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Statistics
Statistics Definitions
90
Mathematics
Undergraduate 1
07/11/2019
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Term
'
 Definition
complement
Term
0!
 Definition
1
Term
1!
 Definition
1
Term
2 mathematical proofs that A and B are independent events
 Definition
P(A and B)=P(A)*P(B)
P(A|B)=P(A)
Term
Addition Rule
 Definition
"or" means "add", with an adjustment if they're not mutually exclusive
P(A or B)=P(A)+P(B)
P(A or B)=P(A)+P(B)-P(A and B)
Term
Combinations, nCr
 Definition
n!/r!*(n-r)!
n = total number of items
r = the amount you select
Term
Complement of A
 Definition
all possible outcomes that aren't A
Term
Complement of A (formula)
 Definition
A'=1-A
Term
Conditional probability
 Definition
the probability that B will happen, given that A has already happened
Term
Continuous variable
 Definition
variable whose value is obtained by measuring
Term
Counting principle
 Definition
in a series of events, the number of ways that the series of events can occur is a product of the number of ways each individual event can occur
multiply the number of ways that each event can occur
Term
Critical value
 Definition
number at the border of the rejection region
Term
Discrete variable
 Definition
variable whose value is obtained by counting
Term
Empirical
 Definition
based on real-life trials, tests, experiments
Term
Empirical Rule, or Chebyshev's Theorem
 Definition
68% of the values of x fall between -1 and 1 standard deviation from the mean
95% of the values of x fall between -2 and 2 standard deviations from the mean
99.7% of the values of x fall between -3 and 3 standard deviations from the mean
Also known as The 68-95-99.7 Rule
Term
Expected value
 Definition
exp=Σ[P(x)*x]
exp=[P(x1)*x1]+[P(x2)*x2]+P(x3)*x3]...etc.
Σ=sum
x=a possible value
x1=the first possible value, x2=the second possible value, etc.
Term
Experimental probability
 Definition
another term for relative frequency
probability in real life, in the experiment
Term
Factorial
 Definition
multiplying in descending order
4*3*2*1
10*9*8*7*6*5*4*3*2*1
etc.
Term
Factorial Table
 Definition
1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5,040
8!=40,320
9!=362,880
10!=3,628,800
You can add 10! to the end of higher factorials, i.e. 15!=15*14*13*12*11*3,628,800
Term
Formula for:
"If 3 people line up, what is the probability that the first one is a girl, the second one is a boy, and the third one is a girl?"
 Definition
multiplication rule, multiply the probabilities of each outcome
P(GBG)=P(G)*P(B)*P(G)
Term
Four reasons why this is a binomial experiment
 Definition
1) fixed number of trials, given by n
2) probabilities are independent from trial to trial
3) trials are independent and occur under identical conditions
4) only two possible outcomes, success or failure

acronym FIIT
F = fixed (number of trials)
I = independent (probabilities)
I = independent (trials)
T = two (possible outcomes)
Term
H0
 Definition
null hypothesis, a statement about the population that is assumed to be true unless there is convincing evidence to the contrary; the "default hypothesis", the status quo
We don't accept H0, we only "fail to reject" it.
Term
Ha
 Definition
alternative hypothesis, a statement about the population that contradicts the null hypothesis and is accepted as true only if there is significant evidence in favor of it. We don't reject Ha, we only "fail to accept" it.
Term
IQR (definition and formula)
 Definition
Interquartile range
Q3-Q1
Term
Inferential statistics
 Definition
statistics regarding the probability that something will happen
Term
Law of large numbers
 Definition
as the number of repetitions of an experiment is increased, the relative frequency becomes closer and closer to the theoretical probability
Term
M
 Definition
median, Q2, 50th percentile, value that 50% of the values fall below
Term
Mathematical proof of an outlier
 Definition
outlier < Q1-(1.5)IQR
outlier > Q3+(1.5)IQR
Term
Mathematical proof that A an B are mutually exclusive events
 Definition
P(A and B)=0
Term
Mean (formula)
 Definition
μ=(Σx)/n
Term
Mean (probability distribution)
 Definition
μ=Σ[x*P(x)]
Term
Mean -- binomial probability distribution
 Definition
μ=np
μ=mean
n=number of trials
p=probability of success, or P(success)
Term
Minimum sample size for the Central Limit Theorem
 Definition
30
Term
Multiplication rule
 Definition
"and" means "multiply", with an adjustment if they're dependent
P(A and B)=P(A) * P(B)
P(A and B)=P(A|B) * P(B)
Term
Number of different orders in which you could select every item in the sample space
 Definition
n!
n=number of items in the sample space
Term
P(A and B) -- dependent
 Definition
P(A|B) * P(B)
Term
P(A and B) -- independent
 Definition
P(A) * P(B)
Term
P(A or B) -- mutually exclusive
 Definition
P(A)+P(B)
Term
P(A or B) -- not mutually exclusive
 Definition
P(A)+P(B)-P(A and B)
Term
P(A')
 Definition
1-P(A)
Term
P(A|B)
 Definition
P(A and B)/P(B)
Term
P(E)
 Definition
E/S
E=number of times event occurred
S=sample space
Term
PDF
 Definition
probability distribution function
Term
Permutations with repetition
 Definition
nˆr
n to the power of r
n = total number of items
r = the number of items you select
Term
Permutations without repetition
 Definition
nPr
n!/(n-r)!
n = total number of items
r = the number of items you select
Term
Probability of winning the lottery
 Definition
number of tickets you bought/nCr
Term
Q1
 Definition
25th percentile
value that 25% of the values fall below
Term
Q3
 Definition
75th percentile
value that 75% of the values fall below
Term
Relative frequency, long-term relative frequency
 Definition
probability in real life, result of the experiment
Term
Sampling distribution
 Definition
the distribution of the means of all possible samples of the same size that could be taken from a population
Term
Standard deviation (normal distribution)
 Definition
σ=√{Σ[(x-μ)²]}
Term
Standard deviation (probability distribution)
 Definition
σ=√{Σ[(x-μ)² * P(x)]}
σ=standard deviation
Σ=sum
x=a value that occurred
μ= mean
Term
Standard deviation (sampling distribution)
 Definition

