formulas
Statistics Formula Sheet Name
Formula
Variables
When to Use
Sample Mean
n= # of observations
When calculating sample mean
Sample Variance
= the mean n= # of observ.
When measuring variability/ spread in our data
Standard Deviation
= the mean n= # of observ.
When measuring variability/ spread in our data
Machine Formula
= the mean n= # of observ.
Alternate for Calculating samp. variance
Axiom 3
For mutually Exclusive events
De Morgans Laws
Always true
Theorem 3
Always true
Theorem 6 “Additive Rule”
When
Basic Theorem
When sample has a finite # of equally likely outcomes
Multiplication Rule of Counting
When you select one object from a number of groups
# of ways to select one object from a number of sets containing n pts= (n1)x(n2)…
Combination Counting
n choose k
When finding how many possible comb. of events
n! = n factorial Fish in the Lake
a= # obj. you want x= # you choose from a N= total # obj. n= total # you choose
Conditional Probability
*rearrange to condition backwards*
More Conditional Probability
If P(A) dn = o
Always true
Bayes’ Theorem
Let A be any event, let B1 and B2 be two events such that:
Allows you to calc. the reverse conditional prob. when you know the backward cond. probs. P(A) formula= the law of total probability
Independence
P(A|B)=P(A) P(B|A)=P(B) P(A B)=P(A)P(B)
When these statements are true, events indep.
Law Of Total Probability
The Binomial Distribution
n PX (x) = P(X = x) = p x (1− p) n−x x
X= r.v X= value
Use when there are two possible outcomes, trials are independent
€ Expected Value of X
E(X) = ∑ xP(X = x) If X has a discrete uniform distribution, then: n
n
€E(X) = a P(X = a ) = 1 a = a(bar) ∑i ∑i i n i=1 i=1
€
Also described as the expectation or the mean
Population Variance
E[(X − µx ) 2 ] = ∑ (X − µx ) 2 Px (x) all x
Denoted by Var(X) or sigma squared x
Use first part for all random variables, the second part (with sigma) for discrete r.v.s
€ Probability Density Function
Changing a r.v into a z-score “Process € of Standardization”
x −µ σ ~ N(0,1) z=
€ x(bar) − µ = z ~ N(0,1) σ n
Central Limit Theorem
€
Confidence Intervals
Use to define continuous random variables
1 1 − 12σ 2 (X − µ )2 f X (x) = e 2Π σ for − ∞ ≤ X ≤ +∞
σ {L,R} = X(bar) ± zα / 2 n With n small (less than 30), replace z with t, calculating the degrees of freedom= N-1
€ Population Mean
E(X 2 ) = µx = ∑ xPx (x) all x
Population Variance
var(x) = E(X 2 ) − µx 2 E(X 2 ) = ∑ xPx (x)
€
all x
or E(X 2 ) = np Var(X) = np(1− p) = npq Population Standard Deviation
σ x = np(1− p)
€
€
Z~N(0,1) is a standard normal random variable
Therefore you must chance the sigma value to take into account the sample size
100(1- alpha)= degree of confidence
Formula
Variables
When to Use
Sample Mean
n= # of observations
When calculating sample mean
Sample Variance
= the mean n= # of observ.
When measuring variability/ spread in our data
Standard Deviation
= the mean n= # of observ.
When measuring variability/ spread in our data
Machine Formula
= the mean n= # of observ.
Alternate for Calculating samp. variance
Axiom 3
For mutually Exclusive events
De Morgans Laws
Always true
Theorem 3
Always true
Theorem 6 “Additive Rule”
When
Basic Theorem
When sample has a finite # of equally likely outcomes
Multiplication Rule of Counting
When you select one object from a number of groups
# of ways to select one object from a number of sets containing n pts= (n1)x(n2)…
Combination Counting
n choose k
When finding how many possible comb. of events
n! = n factorial Fish in the Lake
a= # obj. you want x= # you choose from a N= total # obj. n= total # you choose
Conditional Probability
*rearrange to condition backwards*
More Conditional Probability
If P(A) dn = o
Always true
Bayes’ Theorem
Let A be any event, let B1 and B2 be two events such that:
Allows you to calc. the reverse conditional prob. when you know the backward cond. probs. P(A) formula= the law of total probability
Independence
P(A|B)=P(A) P(B|A)=P(B) P(A B)=P(A)P(B)
When these statements are true, events indep.
Law Of Total Probability
The Binomial Distribution
n PX (x) = P(X = x) = p x (1− p) n−x x
X= r.v X= value
Use when there are two possible outcomes, trials are independent
€ Expected Value of X
E(X) = ∑ xP(X = x) If X has a discrete uniform distribution, then: n
n
€E(X) = a P(X = a ) = 1 a = a(bar) ∑i ∑i i n i=1 i=1
€
Also described as the expectation or the mean
Population Variance
E[(X − µx ) 2 ] = ∑ (X − µx ) 2 Px (x) all x
Denoted by Var(X) or sigma squared x
Use first part for all random variables, the second part (with sigma) for discrete r.v.s
€ Probability Density Function
Changing a r.v into a z-score “Process € of Standardization”
x −µ σ ~ N(0,1) z=
€ x(bar) − µ = z ~ N(0,1) σ n
Central Limit Theorem
€
Confidence Intervals
Use to define continuous random variables
1 1 − 12σ 2 (X − µ )2 f X (x) = e 2Π σ for − ∞ ≤ X ≤ +∞
σ {L,R} = X(bar) ± zα / 2 n With n small (less than 30), replace z with t, calculating the degrees of freedom= N-1
€ Population Mean
E(X 2 ) = µx = ∑ xPx (x) all x
Population Variance
var(x) = E(X 2 ) − µx 2 E(X 2 ) = ∑ xPx (x)
€
all x
or E(X 2 ) = np Var(X) = np(1− p) = npq Population Standard Deviation
σ x = np(1− p)
€
€
Z~N(0,1) is a standard normal random variable
Therefore you must chance the sigma value to take into account the sample size
100(1- alpha)= degree of confidence
