Scholarship Statistics

Time series: Main features

Seasonality, Trend, Unusual (noise, residuals), Cycles (periodical but not seasonal variations) (STUC)

Inference: Main features

Shape (linear, parabolic, etc.)
Association (positive, negative, etc.)
Strength of relationship
Scatter (constant, non-constant, etc.)

Bivariate: Main features

Trend, Association, Strength, Groups, Unusual (TASGU)

σ

standard deviation

λ

variance (σ^2)

μ

mean

E(x)

Mean (expected value) of x

nCr

Choosing r terms out of n terms

normal distribution: z

Represents how far away the value is from the mean (i.e. larger value, less similar to the mean, z=0, value=mean)

x

value that the random variable could take

P(X=x)

the probability that the random variable X takes the value x

values of σ and μ when finding z in calculator

SD=1, M=0

discrete

Distinct, separate

continuous

measurable without interruption

Poisson Distribution characteristics

Discrete, events occur Randomly, events are Independent of each other, probability of event is Proportional to size of interval, events DON'T occur simultaneously (DRIPS), mean=variance

Binomial distribution characteristics

number of trials is Fixed, events are Independent, probability of Success (p/π) is the same, only Two outcomes (FIST)

Normal distribution characteristics

Bell shape, Continuous, no upper or lower Limit, never touches x Axis, most data is Clustered around central value, Symmetrical (B.CLACS)

rectangular distribution characteristics

Continuous, Rectangular shape, probability is found through Area of distribution, Max and Min are given but no mode (CRAM)

Triangular distribution characteristics

Continuous, Triangular shape, Max, Min, and Mode are given (CTM)

area of trapezium

(h(a+b))/2 (where a & b are parallel sides)

random variable (expected values) characteristics

Discrete, presented in a Table (DT)

Sum of infinite series

S=a/(1-r)
a = first term
r = common multiplication ratio between terms (eg. 1/2 -> 1/8: 1/2 * 1/4 = 1/8 so r=1/4)

events are independent if

P(A|B)=P(A) or P(A and B)=P(A) * P(B)

events are mutually exclusive if

P(A and B)=0

A is x times as likely as B if

P(A)/P(B)=x

A is x times more likely than B if

(P(A)-(P(B))/P(B)=x

How can a simulation help someone consider whether a particular low result is unlikely to occur?

A simulation would allow the person to see variation in the percentage of results that are the same as or less than that particular low result.
They could then compare their observed low result to the simulated distribution of results to consider the likelihood of the observed low result occuring.

percentage change from 20% -> 22%

10% more likely

percentage change from 20% -> 30%

Increase of 10 percentage points

Reasons data may not be applicable:

1. Data is old and the likelihood of certain things may have changed.
2. The data may be more specific than the claim (eg. claim is for all people, but data is only for people under 30 YO)
3. The probability of something happening depends on a lot of other factors (eg. gender, location, age, etc) so the figure may not necessarily be applicable for a specific individual.

Can you inverse a conditional (A given B) probability?

No. Just because there is a certain likelihood of A happening given that B has already happened doesn't mean that the inverse of that will be true.
eg. there is a 60% chance of passing your driver's test given you are under 20. This doesn't mean that there is a 60% chance of being under 20 given that you passed your driver's test.

Remember:

Use common sense with the questions. Just because a calculation proves/disproves something, still consider whether that is realistic/whether there are other options too.
(eg. events are mutually exclusive as data said so, however they may not be anything stopping someone from doing both in the population, therefore they are technically not mutually exclusive in the population.)

Investigative cycle

PPDAC - Problem/Plan/Data/Analysis/Conclusion

Negative skew has the tail:

On the left/towards the negative side (the tail tells the tale)

Rule of thumb for confidence intervals (single variable):

Confidence interval = 1/sqrt(n) where n is sample size

Margin of error for comparison within a group (comparing 2 variables for the MoE of the difference)

2/sqrt(n)

Margin of error for difference between independent groups:

1.5 * average of both MoE (average of 1/sqrt(n1) and 1/sqrt(n2) where n1 and n2 are sample sizes of groups 1 and 2)

How to check if a coin flip, etc. is biassed:

Run a simulation for the flip and do it 1000 times. If the result you got (eg. 80 heads) occurs very infrequently in the simulation, then it is likely that the coin flip may have been biassed. If it occurs frequently in the simulation, then the flip is likely fair.

A ∩ B

Intersection: Both A and B occur.

A ∪ B

Union: A and/or B occur (i.e. A, B, and A & B included)

A | B

Probability of A occurring, given B has occurred.

P(A')

Probability of A NOT occurring

Complimentary events

P(A) + P(A') = 1#