σ=(√{Σ[(x-xbar)²]})/√n

1. solve x-xbar for all values of x

2. square these differences

3. add up these squares

4. take the square root of this sum

5. take the square root of n

6. divide the square root of the sum by the square root of n
Term
Standard deviation -- binomial probability distribution
 Definition
σ=√(npq)
σ=standard deviation
n=number of trials
p=probability of success, or P(success)
q=probability of failure, or P(failure)
Term
Symmetry Rule of Z-Scores
 Definition
The area to the left of a negative z-score -z is the same size as the area to the right of a positive z-score z. P(x<-z)=P(x>z).
Term
Test statistic
 Definition
(xbar-μ)/σxbar
Term
Test statistic for a test of population mean
 Definition
test=(xbar-μ)/σxbar
Term
Theoretical probability
 Definition
probability one would expect without conducting trials
Term
Type I error
 Definition
Reject H0 when it is true
Term
Type II error
 Definition
Fail to reject H0 when it is not true
Term
Variance (probability distribution)
 Definition
σ²
σ=standard deviation (probability distribution)
∑[(x-μ)²*P(x)]
∑=sum
x=a value that occurred
μ=mean
Term
Variance -- binomial probability distribution
 Definition
σ²=npq
σ²=variance (the variance doesn't have a letter or symbol of its own, is expressed only as "standard deviation squared", or you could just write variance or "var" if that's easier
n=number of trials
p=probability of success, or P(success)
q=probability of failure, or P(failure)
Term
Without replacement (playing cards, marbles, balls, etc.)
 Definition
the sample size S will change
be careful about the denominators when you apply the probability formula P=E/S
the numerator (event) may change too, depending on the thing that was removed
Term
Z-score (normal distribution)
 Definition
z=(x-μ)/σ
Term
confidence interval formula (mean)
 Definition
CI=±1.96(σxbar)
Term
confidence interval formula (proportions)
 Definition
CI=±[1.96(sp)+(0.05/n)]
Term
degrees of freedom
 Definition
df=n-1
don't need to know what "degrees of freedom" actually means
Term
margin of error (definition)
 Definition
difference between the population mean or proportion and the outer limits of the confidence interval
Term
margin of error for a 95% confidence interval for a population proportion (formula)
 Definition
ME=1.96*√{[p(1-p)]/n}
Term
sp, standard deviation of sample proportions
 Definition
sp=√{[p(1-p)]/n}
Term
test statistic (definition)
 Definition
the z-score we use to find the critical value(s) for a hypothesis test
Term
test statistic for a test of population proportion
 Definition
T=(p-p0)/{√[(p0*q0)/n]}
T=test statistic
p=sample proportion
p=null proportion (from the null hypothesis)
q0=null proportion of failure, or 1-p0
n=sample size
Term
test statistic for a test of population standard deviation
 Definition
[(n-1)*s²]/σ²
Term
variance (normal distribution)
 Definition
σ²
Σ[(x-μ)²]
Term
z-score (normal distribution)
 Definition
z=(x-μ)/σ
Term
z-score (sampling distribution)
 Definition
z=(x-μ)/(σ/√n)
Term
z-scores to use for a 90% confidence interval, a 95% confidence interval, and a 99% confidence interval
 Definition
1.645, 1.96, 2.576
Term
α
 Definition
level of significance; maximum probability of xbar occurring in which we can conclude that the occurrence is not a result of sampling error; probability of making a Type I error
Term
μ and xbar
 Definition
μ = mean of the population
xbar = mean of a sample
Term
μp, the mean of sample proportions
 Definition
μp=π π=the population proportion p=the sample proportion

similar to the formula μxbar=μ
Term
μxbar (definition)
 Definition
the mean of all sample means, "the mean of means"
Term
μxbar(formula)
 Definition
μxbar = Σ[xbar*P(xbar)]
Term
σxbar (definition)
 Definition
the standard deviation of all sample means
Term
σxbar (formula)
 Definition
σ/√n
(√{Σ[(x-μ)²]})/√n
Averages computed from samples vary less than do individual measurements on the population, and averages computed from larger samples vary less than do averages computed from smaller samples.
Term
√2
 Definition
1.4142135
Term
∩
 Definition
and
Term
∪
 Definition
or
Term
correlation coefficient
 Definition
a measure of the strength of the relationship between two variables
Term
correlation coefficient (2 formulas)
 Definition

r={∑[(x-xbar)(y-ybar)]}/(√{Σ[(x-xbar)²]}*{Σ[(y-ybar)²]})

 

r=[∑xy-(1/n)*∑x*∑y]/((√{[∑x²-(1/n)*(∑x)²]*[∑y²-(1/n)*(Σy)²]}))

 

PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

Parentheses Hierarchy: Solve () first, then [], then {}, then (())
Term
When does it make sense to calculate the r value?
 Definition
When, based on a visual representation, like a scatter plot or a table, or even just a list of data points (x,y), there appears to be a strong relationship between x and y. If it looks like there is no relationship, you shouldn't bother calculating r.
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