0 = IMPOSSIBLE event
• P = 1 = CERTAIN event
• 0 < P < 1 = POSSIBLE but UNCERTAIN! event

Question 20

special types of probabilities (3)

Answer 20

• probability of an event which is a subset of another event (e.g., probability of even numbers smaller than 4, probability of even numbers) = P(F) ≤ P(E) (F is a subset of E)
• probabilities fo two events are equal = EQUALLY LIKELY events
• if all events in a subset have same probability, P(G) = n of events you want probability for / tot n of events (because P of one event = 1 / tot n of events)

Question 21

special probabilities formula AND particular cases:

• probability of not E
• probability of E or F (events of one set)

Answer 21

• 1 - P(E)

- P(E) + P(F) - P(E and F) — if E and F MUTUALLY EXCLUSIVE, then P(E) + P(F) - 0

Question 22

independent and dependent sets difference and calculation of P(E)

Answer 22

• independent is when the fact that E occurred does not influence probability of F occurring, dependent events are the opposite
• for independent events P(E and F) = P(E)*P(F) is ALWAYS true
• for independent and NOT mutually exclusive sets, P(E or F) = P(E) + P(F) - P(E)*P(F)

Question 23

what is estimation?

Answer 23

rounding a number to make a calculation less precise but quicker/simpler, by using multiples of 10^n

Question 24

different types of estimation (5)

Answer 24

• round down
• round up
• round to nearest multiple of 10^n (depends on whether 10^n-1 in number is 5 or more)
• round to nearest multiple of a certain number (for fractions, multiples of same number so that fraction simplifies), or to nearest square/cube (so that you can solve a square/cube root more easily)
• round to a range (WATCH OUT to understand when expression is minimised and when maximised!!! e.g. to minimise numerator has to be min and denominator max, and in x-y y has to be as large as possible!!!). has UPPER BOUND and LOWER BOUND

Question 25

sequence definition

Answer 25

a function where x (domain) is made of only positive integers

Question 26

sequence main terminology

Answer 26

• sequence
• nth term of a sequence (= value of sequence when n=number)
• domain (can be INFINITE if ALL positive integers up to infinity, FINITE if a finite number of n up to a certain n)
• range (can be infinite or finite but can be finite even if range is infinite…!!)

Question 27

1. series definition

2. types of series

Answer 27

1.
the SUM of terms of a sequence

2.

• infinite series = sum of ALL terms of a sequence
• PARTIAL series = sum of terms of a sequence UP to a point k