what are mean and median?
measures that locate centre of data !!
other names for mean
- average
- ARITHMETIC mean
what is a property of the median?
when is this property “violated”?
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
what are most frequent measures of spread?
- range (simplest)
- standard deviation
- variance
- coefficient of variation (= st dev / | mean | )
what are properties of standard deviation? (2)
- usually if greater spread from mean, larger standard deviation, but not always. sometimes larger spread from mean but smaller st dev
- always non-negative
what is frequency distribution and what can be measured from it?
- one of the MANY ways to show visually how data are spread
- can calculate mean, median, mode, st dev
what to watch out for when calculating statistical measures from graphs?
- 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)!!!!
definition of set
a collection of numbers or objects
how to list elements in set listed and with what conditions, and what are its properties?
- listed inside brackets with commas IF FINITE!! (otherwise no)
- ORDER does NOT matter
important definitions in a set (4) and definitions (4)
- 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!!)
number of elements in a set, two they are calculated in a union and how is this method called
- 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
- venn Diagram definition
- shapers of Venn Diagram
- what can they be used for?
- regions on a plane showing two ore more sets
- classic Venn diagram only if not mutually exclusive/disjoint or subsets
- word problems !!!
what does counting mean in sets?
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
- what is the multiplication principle?
2. what can it be applied to?
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
- formula for combination in sets
- property of it
- formula of a factorial
- properties of factorials
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)!
flow of what counting (in sets / probability) talks about
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)
combination principle of equality
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.
- definition of DISCRETE (rather than continuous)
2. main terminology of DISCRETE probability
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
probability range and meaning (3)
- range = 0 ≤ P ≤ 1
- P = 0 = IMPOSSIBLE event
- P = 1 = CERTAIN event
- 0 < P < 1 = POSSIBLE but UNCERTAIN! event
special types of probabilities (3)
- 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)
special probabilities formula AND particular cases:
- probability of not E
- probability of E or F (events of one set)
- 1 - P(E)
- P(E) + P(F) - P(E and F) — if E and F MUTUALLY EXCLUSIVE, then P(E) + P(F) - 0
independent and dependent sets difference and calculation of P(E)
- 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)
what is estimation?
rounding a number to make a calculation less precise but quicker/simpler, by using multiples of 10^n
different types of estimation (5)
- 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
